Internet-Draft                                                D. Brown
Intended status: Experimental                               BlackBerry
Expires: 2021-04-05                                         2020-10-02

          Elliptic curve 2y^2=x^3+x over field size 8^91+5


  Multi-curve elliptic curve cryptography with curve
  2y^2=x^3+x/GF(8^91+5) hedges a risk of new curve-specific attacks.
  This curve features: isomorphism to Miller's curve from 1985; low
  Kolmogorov complexity (little room for embedded weaknesses of
  Gordon, Young--Yung, or Teske); similarity to a Bitcoin curve;
  Montgomery form; complex multiplication by i
  (Gallant--Lambert--Vanstone); prime field; easy reduction,
  inversion, Legendre symbol, and square root; five 64-bit-word field
  arithmetic; string-as-point encoding; and 34-byte keys.

Status of This Memo

  This Internet-Draft is submitted in full conformance with the
  provisions of BCP 78 and BCP 79.  Internet-Drafts are working
  documents of the Internet Engineering Task Force (IETF).  Note that
  other groups may also distribute working documents as
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  Copyright (c) 2020 IETF Trust and the persons identified as the
  document authors.  All rights reserved.

  This document is subject to BCP 78 and the IETF Trust's Legal
  Provisions Relating to IETF Documents
  ( in effect on the date of
  publication of this document.  Please review these documents
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  translate it into languages other than English.

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1.  Introduction
2.  Requirements Language (RFC 2119)
3.  Use ONLY in multi-curve ECC
4.  Encoding points
4.1.  Point encoding process
4.1.1.  Summary
4.1.2.  Details
4.2.  Point decoding process
4.2.1.  Summary
4.2.2.  Detail
5.  Point validation
5.1.  When to validate
5.1.1.  Mandatory validation
5.1.2.  Simplified validation
5.1.3.  Minimal validation
5.2.  Point validation process
6.  OPTIONAL encodings
6.1.  Encoding scalars
6.2.  Encoding strings as points
7.  IANA considerations
8.  Security considerations
8.1.  Field choice
8.2.  Curve choice
8.3.  Encoding choices
8.4.  General subversion concerns
8.5.  Concerns about 'aegis'
9.  References
9.1.  Normative References
9.2.  Informative References

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Appendix A.  Why 2y^2=x^3+x/GF(8^91+5)?
A.1. Not for single-curve ECC
A.2.  Risks of new curve-specific attacks
A.2.1.  What would be considered a "new curve-specific" attack?
A.2.2.1.  What would be considered a "new" attack?
A.2.2.2.  What is, would be, considered a "curve-specific attack"?
A.2.2.3.  Rarity of published curve-specific attacks
A.2.2.4.  Correlation of curve-specific efficiency and attacks
A.3.  Mitigations against new curve-specific attacks
A.3.1.  Fixed curve mitigations
A.3.1.2.  Existing fixed-curve mitigations
A.3.1.2.  Migitations used by 2y^2=x^3+x/GF(8^91+5)
A.3.2.  Multi-curve ECC
A.3.2.1.  Multi-curve ECC is a redundancy strategy
A.3.2.2.  Whether to use multi-ECC
A.  Benefits of multi-curve ECC
A.  Costs of multi-curve ECC
A.3.2.3.  Applying multi-curve ECC
A.4.  General features of curve 2y^2=x^3+x/GF(8^91+5)
A.4.1.  Field features
A.4.3.  Equation features
A.4.4.  Finite curve features
A.4.4.1.  Curve size and cofactor
A.4.4.2.  Pollard rho security
A.4.4.3.  Pohlig--Hellman security
A.4.4.2.  Menezes--Okamoto--Vanstone security
A.4.4.3.  Semaev--Araki--Satoh--Smart security
A.4.4.4.  Edwards and Hessian form
A.4.4.5.  Bleichenbacher security
A.4.4.6.  Bernstein's "twist" security
A.4.4.7.  Cheon security
A.4.4.8  Reductionist security assurance for Diffie--Hellman
Appendix B.  Test vectors
Appendix C.  Sample code (pseudocode)
C.1.  Scalar multiplication of 34-byte strings
C.1.1.  Field arithmetic for GF(8^91+5)
C.1.2.  Montgomery ladder scalar multiplication
C.1.3.  Bernstein's 2-dimensional Montgomery ladder
C.1.4.  GLV in Edwards coordinates (Hisil--Carter--Dawson--Wong)
C.2.  Sample code for test vectors
C.3.  Sample code for a command-line demo of Diffie--Hellman
C.4.  Sample code for public-key validation and curve basics
C.5.  Elligator i
Appendix D.  Minimizing trapdoors and backdoors
D.1.  Decimal exponential complexity
D.1.1.  A shorter isomorophic curve
D.1.2.  Other short curves
D.1.3.  Converting DEC characters to bits
D.1.4.  Common acceptance of decimal exponential notation

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D.2.  General benefits of low Kolmogorov complexity to ECC
D.2.1.  Precedents of low Komogorov complexity in ECC
D.3.  Risks of low Kolmogorov complexity
D.4.  Alternative measures of Kolmogorov complexity
Appendix E. Primality proofs and certificates
E.1.  Pratt certificate for the field size 8^91+5
E.2.  Pratt certificate for subgroup order

1.  Introduction

  Elliptic curve cryptography (ECC) is now part of several IETF

  Multi-curve ECC can mitigate the risk of new curve-specific attacks
  on ECC.

  This document aims to contribute to multi-curve ECC by describing
  how to use the curve

    2y^2=x^3+x / GF(8^91+5)

  for elliptic curve Diffie--Hellman (ECDH).

  Appendix A expands on why and when 2y^2=x^3+x/GF(8^91+5) is useful
  in multi-curve ECC.

2.  Requirements Language (RFC 2119)

  The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
  document are to be interpreted as described in RFC 2119 [BCP14].

3.  Use ONLY in multi-curve ECC

  An implementation using curve 2y^2=x^3+x/GF(8^91+5) in elliptic
  curve cryptography MUST use it in a combination with other curves,
  such as Curve25519 or NIST P-256 (as a second layer of defense
  against unlikely security failures in the other curves).

  Appendix A expands on why and when 2y^2=x^3+x/GF(8^91+5) is useful
  in multi-curve ECC.

4.  Encoding points

  Elliptic curve cryptography uses points for public keys and raw
  shared secret keys.

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  Abstractly, points are mathematical objects.  For curve 2y^2=x^3+x,
  a point is either a pair (x,y), where x and y are elements of
  mathematical field, or a special point O (whose x and y coordinates
  may be deemed as infinity).

    Note: The special point O should never be used as a key in
    practice.  In theory, point O is needed for the points to form a
    mathematical group.

  For curve 2y^2=x^3+x/GF(8^91+5), the coordinates x and y of the
  point (x,y) are integers modulo 8^91+5, which can be represented as
  integers in the interval [0,8^91+4].

    Note: An implementation will often internally represent the
    x-coordinate as a ratio [X:Z] of field elements.  Each field
    element has multiple such representations, but [x:1] can viewed as
    normal representation of x.  (Infinity can be then represented by

  To interoperably communicate, points must be encoded as byte

  This draft specifies an encoding of finite points (x,y) as strings
  of 34 bytes, as described in the following sections.

    Note: The 34-byte encoding is not injective. Each point is
    generally among a group of four points that share the same byte

    Note: The 34-byte encoding is not surjective.  Approximately half
    of 34-byte strings do not encode a point (x,y).

    Note: In elliptic Diffie--Helman (ECDH), the 34-byte encoding
    works well, despite being neither injective nor surjective.

4.1.  Point encoding process

4.1.1.  Summary

  A point (x,y) is encoded by the little-endian byte representation of
  x or -x, whichever fits into 34 bytes.

4.1.2.  Details

  A point (x,y) is encoded into 34 bytes, as follows.

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  First, ensure that x is fully reduced mod p=8^91+5, so that

    0 <= x < 8^91+5.

  Second, further reduce x by a flipping its sign, as explained next.

   x' =: min(x,p-x) mod 2^272.

  Third, set the byte string b to be the little-endian encoding of the
  reduced integer x', by finding the unique integers b[i] such that
  0<=b[i]<256 and

   (x' mod 2^272) = sum (0<=i<=33, b[i]*256^i).

  Pseudocode can be found in Appendix C.

    Note: The loss of information that happens upon replacing x by -x
    corresponds to applying complex multiplication by i on the curve,
    because i(x,y) = (-x,iy) is also a point on the curve.  (To see
    this: note 2(iy)^2 = -(2y^2) = -(x^3+x) = (-x)^3+(-x).)  In many
    applications, particularly Diffie--Hellman key agreement, this
    loss of information is carried through to the final shared secret,
    which means that Alice and Bob can agree on the same secret 34

  In ECC systems where the original x-coordinate and the decoded
  x-coordinate need to match exactly, the 34-byte encoding is probably
  not usable unless the following pre-encoding procedure is practical:

    Given a point x where x is larger than min(x,p-x), first replace x
    by x'=p-x, on the encoder's side, using the new value x' (instead
    of x) for any further step in the algorithm.  In other words,
    replace the point (x,y) by the point (x',y')=(-x,iy).  Most
    algorithms will also require a discrete logarithm d of (x,y),
    meaning (x,y) = [d] G for some point G.  Since (x',y') = [i](x,y),
    we can replace by d' such that [d']=[i][d].  Usually, [i] can be
    represented by an integer, say j, and we can compute d' = jd (mod

4.2.  Point decoding process

4.2.1.  Summary

  The bytes are little-endian decoded into an integer which becomes
  the x-coordinate.  Public-key validation is done when needed.  If
  needed, the y-coordinate is recovered.

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4.2.2.  Detail

  If byte i is b[i], with an integer value between 0 and 255
  inclusive, then

   x = sum( 0<=i<=33, b[i]*256^i)

    Note: a value of -x (mod p) will also be suitable, and results in
    a point (-x,y') which might be different from the originally
    encoded point.  However, it will be one of the points [i](x,y) or
    -[i](x,y) where [i] means complex multiplication by [i].

  In many cases, such as Diffie--Hellman key agreement using the
  Montgomery ladder, neither the original value of coordinate x (among
  x and -x) nor coordinate y of the point is needed.  In these cases,
  the decoding steps can be considered completed.

    |                                                       |
    |        \  W  / /A\  |R) |N | I |N | /G   !            |
    |         \/ \/ /   \ |^\ | \| | | \| \_7  0            |
    |                                                       |
    |                                                       |
    |  WARNING: Some byte strings b decode to an invalid    |
    |  point (x,y) that does not belong to the curve        |
    |  2y^2=x^3+x.  Some applications would suffer from a   |
    |  severe  attack if they allow use of (x,y) not on     |
    |  the curve.  Such vulnerable applications MUST        |
    |  validate that the decoded point (x,y) is on the      |
    |  curve, as described in Section 5.                    |
    |                                                       |

  In cases where a value for at least one of y, -y, iy, or -iy is
  needed (such as in Diffie--Hellman key agreement using Edwards
  coordinates), a candidate value for y can be obtained by computing a
  square root:

    y = ((x^3+x)/2)^(1/2).

  In some specialized applications (not Diffie--Hellman), it is
  important for the decoded value of x to match the original value of
  x exactly.  In that case, the encoder should use the procedure that
  replaces x by p-x, and adjusts the discrete logarithm appropriately.
  These steps can be done by the encoder, with the decoder doing

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5.  Point validation

  In elliptic curve cryptography, scalar multiplying an invalid public
  key by a private key risks leaking information about the private

    Note: For curve 2y^2=x^3+x over 8^91+5, the underlying attacks are
    slightly milder than is average for a typical elliptic curve.

  To avoid leaking information about the private, the public key can
  be validated, which includes various checks on the public key.

5.1.  When to validate

  This section specifies three strategies (mandatory, simplified, and
  minimal) about deciding when to validate whether a given point (x,y)
  is on the curve 2y^2=x^3+x/GF(8^91+5).

5.1.1.  Mandatory validation

  As a precautionary defense-in-depth, an impelementation MAY opt to
  apply mandatory validation, meaning every public key (and point) is

5.1.2.  Simplified validation

  A small, general-purpose, implementation aiming for high speed might
  not be able to afford the cost of mandatory validation from Section
  4.1.1, because each validation costs about 10% of a scalar

  As a practical middle ground, an impelmentation MAY opt to apply
  simplified validation, which is the rule is that a distrusted public
  key is validated before being scalar multiplied by a static secret

    |   STATIC                                                      |
    |   SECRET                                                      |
    |    KEY      ------\                     _  ___                |
    |     +              )   PUBLIC |\/| | | (_`  |                 |
    |  UNTRUSTED  ------/    KEY    |  | \_/ ._)  |  BE VALIDATED.  |
    |   PUBLIC                                                      |
    |    KEY                                                        |

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    Note: Simplified validation implies that when the secret key is
    ephemeral (for example, used in one Diffie--Hellman transaction),
    the public key need not be validated.

    Note: Simplified validation implies that when the point being
    scalar multiplied is a known valid fixed point, or a previously
    validated public key (including a public key from a certificate in
    which the certification authority has a policy to valid public
    keys), then validation is not needed.

5.1.3.  Minimal validation

  An implementation MAY opt to use minimal validation, meaning doing
  as little point validation as possible, just enough to resist known
  attack against the implementation.

  The curve 2y^2=x^3+x is not twist-secure: using the Montgomery
  ladder for scalar multiplication is not enough to thwart invalid
  public key attacks.

  For example, consider a static hashed-ECDH implementation
  implemented with a Montgomery ladder, such that the static secret
  key is used in at most ten million times hashed-ECDH transactions.
  Even if exposed to invalid points on the twist, the security risk is
  nearly negligible -- so minimal validation would not validate the
  peer's public keys.

5.2.  Point validation process

  Upon decoding a 34-byte string into x, the next step is to compute
  z=2(x^3+x). Then one checks if z has a nonzero square root (in the
  field of size 8^91+5).  If z has a nonzero square root, then the
  represented point is valid, otherwise it is not valid.

  Equivalently, one can check that x^3 + x has no square root (that
  is, x^3+x is a quadratic non-residue).

  To check z for a square root, one can compute the Legendre symbol
  (z/p) and check that is 1.  (Equivalently, one can check that

  The Legendre symbol can be computed using Gauss' quadratic
  reciprocity law, but this requires implementing modular integer
  arithmetic for integral moduli smaller than 8^91+5.

  Instead, one can compute the Legendre symbol using powering in the
  field: (z/p) = z^((p-1)/2) = z^(2^272+2).  This is much slower than
  using quadratic reciprocity, but is perhaps simpler.

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  More generally, in signature applications (such as [B2]), where the
  y-coordinate is also needed, the computation of y, which involves
  computing a square root will generally implicitly include a check
  that x is valid.

  OPTIONAL: In some rare situations, it is also necessary to ensure
  that the point has large order, not just that it is on the curve.

  For points on this curve, each point has large order, unless it has
  torsion by 12.  In other words, if [12]P != O, then the point P has
  large order.

  OPTIONAL: In even rarer situations, it may be necessary to ensure
  that a point P also has a prime order q = ord(G).  The costly method
  to check this is checking that [q]P = O.  An alternative method is
  to try to solve for R in the equation [12]R=P, which involves
  methods such as division polynomials.  To be completed.

6.  OPTIONAL encodings

  The following two encodings are not usually needed to obtain
  interoperability in the typical ECC applications, such as
  Diffie--Hellman (or digital signatures).  In more specialized
  application, these encodings can be useful.

6.1.  Encoding scalar multipliers

  Scalar (integer point multipliers) sometimes need to be encoding as
  byte strings.   Typical examples are the following applications.

   - Digital signature in ECC generallly require scalar encodings.
     This draft does not specify signature algorithms in detail, only
     providing some general suggestions.

   - An implementation needs to store scalars, because scalars are
     used at least twice, and must be stored between these two uses.
     For example, in elliptic curve Diffie--Hellman, Alice has scalar
     a, sends Bob point aG, keeps scalar a until she receives point
     B from Bob, to which she then applies aB.  (If a is ephemeral,
     she then deletes a.)  An implementation is free to use any
     encoding of scalar, but implementation are often constructed in
     modular pieces, and any pieces handling the same scalar need to
     be able to convey the scalar.

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  In Diffie--Hellman implementations based on G which has prime order
  q, where q is approximately p/72, the value of scalar s usually only
  matters mod q.  So, one can reduce s, replacing it by s mod q,
  making s<q.  Since q < 2^267 < 256^34, a value s can be represented
  in 34 bytes.

  Basically, little-endian byte encoding of scalars is recommended,
  for consistency the little-endian byte encoding of field elements.

6.2.  Encoding strings as points

  In niche applications, it may be desired to encode an arbtirary
  string as a point on a curve.  Example reasons to encode arbitrary
  34-byte strings include:

    - Encoding passwords (or their hashes) in a password-authenticated
      key exchange (PAKE).

    - Hiding the fact that ECC is being used.

  To this end, this section sketches a method to reversibly encode
  any 34-byte string as a point.

    Note: To encode variable-length strings as points, one can first
    compute a 34-byte hash of the variable-length string, and then
    encode the hash.  Encoding of variable-length strings is not, and
    cannot be, reversible.

    Note: The point decoding scheme of Section 4.2 does not suffice to
    encode strings, because only about half of all 34-byte strings are

    Note: The string-as-point encoding has the the property that only
    about half of all points are decodable as 34-bytes strings.
    Encoding a uniformly distributed 34-byte string as a point yields
    non-uniformly distributed points.

  The encoding is called Elligator i.

    Note: The Elligator i encoding is a minor variation of the
    Elligator 2 construction [Elligator], introduced in [B1].  A minor
    variation is necessary because Elligator 2 fails for curves with
    j-invariant 1728, and curve 2y^2=x^3+x has j-invariant 1728.

  Fix a square root i of -1 in the field in GF(8^91+5).  For example,
  2^(8^89+1) mod 8^91+5.

  To encode a 34-byte string b,

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    1. Let b represent a field element r, using little-endian base

    2. Compute x = i-3i/(1-ir^2).  Let j=1.

    3. If 2y^2=x^3+x has no solution y, then replace x by x+i and j by

    4. Find two solutions y[1] and y[2] to 2y^2=x^3+x, such that

    5. Compute y=y[j].

  Now (x,y) is a point on the curve 2y^2=x^3+x.

  The Elligator i encoding is reversible, because it has the decoding
  sketched below.

  If y>p-y, replace x by x-i.  Solve for s = -i - 3/(i-x).  Let r =
  sqrt(s).  If r > p-r, replace r by p-r.  Write r in little-endian
  base 256 to get a 34-byte string b.

    Note: Just to illustrate a constrast between Elligator i encoding
    and the normal point encoding, consider the useless example of
    applying both encodings.  Start with 34-byte string b.  Apply
    Elligator i encoding to get a point (x,y).  Apply the point
    encoding to (x,y) to get a 34-byte string b'.  In summary,
    b'=encode(encode(b)).  The byte string b' has no significant
    relation to b.  The map b->b' from 34-byte strings to themselves
    is lossy (non-injective) with ratio ~4:1, and the image set is
    about one quarter of all 34-byte strings.

7.  IANA considerations

  This document requires no actions by IANA, yet.

8.  Security considerations

  No cryptographic algorithm is without risk.

  Possible security risks of 2y^2=x^3+x/GF(8^91+5) are listed in this

  Risk is difficult to estimate, especially aginst possible unknown
  attacks.  Relative risk is slightly easier to estimate, if a
  comparable cryptographic system is available as a benchmark.

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  The security risks of 2y^2=x^3+x/GF(8^91+5) are compared to the
  risks of a typical generic curve in ECC, or to the risks of specific
  well-established curves in ECC (such as NIST P-256 and Curve25519).

    Note: Because 2y^2=x^3+x/GF(8^91+5) MUST be used only in
    multi-curve ECC, comparison to other curves is mainly for the
    purposes of benchmarking, and for selection among selection of a
    secondary or tertiary cuve in a multi-curve ECC implementation.

    Note: For possible security benefits of 2y^2=x^3+x/GF(8^91+5), see
    Appendix A.

8.1.  Field choice

  The field 8^91+5 has the following risks.

  - 8^91+5 is a special prime.  As such, it is perhaps vulnerable to
    some kind of attack.  For example, for some curve shapes, the
    supersingularity depends on the prime, and the curve size is
    related in a simple way to the field size, causing a potential
    correlation between the field size and the effectiveness of an
    attack, such as the Pohlig--Hellman attack.  In summary, field
    size is positively correlated to some known attacks, and perhaps a
    special field size is positively correlated to a potential attack.

    Nonetheless, many other standard curves, such as the NIST P-256
    and Curve25519, also use special prime field sizes.  In this
    regard, all these special field curves have a similar risk.

    Yet other standard curves, such as the Brainpool curves, use
    pseudorandom field sizes, reducing their risk to potential
    special-field attack.

  - 8^91+5 arithmetic implementation, while implementable in five
    64-bit words, has some risk of overflowing, or of not fully
    reducing properly.  A smaller field, such as that used in
    Curve25519, should simpler reduction and overflow-avoidance

  - 8^91+5, by virtue of being well-above 256 bits in size, risks its
    user doing extra, and perhaps unnecessary, computation to protect
    their 128-bit keys, whereas smaller curves might be faster (as
    expected) yet still provide enough security.  In other words, the
    extra computational cost for exceeding 256 bits is wasteful, and
    partially a form of denial of service.

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  - 8^91+5 is smaller than some other six-symbol primes: 8^95-9,
    9^99+4 and 9^87+4.  Therefore, arguably, 8^91+5 fails to
    absolutely maximize field size relative to decimal exponential
    complexity.  In particular, curves defined over larger field size
    have better Pollard rho resistance (of the ECDLP).

    Nonetheless, the primes 9^99+4 and 9^87+4 are not close to a power
    of two, so probably suffer from about two time slower
    implementation than 8^91+5, which is a significant runtime cost,
    and perhaps also a security risk (due to implementation bugs).

    The prime 8^95-9 is, just like 8^91+5, very close to a power of
    two.  It may thus have efficiency comparable to 8^91+5 for basic
    field arithmetic operations, such as addition, multiplication and
    reduction.  The field 8^95-9 is a little larger, but is likely
    also implementable using five 64-bit words.  Being larger, 8^95-9
    has a slightly greater risk than 8^91+5 of leading to an
    arithmetic overflow implementation fault in field arithmetic.
    Field size 8^95-9 has much less simple powering algorithms for
    computing field inverses, Legendre symbols, and square roots: so
    these operations, often important for ECC, may require more code,
    more runtime, and perhaps more risk of implementation bugs.

  - 8^91+5 is smaller than 2^283 (the field size for curve sect283k1
    [SEC2], [Zigbee]), and many other five-symbol and four-symbol
    prime powers (such as 9^97).  It provides less resistance to
    Pollard rho than such larger prime powers.  Recent progress in the
    elliptic curve discrete logarithm problem, [HPST] and [Nagao], is
    the main reason to prefer prime fields instead of power of prime
    fields.  A second reason to prefer a prime field (including the
    field of size 8^91+5) over small characteristic fields is the
    generally better software speed of large characteristic field.
    (Better software speed is mainly due to general-purpose hardware
    often having dedicated fast multiplication circuits:
    special-purpose hardware should make small characteristic field

  - The Kolmogorov complexity of 8^91+5 as six symbols is only minimal
    for decimal exponential complexity: but it is not minimal if other
    types of complexity measures are allowed.  For example, if we
    allow the exclamation mark for the factorial operation -- which is
    quite standard notation! -- primes larger than 8^91+5 expressible
    in fewer symbols.  For example, 94!-1 is a 485-bit prime number,
    expressible in five symbols.  Such numbers, so far as I know, are
    not close to a power of two, so would have similar inefficiency
    and implementability defects to primes like 9^99+4 and 9^87+4.
    Such inefficiencies could resaonably by the curve choice criteria,
    ruling out such primes.

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    Arguably, in traditional mathematical notation, the symbol '^' is
    not actually written, with operation being marked by the use of
    superscripts.  In this view, using an ASCII character count
    arguably gives unduly low weight to the factorial operation as
    compared to exponentiation.

  See [B1] for further discussion about the relative merits of 8^91+5.

8.2.  Curve choice

  A first risk of using 2y^2=x^3+x is the fact that it is a special
  curve.  It is special in having complex multiplication leading
  to an efficient endomorphism.  Miller, in 1985, already suggested
  exercising prudence when considering such special curves.  Gallant,
  Lambert and Vanstone found ways to slightly speed up Pollard rho
  given such an endomorphism, but no other attacks have been found.

  Menezes, Okamoto and Vanstone (MOV) found an attack on special
  elliptic curves, of low embedding degree.  The curve
  2y^2=x^3+x/GF(8^91+5) is not vulnerable to their attack, but if one
  changes the underlying to some different primes, say p', the
  resulting curve 2y^2=x^3+x/GF(p') is vulnerable to their attack for
  about half of all primes.  Because the MOV was later than Miller's
  caution from 1984, Miller's prudence seems prescient.  Perhaps he
  was also prescient about yet other potential attacks (still
  unpublished), and these attacks might affect 2y^2=x^3+x/GF(8^91+5).

  Many other standard curves, NIST P-256 [NIST-P-256], Curve25519,
  Brainpool [Brainpool], do not have any efficient complex
  multiplication endomorphisms.  Arguably, these curves comply to
  Miller's advice to be prudent about special curves.

  Yet other (fairly) standard curves do, such as NIST K-283 (used in
  [Zigbee]) and secp256k1 (see [SEC2] and [BitCoin]).  Furthermore, it
  is not implausible [KKM] that special curves, including those
  efficient endomorphisms, may survive an attack on random curves.

  A second risk of 2y^2=x^3+x over 8^91+5 is the fact that it is not
  twist-secure.  What may happen is that an implementer may use the
  Montgomery ladder in Diffie--Hellman and re-use private keys.  They
  may think, despite the (ample?) warnings in this document, that
  public key validation in unnecessary, modeling their implementation
  after Curve25519 or some other twist-secure curve.  This implementer
  is at risk of an invalid public key attack.  Moreover, the
  implementer has an incentive to skip public-key validation, for
  better performance.  Finally, even if the implementer uses
  public-key validation, then the cost of public-key validation is

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  A third risk is a biased ephemeral private key generation in a
  digital signature scheme.  Most standard curves lack this risk
  because the field size is close to a power of two, and the cofactor
  is a power of two.  Curve 2y^2=x^3+x over 8^91+5 has a base point
  order which is approximately a power of two divided by nine (because
  its cofactor is 72=8*9.)  As such, it is more vulnerable than
  typical curves to biased ephemeral keys in a signature scheme.

  A fourth risk is a Cheon-type attack.  Few standard curves address
  this risk, and 2y^2=x^3+x over 8^91+5 is not much different.

  A fifth risk is a small-subgroup confinement attack, which can also
  leak a few bits of the private key.   Curve 2y^2=x^3+x over 8^91+5
  has 72 elements whose order divides 12.

8.3.  Encoding choices

  To be completed.

  As in all ECC, projective coordinates are not suitable as the final
  representation of an elliptic curve point, for two reasons.

  - Projective coordinates for a point are generally not unique: each
    point can be represented in projective coordinates in multiple
    different ways.  So, projective coordinates are unsuitable for
    finalizing a shared secret, because the two parties computing the
    shared secret point may end up with different projective

  - Projective coordinates have been shown to leak information about
    the scalar multiplier [PSM], which could be the private
    key.  It would be unacceptable for a public key to leak
    information about the private key.  In digital signatures, even a
    few leaked bits can be fatal, over a few signatures

  Therefore, the final computation of an elliptic curve point, after
  scalar multiplication, should translate the point to a unique
  representation, such as the affine coordinates described in this

  For example, when using a Montgomery ladder, scalar multiplication
  yields a representation (X:Z) of the point in projective
  coordinates.  Its x-coordinate is then x=X/Z, which can be computed
  by computing the 1/Z and then multiplying by X.

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  The safest, most prudent way to compute 1/Z is to use a side-channel
  resistant method, in particular at least, a constant-time method.
  This reduces the risk of leaking information about Z, which might in
  turn leak information about X or the scalar multiplier.  Fermat
  inversion, computation of Z^(p-2) mod p, is one method to compute
  the inverse in constant time (if the inverse exists).

8.4.  General subversion concerns

  Although the main motivation of curve 2y^2=x^3+x over 8^91+5 is to
  minimize the risk of subversion via a backdoor ([Gordon], [YY],
  [Teske]), it is only fair to point out that its appearance in this
  very document can be viewed with suspicion as an possible effort at
  subversion (via a front-door).  (See [BCCHLV] for some further

  Any other standardized curve can be view with a similar suspicion
  (except, perhaps, by the honest authors of those standards for whom
  such suspicion seems absurd and unfair).  A skeptic can then examine
  both (a) the reputation of the (alleged) author of the standard,
  making an ad hominem argument, and (b) the curve's intrinsic merits.

  By the very definition of this document, the reader is encouraged to
  take an especially skeptical viewpoint of curve 2y^2=x^3+x over
  8^91+5.  So, it is expected that skeptical users of the curve will

  - use the curve for its other merits (other than its backdoor
    mitigations), such as efficient endomorphism, field inversion,
    high Pollard rho resistance within five 64-bit words, meanwhile
    holding to the evidence-supported belief ECC that is now so mature
    that worries about subverted curves are just far-fetched nonsense,

  - as an additional of layer of security in addition to other
    algorithms (ECC or otherwise), as an extra cost to address the
    non-zero probability of other curves being subverted.

  To paraphrase, consider users seriously worried about subverted
  curves (or other cryptographic algorithms), either because they
  estimate as high either the probability of subversion or the value
  of the data needing protection.  These users have good reason to
  like 2y^2=x^3+x over 8^91+5 for its compact description.
  Nevertheless, the best way to resist subversion of cryptographic
  algorithms seems to be combine multiple dissimilar cryptographic
  algorithms, in a strongest-link manner.  Diversity hedges against
  subversion, and should the first defense against it.

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    Note: For any form of ECC, finite field multiplication can be
    achieved most quickly by using hardware integer multiplication
    circuits.  It is critical that those circuits have no bugs or
    backdoors.  Furthermore, those circuits typically can only
    multiply integers smaller than the field elements.  Larger inputs
    to the circuits will cause overflows.  It is critical to avoid
    these overflows, not just to avoid interoperability failures, but
    also to avoid attacks where the attackers supply inputs likely
    induce overflows [bug attacks], [IT].

8.5.  Concerns about 'aegis'

  The exact curve 2y^2=x^3+x/GF(8^91+5) was (seemingly) first
  described to the public in 2017 [AB].  So, it has a very low age, at
  least compare to more established curves.

  Furthermore, it has not been submitted for a publication with peer
  review to any formally peer-reviewed academic cryptographer forum
  such as the IACR conferences like Crypto and Eurocrypt.  So, it has
  most like been reviewed by very few eyes.

  Arguably, other reviewers have little incentive to study it
  critically, for several reasons.  The looming threat of a quantum
  computer has diverted many researchers towards studying post-quantum
  cryptography, such as supersingular isogeny Diffie--Hellman.  The
  past disputes over NIST P-256 and Curve25519 (and several other
  alternatives) have perhaps tired some reviewers, many of whom
  reasonably wish to concentrate on deployment of ECC.

  So, under the metric of aegis, as in age times eyes (times
  incentive), 2y^2=x^3+x/GF(8^91+5) scores low.  Counting myself (but
  not quantifying incentive) it gets an aegis score of 0.1 (using a
  rating 0.1 of my eyes factor in the aegis score: I have not
  discovered any major ECC attacks of my own.)  This is far smaller
  than my estimates (see below) some more well-studied curves.

  Nonetheless, the curve 2y^2=x^3+x over 8^91+5 at least has some
  similarities to some of the better-studied curves with much higher

  - Curve25519: has field size 8^85-19, which a little similar to
    8^91+5; has equation of the form by^2=x^3+ax+x, with b and a
    small, which is similar to 2y^2=x^3+x.  Curve25519 has been around
    for over 10 years, has (presumably) many eyes looking at it, and
    has been deployed thereby creating an incentive to study.  An
    estimated aegis for Curve25519 is 10000.

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  - NIST P-256: has a special field size, and maybe an estimated aegis
    of 200000.  (It is a high-incentive target.  Also, it has received
    much criticism, showing some intent of cryptanalysis.  Indeed,
    there has been incremental progress in finding minor weakness
    (implementation security flaws), suggestive of actual
    cryptanalytic effort.)  The similarity to 2y^2=x^3+x over 8^91+5
    is very minor, so very little of the P-256 aegis would be relevant
    to this document.

  - secp256k1: has a special field size, though not quite as special
    as 8^91+5, and has special field equation with an efficient
    endomorphism by a low-norm complex algebraic integer, quite
    similar to 2y^2=x^3+x.  It is about 17 years old, and though not
    studied much in academic work, its deployment in Bitcoin has at
    least created an incentive to attack it.  An estimated aegis for
    secp256k1 is 10000.

  - Miller's curve: Miller's 1985 paper introducing ECC suggested,
    among other choices, a curve equation y^2=x^3-ax, where a is a
    quadratic non-residue.  Curve 2y^2=x^3+x is isomorphic to
    y^2=x^3-x, essentially one of Miller's curves, except that a=1 is
    a quadratic residue.  Miller's curve may not have been studied
    intensely as other curves, but its age matches that ECC itself.
    Miller also hinted that it was not prudent to use a special curve
    y^2=x^3-ax: such a comment may have encouraged some cryptanalysts,
    but discouraged cryptographers, perhaps balancing out the effect
    on the eyes factor the aegis.  An estimated aegis for Miller's
    curves is 300.

  Obvious cautions to the reader:

  - Small changes in a cryptographic algorithm sometimes cause large
    differences in security.  So security arguments based on
    similarity in cryptographic schemes should be given low priority.

  - Security flaws have sometimes remained undiscovered for years,
    despite both incentives and peer reviews (and lack of hard
    evidence of conspiracy).  So, the eyes-part of the aegis score is
    very subjective, and perhaps vulnerable false positives by a herd
    effect.  Despite this caveat, it is not recommended to ignore the
    eyes factor in the aegis score: don't just flip through old books
    (of say, fiction), looking for cryptographic algorithms that might
    never have been studied.

9.  References

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9.1.  Normative References

  [BCP14] Bradner, S., "Key words for use in RFCs to Indicate
          Requirement Levels", BCP 14, RFC 2119, March 1997,

9.2.  Informative References

  To be completed.

  [AB] A. Allen and D. Brown.  ECC mod 8^91+5, presentation to CFRG,

  [AMPS] Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, and
     Juraj Somorovsky.  Prime and Prejudice: Primality Testing Under
     Adversarial Conditions, IACR ePrint,
     2018. <>

  [B1] D. Brown.  ECC mod 8^91+5. IACR ePrint, 2018.

  [B2] D. Brown.  RKHD ElGamal signing and 1-way sums. IACR ePrint,
     2018. <>

  [B3] D. Brown.  Rolling up sleeves when subversion's in the field?
     IACR eprint, 2020. <>

  [KKM] A. Koblitz, N. Koblitz and A. Menezes.  Elliptic Curve
     Cryptography: The Serpentine Course of a Paradigm Shift, IACR
     ePrint, 2008.  <>

  [BCCHLV] D. Bernstein, T. Chou, C. Chuengsatiansup, A. Hulsing,
     T. Lange, R. Niederhagen and C. van Vredendaal.  How to
     manipulate curve standards: a white paper for the black hat, IACR
     ePrint, 2014. <>

  [Elligator] (((To do:))) fill in this reference.

  [NIST-P-256] (((To do:))) NIST recommended 15 elliptic curves for
     cryptography, the most popular of which is P-256.

  [Zigbee] (((To do:))) Zigbee allows the use of a
     small-characteristic special curve, which was also recommended by
     NIST, called K-283, and also known as sect283k1.  These types of
     curves were introduced by Koblitz.  These types of curves were
     not recommended by NSA in Suite B.

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  [Brainpool] (((To do:))) the Brainpool consortium (???) recommended
     some elliptic curves in which both the field size and the curve
     equation were derived pseudorandomly from a nothing-up-my-sleeve

  [SEC2] Standards for Efficient Cryptography.  SEC 2: Recommended
     Elliptic Curve Domain Parameters, version 2.0, 2010.

  [IT] T. Izu and T. Takagi.  Exceptional procedure attack on elliptic
     curve cryptosystems, Public key cryptography -- PKC 2003, Lecture
     Notes in Computer Science, Springer, pp. 224--239, 2003.

  [PSM] (((To do:))) Pointcheval, Smart, Malone-Lee.  Projective
     coordinates leak.

  [BitCoin] (((To do:))) BitCoin uses curve secp256k1, which has an
     efficient endomorphism.

  [Bleichenbacher] To do: Bleichenbacher showed how to attack DSA
      using a bias in the per-message secrets.

  [Gordon] (((To do:))) Gordon showed how to embed a trapdoor in DSA

  [HPST] Y. Huang, C. Petit, N. Shinohara and T. Takagi.  On
     Generalized First Fall Degree Assumptions, IACR ePrint 2015.

  [Nagao] K. Nagao.  Equations System coming from Weil descent and
     subexponential attack for algebraic curve cryptosystem, IACR
     ePrint, 2015.  <>

  [Teske] E. Teske.  An Elliptic Curve Trapdoor System, IACR ePrint,
     2003.  <>

  [YY] (((To do:))) Yung and Young, generalized Gordon's ideas into
     Secretly-embedded trapdoor ... also known as a backdoor.

Appendix A.  Why 2y^2=x^3+x/GF(8^91+5)?

  This sections says why curve 2y^2=x^3+x/GF(8^91+5) can improve ECC,
  if used properly in multi-curve ECC.

    Note: Later sections (especially 4, 5, 6, 8, A, B, C, and D) cover
    some relatively routine ECC details about how to use

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A.1. Not for single-curve ECC

  Curve 2y^2=x^3+x/GF(8^91+5) SHOULD NOT be used in single-curve ECC.

  It is riskier than other IETF-approved curves, such as NIST P-256
  and Curve25519, for at least the following reasons:

    - it is newer, so riskier, all else equal, and

    - it is special, with complex multiplication by i: consensus
      continues to agree with Miller's original 1985 opinion that
      using (such) special curves is not "prudent".

  Koblitz, Koblitz and Menezes [KKM] somewhat dissent from the
  consensus against special curves.  They list several plausible cases
  of special curves -- including some with complex multiplication --
  that they argue might well be safer than random curves.  (Others go
  even further, dismissing prudence against special curves as myth

  Despite this dissent, this report adheres to the consensus, which is
  to prefer other curves for single-curve ECC.

  The relative newness of 2y^2=x^3+x/GF(8^91+5) is not entire.  The
  curve equation is isomorphic to one proposed by Miller in 1985,
  making it older than the isomorphism class of curve equations in
  NIST P-256 or Curve25519.  The field size, the prime 8^91+5=2^273+5,
  is a prime likely to have been considered before the field size
  primes NIST P-256 or Curve25519, but probably not in an application
  to ECC (i.e. probably in surveys of special primes).

A.2.  Risks of new curve-specific attacks

  A risk for all ECC is new curve-specific attacks, especially attacks
  on the elliptic curve discrete logarithm problem.  A new
  curve-specific attack could break any ECC using the affected curves.

  The main benefit to ECC of curve 2y^2=x^3+x/GF(8^91+5) is to reduce
  this risk in multi-curve variant of ECC.

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    Note: an arguably larger risk, a quantum computer capable of
    running Shor's algorithm, looms over all of ECC.  The probability
    of this risk is basically independent of the probability new
    curve-specific attack, but the impacts are heavily dependent, if a
    quantum attack impacts ECC, then the new curve-specific attacks
    are totally moot.  Also, even if no quantum attack on ECC emerges,
    but PQC supplements or replaces ECC, then a new curve-specific
    attack becomes much more tolerable.  For sake of argument, suppose
    probabilities 1% for a new curve-specific attack by 2030, and 10%
    for a quantum-attack on ECC by 2030.  Addressing the 10%
    probability risk is more urgent, but there is still a 90% chance
    that of no-quantum-attack.  Assuming that PQC is combined with ECC
    (instead of replacing it) and assuming that the 10% and 1%
    probabilities above are formally independent, then there is 0.9%
    probability that new-curve specific on ECC by 2030 would affect
    PQC+ECC systems, reducing their security to that of PQC only.

A.2.1.  What would be considered a "new curve-specific" attack?

  The idea of new curve-specific attacks is now discussed.  The
  purpose is to remind the reader of the risks, by comparison to past
  curve-specific attacks, so that a user can estimate the benefits of
  addressing the risk.  Ultimately, the reader should make an informed
  as possible decision whether the extra cost of multi-curve is

A.2.2.1.  What would be considered a "new" attack?

  The "new" in "new curve-specific attack" means hypothetical and not
  yet published, and hence, either future or hidden.  This
  contemplates an adversary with superior cryptanalytic capability
  than current state-of-the-art knowledge.

A.2.2.2.  What is, would be, considered a "curve-specific attack"?

  The "curve-specific" in "new curve-specific attakc" means that the
  following conditions on the attack are true

    - it affects almost ECC algorithms using the specific curve
      (typically, if the discrete logarithm problem is easy for that
      curve, or in some cases, the decision Diffie--Hellman problem),

    - it does not affect ECC using at least one other curve
      (typically, many other curves), and

    - it would not affect a generic group of the same size of the
      secure ECC group.

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    Note: For example, the naive Pollard-rho attack is not
    "curve-specific" because it fails the second condition and third
    condition (it affects all curves and all generic groups of equal
    or smaller size than the attacked curve).  The Pohlig--Hellman
    attack (on smooth order groups) is not curve-specific because it
    fails the third condition.

    Note: A side-channel attack on an ECC implementation is not
    necessarily "curve-specific" in the strict sense above, if
    another ECC implementation using the same curve resists the
    attack.  Some curves may be more prone than others to side-channel
    attacks, here we refer to that situtation "curve-specific

  Prime-field curves were affected by two curve-specific attacks (on
  the discrete logarithm): the MOV attacks, and the SASS attack, both
  from before 2001.  For the decision Diffie--Hellman problem, a
  generalization of the MOV attack can be considered as

  For non-prime-field curves, more recent curve-specific attacks have
  been discovered, some asymptotically polynomial-time.  (To be

A.2.2.3.  Rarity of published curve-specific attacks

  To be completed.

  The known curve-specific attacks against prime-field curves are rare
  in the sense of having negligible probability of affecting a random
  curve (over a given prime-field).

  Some of these are attacks are also field-specific too.
  These attacks somewhat rare among all possible non-prime-field
  curves (though in some cases the probability among certain class of
  curves is non-negligible).

  If the rarity of the known curve-specific attacks carries over to
  any new curve-specific attacks, then truly random curves should
  resist the new curve-specific attacks, except with negligible
  probability.  Honestly generated, non-random curves should also
  resist the new curve-specific attacks, except in the unfortunate
  case the new curve-specific attack is correlated with the honest
  curve generation criteria.

A.2.2.4.  Correlation of curve-specific efficiency and attacks

  To be completed.

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  Many of the known curve-specific attacks affected previously
  proposed curves, and presumably honestly generated curves.  For
  example, supersingular curves were proposed for their slightly
  greater efficiency over ordinary curves, but then turned out to be
  vulnerable to the MOV attack.  (Similarly, curves vulnerable to the
  SASS attack were proposed for slight efficiencies, before the SASS
  attack was published.)  So, such correlations are not only
  plausible, but the real-world pattern for ECC.  Accidents have
  already happened for such non-random curves.

  Worse yet, if a non-random curve is chosen maliciously, a
  correlation between a hidden curve-specific attack and some sensible
  curve generation criteria might well make it possible for a
  maliciously chosen non-random curve to be made vulnerable to a
  hidden curve-specific attack.

A.3.  Mitigations against new curve-specific attacks

  Because the risk of new curve-specific attack is nonzero, applying
  mitigations against the risk potentially improves security, albeit
  at some cost.

A.3.1.  Fixed curve mitigations

  Often, a single fixed curve is used across a system of ECC users,
  generally for reasons of efficiency.  This exposes the system to the
  nonzero risk of new curve-specific attacks.

A.3.1.2.  Existing fixed-curve mitigations

  Some of the better established fixed curve have sensibly included
  mitigations against the nonzero risk of new curve-specific attacks.

  - NIST curve P-256 has coefficients derived from the ouptut of
    SHA-1, perhaps aiming to avoid any new curve-specific weakness
    that would appply rarely to random curves, although inadequately
    so, because the seed input to the hash is utterly inexplicable,
    and plausibly manipulable.

  - Bernstein's Curve25519 results from a "rigid", non-random design
    process, favoring efficiency over all else, perhaps eliminating
    intentional subversion towards a new curve-specifc weakness.

  - Brainpool's curves are derived using hash functions applied to
    nothing-up-my-sleeve numbers, perhaps aiming to mitigate both
    intentional subversion and accidental rare weakness.

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    Note: A reasonable inference from these curves is that risk of new
    curve-specific attacks warranted the mitigations used (as listed
    above).  The risk may be less now that further time has passed,
    because no other curve-specific attacks against prime-field curves
    arose in the interim.  The risk is still not zero, so the
    mitigations may still be warranted.

A.3.1.2.  Migitations used by 2y^2=x^3+x/GF(8^91+5)

  The curve 2y^2=x^3+x/GF(8^91+5) includes similar fixed-curve
  mitigations against the risk of new curve-specific attacks:

  - a short description (low Kolmogorov compelxity), aiming to have
    little wiggle for an intentional embedded weakness (somewhat like
    a nothing-up-my-sleeve number used in the Brainpool curves),

  - a set of special efficiencies, such as a curve endomorphism,
    Montgomery form, and fast field operation (somewhat like the
    "rigid" properties of Curve25519 favor efficiency as a mitigation
    to fight off intentional embedded weakness),

  - a prime field, to stay clear of recent curve-specific attacks on
    non-prime-field ECC.

  These mitigations do not suffice to justify its use in single-curve
  ECC (instead of more established non-special curves).

    Note: The mitigations above, like those of NIST P-256 and
    Curve25519, have a cost which consists mostly of a one-time
    computation.  The mitigations are somewhat warranted, even if
    multi-curve ECC, because the aim of multi-curve is to hedge the
    risk of curve-specific attacks, so it makes sense for each
    individual curve to include mitigations against this risk.

A.3.2.  Multi-curve ECC

  This section further motivates the value of multi-curve ECC over
  single-curve ECC, but does specify a detailed way to do multi-curve

  Multi-curve ECC is only really effective if used with a diverse set
  of curves.  Multi-curve ECC SHOULD use a set of curves including the
  three curves:

      NIST P-256, Curve25519, and 2y^2=x^3+x/GF(8^91+5).

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  Multi-curve ECC aims to further mitigate the risk of curve-specific
  attack, by securely combining a diverse set of curves.  The aim is
  that at least one of the curves used in multi-curve ECC resists a
  new curve-specific attack (if a new attack ever appears).  This aim
  is only plausible if the set of curves used is diverse, in features
  or in authorship.

  This curve contributes to the diversity necessary for multi-curve
  ECC, with special technical features distinct from established
  curves NIST P-256 and Curve25519 (and Brainpool):

    - complex multiplication by i (low discrimiant, rather than high),

    - a greater emphasis on low Kolmogorov descriptional complexity
      (rather than hashed coefficient or efficiency).

A.3.2.1.  Multi-curve ECC is a redundancy strategy

  Multi-curve ECC is an instance of a strategy often called
  redundancy, applied to ECC.  Redundancy is quite general in that it
  can be applied to other types of cryptography, to other types of
  information security, and even to safety systems.  Other names for
  redundant strategies include:

    strongest-link, defense-in-depth, hybrid, hedged, composite,
    fail-safe, diversified, resilient, belt-and-suspenders, fault
    tolerant, robust, multi-layer, robustness, compound, combination,

A.3.2.2.  Whether to use multi-ECC

  Multi-curve ECC mitigates the risk of new curve-specific attacks, so
  ought to be used instead of single-curve ECC if affordable, such as

    - the privacy of the data being protected has higher value than
      the extra cost of multi-curve ECC, which may be the case for at
      least financial, medical, or personally-identifying data, and

    - ECC is only a tiny portion of the overall system costs, which
      would be the case if the data is human-generated or high-volume,
      or if ECC is combined with slow or large post-quantum
      cryptography (PQC).

A.  Benefits of multi-curve ECC

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  The benefit of multi-curve ECC is difficult to quantify.  The aimed
  benefit over single-curve ECC is extra security, in the event of a
  signficant curve-specific attack.

  No extra security results if all the curves used are the same.  The
  curves must be diverse, so that a potential attack on one is somehow
  unlikely to affect the other.  This diversity is difficult to
  assess.  Intuitively, a geometric metaphor of a polygon for the
  space of all choices might help.  Maximally distant points in a
  polygon tend to be vertices, the extremities of the polygon.
  Translating this intuition suggests choosing curves at the extremes
  of features.

    Note: By contrast, in a single-curve ECC, the geometric
    metaphor suggests a central internal point, on the grounds that
    each vertex is more likely to be affected to a special attack.
    Carrying this over to multi-curve suggests that a diverse set
    ought to include a non-extreme curve too.

  As always, the benefit of security is really the negative of the
  cost of an attack, including the risk.

  The contextual benefit of multi-curve ECC therefore depends very
  much on the application, involving the assessing both the
  probability of attack, and the impact of the attack.

  Higher value private data has greater impact if attacked, and
  perhaps also higher probability, if the adversary is more motivated
  to attack it.

  Low probability of attacks are mostly inferred through failed but
  extensive cryptanalysis efforts.  Normally, this is only intuited,
  but approaches to quantifiably estimate these probabilities is
  possible too, under sufficiently strong assumptions.

  To be completed.

A.  Costs of multi-curve ECC

  The cost of multi-curve ECC is fairly easy to quantify (easier than
  quantifying the benefit).

  The cost of multi-curve is meant to be compared to the cost of
  single-curve ECC.

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  The cost ratio is approximately the number of curves used.  The cost
  difference depends on the devices implementing the ECC.

  For example, on a current personal computer, the extra cost per ECC
  transaction can include up to 1 millisecond of runtime and sending
  an extra 30 bytes or more.  In low-end devices, the time may be
  higher due to slower processors.

  The contextual cost of ECC depends on the application context.  In
  some applications, such as personal messages between two users, the
  cost (milliseconds and a few hundred bytes) is affordable relative
  to the time users spent writing and reading the messages.  In other
  applications, such as automated inter-device communication with
  frequent brief messages, single-curve ECC may already be a
  bottleneck, costing most of the run-time.

A.3.2.3.  Applying multi-curve ECC

  For key establishment, NIST recently proposed (in a draft amendment
  to Special Publication 800-133 on key derivation) a mechanism to
  support deriving a single symmetric key from the result of multiple
  key establishments.  In summary, the mechansim is that the raw ECDH
  shared secrets would be concatenated and fed into a hash-based key
  derivation function.

  An alternative would be to XOR multiple shared symmetric-key

  So, multi-curve elliptic curve Diffie--Hellman (ECDH) key agreement
  could use one of these mechanism to derive a single key from
  multi-curve ECDH.

  A mechanism to support sending more than one ECDH public key
  (usually ephemeral), with an indication of the curve for each ECDH
  key, would also be needed.

  For signatures, the simplest approach is to attach multiple
  signatures to each message.  (For signatures providing message
  recovery, then an approach is to apply the results, with outer
  signatures recover the inner signed message, and so on.)

A.4.  General features of curve 2y^2=x^3+x/GF(8^91+5)

  This subsection describes some general features of the curve


  presuming a familiarity with elliptic curve cryptography (ECC).

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  Each of a set of well-established features, such as Pollard rho
  security or Mongtomgery form, for ECC in general are evaluated and
  summarized for the specific curve 2y^2=x^3+x/GF(8^91+5).

    Note: Interoperable ECC requires a few more details than are
    deducible from mathematical description 2y^2=x^3+x/GF(8^91+5) of
    the curve, such encoding points as byte strings.  These details
    are discussed in Sections 4, 5, and 6.

A.4.1.  Field features

  The curve's field of definition, GF(8^91+5), is a finite field, as
  is always the case in ECC.  (Finite fields are Galois field, and the
  field of size is p is written as GF(p).)

  The field size is the prime p=8^91+5.  (See the appendix for a
  Pratt primality certificate.)

  In hexadecimal (base 16, big-endian) notation, the number 8^91+5 is


  with with 67 zeros between 2 and 5.

  The most recent known curve-specific attacks on
  prime-field ECC are from 2000.

  Prime fields in ECC tend be more efficient in software than in

  The prime p is very close to a power of two.  Primes very close to a
  power of two are sometimes known as Crandall primes.  Reduction
  modulo p is more efficient for Crandall primes than for most other
  primes (or at least random primes).  Perhaps Crandall primes are
  more resistant to side-channel attacks or implementation faults than
  than most other primes.

  The fact that p is slightly larger than a power of two -- rather
  than slightly lower -- means that powering algorithms to compute
  inverses, Legendre symbols, and square roots are simpler and
  slightly more efficient (than would be for prime below a 2-power).

A.4.3.  Equation features

  The curve equation 2y^2=x^3+x has Montgomery form,


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  with (a,b) = (0,2).  This permits the Montgomery ladder scalar point
  multiplication algorithm to be used, which makes it relatively
  efficient, and also easier to protect against side channels.

  The curve 2y^2=x^3+x has complex multiplication by i, given an

   (x,y) -> (-x,iy).

    Note: Strictly speaking, over some fields, the curve would be
    supersingular, in which the term "complex mutliplication" is not
    used, because the curve then has quaternionic multiplication.

  The endomorphism permits the Gallant--Lambert--Vanstone (GLV) scalar
  multiplication algorithm, which makes it relatively efficient.  (The
  GLV method can also be combined with Bernstein's two-dimensional
  variant of the Montgomery ladder algorithm.)

  The curve has j-invariant 1728, because it has complex
  multiplication by i.

    Note: The j-invariants 0 and 1728 are special in that the curves
    with these j-invariants have more than two automorphisms.
    (Relatedly, over complex numbers, the moduli space of elliptic
    curves is an orbifold, with exactly two non-smooth points, at j=0
    and j=1728.)

A.4.4.  Finite curve features

  This section describes features of 2y^2=x^3+x/GF(8^91+5) as a finite
  curve consisting, the points (x,y) for x,y in GF(p), and also the
  point at infinity.  In other words, these features are specific to
  the combination of both the finite field and the curve equation.

    Note: In algebraic geometry, these points are said to rational
    over k=GF(p), and the set of rational points written as E[k] =
    (2y^2=x^3+x)[GF(8^91+5)], to distinguish from points with
    coordinates in the alebraic closure of k=GF(p).

  Many security properties, and a few performance properties, of ECC
  are specific to a finite curve.

A.4.4.1.  Curve size and cofactor

  The curve (of points rational over GF(8^91+5)) has size (order) 72q
  for a large prime q, which is, in hexadecimal,


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    NOTE: Appendix E has a Pratt primality certifcate for q.

  So, the curve has cofactor 72.

  The curve size can verified by implementing the curve's elliptic
  curve arithmetic, and scalar multiplying random points on the curve
  by the claimed size.  It can be partially verified using the complex
  multiplication theory, and a little big integer arithmetic.

  The prime p=8^91+5 has p=1 mod 4, so a theorem of Fermat says there
  exist integers u and v such that p=u^2+v^2.  Numbers u and v can
  found using a special case of Cornacchia's algorithm, and are listed
  further below.

  Complex multiplication theory says that a curve with complex
  multiplication by i has size s=(u+1)^2+v^2 = p+2u+1.  By negation
  and swapping u and v, there are four possible sizes, p+2u+1, p-2u+1,
  p+2v+1, p-2v+1 (sometimes known as the twist sizes).

  Curve 2y^2=x^3+x/GF(8^91+5) has one of these four sizes.  In this
  case, its size s is divisible by 72, and has large prime factor q =
  s / 72.

  The following 'bc' program includes values for u and v applicable to
  2y^2=x^3+x/GF(8^91+5), verifies these calculations, and outputs q.

    p = 8^91+5
    u = 104303302790113346778702926977288705144769
    v = 65558536801757875228360405858731806281506
    if ( p != u^2+v^2 ) { "u and v incorrect" ; halt }
    s = (u+1)^2 + v^2
    if ( 0 != (s % 72)) { "size not divisible by 72" ; halt}
    q = s/72

    Note: Theory only indicates that s has one of four values, so an
    extra step is needed to verify which of the four values is the
    size.  Scalar multiplication by s is a general method.  A faster
    method, specific to 2y^2=x^3+x/GF(8^91+5), is to show that only
    one of the four candidate sizes is divisible by 3, and then
    demostrate a point of order 3 on this curve.  Symbolic calculation
    with elliptic curve arithmetic show that the point (x,y) has order
    3 if 3x^4 + 1 = 0 in GF(p).  The big integer calculation
    (-(1+2p)/3)^((p-1)/4) = 1 mod p shows that such an x exists in

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    Note: The Schoof--Elkies--Atkin (SEA) point-counting algorithm can
    compute the size of any general curve, but is slower than methods
    for some special curves, which is why Miller suggested special
    curves 1985.

A.4.4.2.  Pollard rho security

  The prime q is 267-bit number.  The Pollard rho algorithm for
  discrete logarithem to the base G (or any order q point) takes
  (proportional to) sqrt(q) ~ 2^133 elliptic curve operations.  The
  curve provides at least 2^128 security against Pollard rho attacks,
  with about 5 bits to spare.

    Note: Arguably, the fact ECC operations are slower than
    symmetric-key operartions (such as hashing or block ciphers),
    means that ECC security should be granted a few extra bits,
    perhaps 5-10 bits, of security when trying to match ECC security
    with symmetric-key security.  In this case, one might say that
    2y^2=x^3+x/GF(8^91+5) resists Pollard-rho with 2^140 security,
    providing 12 bits of extra security.  The extra security can be
    viewed as a safety margin for error, or as an excessive to the
    extent the smaller, and faster curves would more than suffice to
    match 2^128 security of SHA-256 and AES-128.

  Gallant, Lambert, Vanstone, show how to speed up Pollard rho
  algorithms when the group has an extra endormorphism, which would
  apply to 2y^2=x^3+x.  The speed-up here amounts to a couple of bits
  in the security,

A.4.4.3.  Pohlig--Hellman security

  The small cofactor means the curve effectively resists
  Pohlig--Hellman attack (a generic algorithm to solve discrete
  logarithms in any group in time sqrt(m) where m is the largest
  prime factor of the group size).

    Note: Consensus in ECC is to recommend a small factor, such as 1,
    2, 4, or 8, despite the fact that, for random curves, the typical
    cofactor is approximately p^(1/3), which is much larger.  The
    small cofactor helps resists Pohlig--Hellman without increasing
    the field size.  (A larger field size would be less efficient.)

A.4.4.2.  Menezes--Okamoto--Vanstone security

  The curve has a large embedding degree.  More precisely, the curve
  size 72q has q with embedding degree (q-1)/2.

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  This means that the discrete logarithms to base G (a point of order
  q) resist Menezes--Okamoto--Vanstone attack.

  The large embedding degree also means that that no feasible pairings
  exist that could be used solve the decision Diffie--Hellman problem
  (for points of order q).  Similarly, the larger embedding degree
  also means, it cannot be used for pairing-based cryptography (and it
  would already too small to be used for pairing-based cryptography).

    Note: Intuitively, a near-miss or a close-call could describe this
    curve's resistance to the MOV attack.  For about half of all primes
    P, then curve 2y^2=x^3+x is supersingular over GF(P), with
    embedding degree 2, making them vulnerable to the MOV attack
    reduces the elliptic curve discrete logarithm to the finite field
    discrete logarithm over GF(P^2).  Miller suggested in 1985 to use
    isomorphic equations, y^2=x^3-ax, without knowing about the 1992
    MOV attack.  These special curves would then be vulnerable with
    ~50% chance of being, depending on the prime P.  This curve was
    chosen in full knowledge of the MOV attack.

    Note: The near-miss or close-call intuition is misleading, because
    many cryptographic algorithms become insecure based on the
    slightest adjustment to the algorithm.

    Note: The non-supersingularity means that the endomorphism ring is
    commutative.  For this curve the endomorphism ring is isomorphic
    to the ring Z[i] of Gaussian integers.

A.4.4.3.  Semaev--Araki--Satoh--Smart security

  The fact that the curve size 72q does not equal p, means that the
  curve resists the Semaev--Araki--Satoh--Smart attack.

A.4.4.4.  Edwards and Hessian form

  The cofactor 72 is divisible by 4, so the curve isomorphic to a
  curve with an Edwards equation, permitting implementation even more
  efficient than the Montgomery ladder.

  The Edwards form makes possible the Gallant--Lambert--Vanstone
  method that used the efficient endomorphism.

  The cofactor 72 is also divisible by 3, so the curve is isomorphic
  to a curve with a Hessian equation, which is another type of
  equation permmitting efficient implementation.

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    Note: It is probably too optimisitic and speculative to hope that
    future research will show how to take advantage by combining the
    efficiencies of Edwards and Hessian curve equations.

A.4.4.5.  Bleichenbacher security

  Bleichenbacher's attack against faulty implementations
  discrete-log-based signatures fully affects 2y^2=x^3+x/GF(8^91+5),
  because the base point order q is not particularly close to a power
  of two.  (Some other curves, such as NIST P-256 and Curve25519, have
  the base point order is close to a power of two, which provides
  built-in resistant to Bleicenbacher's faulty signature attack.)

    Note: Bleichenbacher's attack exploits the signature implmentation
    fault of naively reducing uniformly random bit strings modulo q,
    the order of the base point, which results in a number biased
    towards the lower end of the interval [0,q-1].

  So, q-uniformization of the pre-message secret numbers is critical
  for signature applications of 2y^2=x^3+x/GF(8^91+5).  Various
  uniformization methods are known, such as reducing extra large
  numbers, repeated sampling, and so on.

A.4.4.6.  Bernstein's "twist" security

  Unlike Curve25519, curve 2y^2=x^3+x/GF(8^91+5) is not
  "twist-secure", so a Montgomery ladder implementation for static
  private keys often requires public-key validation, which is
  achievable by comptuation of a Legendre symbol related to the
  received public key.

  In particular, a Montgomery ladder x-only implementation that does
  not implement public-key validation will process a value x for which
  no y satsifying the equation exists in GF(p).  More precsiely, a y
  does exist, but it belongs to the extension field GF(p^2).  In this
  case, the Montgomery ladder treats x as though it were (x,y) where x
  is GF(p) but y is not.  Such points belong to a "twist" group, and
  this group has order:

    2^2 * 5 * 1526119141 * 788069478421 * 182758084524062861993 *

  An adversary can exploit this, by finding such invalid x that
  correspond to a lower order group element, and thereby try to learn
  partial information about a static private key used by a
  non-validating Montgomery ladder implementation.

A.4.4.7.  Cheon security

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  Niche applications in ECC involve revealing points [d^e]G for one
  secret number d, and many different integer e, or at least one large
  e.  One way such points could be reveal is in protocols that employ
  a static Diffie--Hellman oracle, a function to compute [d]P from any
  point P, which might be applied e times, if e is reasonably small.

  Typical ECDH, to be clear, would never reveal such points, for at
  least two reasons:

    - ECDH is ephemeral, so that the same d is never re-used across
      ECDH sessions (because d is used to compute [d]G and [d]Q, and
      then discarded),

    - ECDH is hashed, so though P=[d]G is sent, the point [d]Q is
      hashed to get k = H([d]Q), and then [d]Q is discarded, so the
      fact that hash is one-way means that k should not reveal [d]Q,
      if k is ever somehow revealed.

  The Brown--Gallant--Cheon q-1 algorithm finds d, given [d^e]G, if
  e|(q-1).  It uses approximately sqrt(q/e) elliptic curve operations.
  The Cheon q+1 algorithm finds d, given all the points [d]G, [d^2]G,
  ..., [d^e]G, if e|(q+1), and takes a similar amount of computation.
  These two algorithms rely on factors e of q-1 or q+1, so the
  factorization of these numbers affects the security against the

  Cheon security refers to the ability to resist these algorithms.

  It is possible seek out special curves with relatively high Cheon
  security, becasue q-1 and q+1 have no suitable factors e.

  The curve 2y^2=x^3+x/GF(8^91+5) has typical Cheon security in terms
  of the factorization of q-1 and q+1.  Therefore, in the niche
  applications that reveal the requisite points, mitigations ought to
  be applied, such as limiting the rate of revealing points, or using
  different value d as much as possible (one d per recipient).

  For 2y^2=x^3+x/GF(8^91+5) the factorization of q-1 and q+1 are:

    q-1 = 2^3 * 101203 * 23810182454264420359 *

    q+1 = 2 * 3 * 11 * 21577 * 54829 * 392473 * 854041 *

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  The q-1 and q+1 algorithms convert an oracle for function P -> [d]P
  into a way to find d.  This may be viewed as a reduction of the
  discrete logarithm problem to the problem of computing the function
  P -> [d]P for the target d.  In other words, computing P -> [d]P is
  almost as difficulty as solving the discrete logartithm problem.  In
  many systems with a static Diffie--Hellman secret d, computing the
  function P -> [d]P needs to be difficult, or the security will be
  defeated.  In these case, an efficient q-1 or q+1 algorithm provides
  a security assurance, that the computing P -> [d]P without knowing d
  is about as hard as solving the discrete logarithm problem.

  To be completed.

A.4.4.8  Reductionist security assurance for Diffie--Hellman

  A series of  research work, from den Boer, from Maurer and Wolf, and
  from Boneh and Lipton, shows that Diffie--Hellman oracle can be used
  to solve a discrete logarithm, under certain conditions.  In other
  words, the discrete logarithm problem can sometimes be reduced to
  the Diffie--Hellman problem.

  This can be interpreted as a security assurance that Diffie--Hellman
  problem is at least as hard the discrete logarithm problem, albeit
  perhaps with some gap in the difficulty.  This formalized security
  assurance supplements the standard conjecture that the
  Diffie--Hellman problem is at least as hard as the discrete
  logarithm.  (A contrarian view is that special conditions under
  which such a reduction algorithm is possible might coincide with
  special conditions under which the discrete logarithm problem is

  The general idea is to consider a Diffie--Hellman oracle in a group
  of order q to provide multiplication in a special representation
  field of order q.  Recovering the ordinary field representation from
  the special field representation amounts to solving the discrete
  logarithm problem.

  To receover the ordinary representation, the idea is to construct an
  auxiliary group of smooth order, where the group is an algebraic
  groups over the field of size q.  Solving a discrete logarithm in
  the auxiliary group is possible using the Pohlig--Hellman problem,
  and solving the discrete logarithm in the auxiliary reveals the
  ordinary representation of the field, which, as already noted
  reveals the discrete logarithm in the original group.

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  The most obvious auxiliary groups have orders q-1 and q+1, but these
  are not smooth numbers.  The next most obvious auxiliary are
  elliptic curve groups with complex multiplication by i, but none of
  these four group have smooth orders either.

  A peculiar strategy to show the existence of an auxiliary group of
  smooth order without having any effective means of constructing the
  group.  This can be done by finding a smooth number in the Hasse
  interval of q.

  To be completed.

Appendix B.  Test vectors

  The following are some test vectors.


  The test vectors are explained as follows.  (Pseudocode generating
  them is supplied in Appendix C.2.)

  Each line is 34 bytes, representing a non-negative 272-bit integer.
  The integer encoding is hexadecimal, with most significant hex
  digits on the left, which is to say, big-endian.

    Note: Public keys are encoded as 34-byte strings are
    little-endian.  Encoded public keys reverse the order of the bytes
    found in the test vectors.  The pseudocode in Appendix C.2 should
    make this clear: since bytes are printed in reverse order.

  Each integer is either a scalar (a multiplier of curve points), or
  the byte representation of a point P through its x-coordinate or the
  x-coordinate of iP (which is the the mod 8^91+5 negation of the
  x-coordinate of P).

  The first line is a scalar integer x.  Its nonzero bytes are the
  ASCII representation of the string "TEST 2y^2=x^3+x/GF(8^91+5)",
  with the byte order reversed.  As a private key, this value of x
  would be totally insecure, because it is too small, and like any
  test vector, it is public.

  The second line is a representation of G, a base point on the curve.

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  The third line is the representation of z = xG.

  The fourth and fifth lines represent updated values of x and z,
  obtained after application of the following 100000 scalar

  A loop of 50000 iterations is performed.  Each iteration consists of
  two re-assignments: z = xz and x = zG via scalar multiplications.
  In the second assignment, the byte representation of the input point
  z is used as the byte representation of an scalar.  Similarly, the
  output x is the byte representation of the point, which is will used
  as as the byte representation of the scalar.

  The purpose of the large number of iterations is to catch a bug that
  has probability larger than 1/100000 of arising on pseudorandom
  inputs.  The iterations do nothing to find rarer bugs (such as those
  that an adversary can invoke), or silent bugs (side channel leaks).

  The sixth and seventh lines are equal to each other.  As explained
  below, the equality of these lines represents the fact the Alice and
  Bob can compute the same shared DH secret.  The purpose of these
  lines is not to catch any more bugs, but rather a sanity check that
  Diffie--Hellman is likely to work.

  Alice initializes her DH private key to x, as already computed on
  the fourth line of the test vectors (which was the result of 100000
  iterations).  She then replaces this x by x^900 mod q (where q is
  the prime which is the order of the order of the base point G).

  Bob sets his private key y as follows.  He begins with y being the
  34-byte ASCII string whose initial characters are "yet another test"
  (not including the quotes, of course).  He then reverses the order
  of bytes, considers this to be a scalar, and reassigns y to yG.
  (So, the y on the left is new, the y on the right is old, they are
  not the samem, after the assignment.)  Another reassignment is done,
  as y -> yy, where the on the right side of the equation one y is
  treated as a scalar, the other as a point.  Finally, Bob's replaces
  y by y^900 mod order(G), similarly to Alice's transformation.

  The test code in C.2 does not compute x^900 directly.  Instead it
  uses 900 scalar multiplication by x, to achieve multiplication by
  x^900.  The same is done for y^900.

  Both lines are xyG.  The first can be computed as y(xG), and the
  second as x(yG).  The equality of the two lines can be used to
  self-test an implementation, even if the implementation being tested
  disagrees with the test vectors above.

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Appendix C.  Sample code (pseudocode)

  This section has sample C code that illustrates well-known elliptic
  algorithms, adaptations specific to 2y^2=x^3+x/GF(8^91+5).

  As a warning: the sample code has not been fully hardened against
  side channels or any other implementation attacks; also, no
  independent party has reivewed the sample code.

    Note: The quality of the sample code is similar to pseudocode, not
    reference code, or software.  It compiles and runs on my personal
    devices, but has not otherwise been tested for quality.

    Note: Non-standard C language extensions are used the sample code:
    the type __int128, available as an C language extension in the GNU
    C compiler (gcc).

    Note: Non-portable C is used (beyond the non-standard C), for
    convenience.  Two's complement integer representation of integers
    is assumed.  Bit-shifts negative integers are used, in a way that
    considered non-portable under strict C, even though commonly used

    Note: Manually minified C is used: to reduce line and character
    counts, and also to (arguably) aid objective code inspection by
    cramming as much code into a single screen and by not misleading
    reviewers with long comments or variable names.

    Note: Automated tools, such as indent (used as in "gcc -E pseudo.c
    | indent"), can partially revert the C sample code spacing to a
    more conventional style, though other aspects of minification are
    not so easy to remove.

    Note: The minification is not total.  It tries to organize the
    code into meaningful units, such as placing single short functions
    on one line or placing all variable declarations on the same line
    with the function parameters.  Python-like indentation is kept.
    (Per Lisp styling, the code clumps closing delimiters (that mainly
    serve the compilers.))

    Note: Long sequence expressions, using the C comma operator, in
    place of multiple expression statements, which would be more
    conventional and terminated by semicolons, save some braces in
    control statements, such as "for" loops and "if" conditionals, and
    enable extra intializations in declarations.

C.1.  Scalar multiplication of 34-byte strings

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  The sample code for scalar multiplication provides an interface for
  scalar multiplication.  A function "mulch" takes as input 3 pointer
  to unsigned character strings.  The first input is the location
  of the result, the second is the muliplier, and the third is the
  base point.

   Note: The input ordering follows the convention of C assignment
   expressions z=x*y.

   Note: The function name "mulch" is short for multiply charcater

  Mulch returns a Boolean value, indicating success or failure.
  Failure is returned only if validation is requested, and the base
  point is invalid.

  Requesting validation is done implicitly, by comparison of pointers.
  Validation is requested unless the base point is the known valid
  base point G, or if the scalar multiple (2nd input) and the output
  (1st input) pointers are equal, meaning that the scalar multiple
  will be overwritten.

    Note: The motivation here for implicitly requesting validation is
    that if the scalar multiple is really ephemeral, the caller should
    be willing, and eager, to overwrite it as soon as possible, in
    order to achieve forward secrecy.  In this case, the need for
    input validation is usually negligible.

  The sample code is to be considered as a single file, pseudo.c.

  The file pseudo.c has two sections.  The first section implements
  arithmetic for the field GF(8^91+5).  The second section implemetns
  Montgomery's ladder for curve 2y^2=x^3+x.  The two sections are not
  entirely independent.  In particular, the field arithmetic section
  is not general-purpose, and could produce errors if used for
  different elliptic curve algorithms, such as Edwards coordinates.

    Note: The scalar muliplication sample code pseudo.c file is
    included into 3 other sample (using a the C preprocessor directive
    #include "pseudo.c").

    Note: Compiler optimizations make a large difference when used on
    the field arithmetic (for versions of the sample code where the
    field and curve arithmetic are in separate source files).  This
    suggests that field arithmetic efficiency has room for further
    improvement by hand assembly.  (The curve arithmetic might be
    improved by re-writing the source code.)  In case, the sample code
    should not be considered to fully optimized.

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    Note: Montgomery's ladder might not be the fastest scalar
    multiplication algorithm for 2y^2=x^3+x/GF(8^91+5).  Experimental
    C implementations using Bernstein's 2-D ladder algorithm seem
    about ~10% faster.  The experimental code somewhat more
    complicated, and thus more likely to vulnerable to side channels
    or overflows.  Even more aggressive C code seems about ~20%
    faster, using Edwards coordinates, Hisil--Carter--Dawson--Wong,
    and Gallant--Lambert--Vanstone, and pre-computed windows.  Again,
    these faster methods are more complicated, and may be more
    vulnerable implementation attacks.  The 10% and 20% gains may be
    lost upon more thorough hardening against implemenatioon attacks,
    or upon more thorough hand-assembly optimizations.

  To be completed.

C.1.1.  Field arithmetic for GF(8^91+5)

  The field arithmetic sample code, is the first part of the file
  pseudo.c.  It implements the field operations used in the Montgomery
  ladder algorithm for elliptic curve 2y^2=x^3+x.  For example, point
  decompression is not used in Montgomery ladders, so the square root
  operation is not included the sample code.  (The Legendre symbol
  computation is included for validation, and is quite similar to the
  square root operation.)

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  #define RZ return z
  #define F4j i j=5;for(;j--;)
  #define FIX(j,r,k) q=z[j]>>r, z[j]-=q<<r, z[(j+1)%5]+=q*k
  #define CMP(a,b) ((a>b)-(a<b))
  #define XY(j,k) x[j]*(ii)y[k]
  #define R(j,k) (zz[j]>>55*k&((k<2)*M-1))
  #define MUL(m,E)\
    zz[0]= m(0,0)E(1,4)E(2,3)E(3,2)E(4,1),\
    zz[1]= m(0,1)m(1,0)E(2,4)E(3,3)E(4,2),\
    zz[2]= m(0,2)m(1,1)m(2,0)E(3,4)E(4,3),\
    zz[3]= m(0,3)m(1,2)m(2,1)m(3,0)E(4,4),\
    zz[4]= m(0,4)m(1,3)m(2,2)m(3,1)m(4,0);\
    z[0]=R(0,0)-R(4,1)*20-R(3,2)*20, z[1]=R(1,0)+R(0,1)-R(4,2)*20,\
    z[2]=R(2,0)+R(1,1)+R(0,2),       z[3]=R(3,0)+R(2,1)+R(1,2),\
    z[4]=R(4,0)+R(3,1)+R(2,2);       z[1]+=z[0]>>55; z[0]&=M-1;
  typedef long long i;typedef i*f,F[5];typedef __int128 ii,FF[5];
  i M=((i)1)<<55;F O={0},I={1};
  f fix(f z){i j=0,q;
    for(;j<5*2;j++) FIX(j%5,(j%5<4?55:53),(j%5<4?1:-5));
    z[0]+=(q=z[0]<0)*5; z[4]+=q<<53; RZ;}
  i cmp(f x,f y){i z=(fix(x),fix(y),0); F4j z+=!z*CMP(x[j],y[j]); RZ;}
  f add(f z,f x,f y){F4j z[j]=x[j]+y[j]; RZ;}
  f sub(f z,f x,f y){F4j z[j]=x[j]-y[j]; RZ;}
  f mal(f z,i s,f y){F4j z[j]=y[j]*s; RZ;}
  f mul(f z,f x,f y){FF zz; MUL(+XY,-20*XY); {F4j zz[j]=0;} RZ;}
  f squ(f z,f x){mul(z,x,x); RZ;}
  i inv(f z){F t;i j=272; for(mul(z,z,squ(t,z));j--;) squ(t,t);
    return mul(z,t,z), (sub(t,t,t)), cmp(O,z);}
  i leg(f y){F t;i j=270; for(squ(t,squ(y,y));j--;) squ(t,t);
    return j=cmp(I,mul(y,y,t)), (sub(y,y,y),sub(t,t,t)), (2-j)%3-1;}

  Field elements are stored as five-element of arrays of limbs.  Each
  limb is an integer, possibly negative, with array z representing

    z[0] + z[1]*2^55 + z[2]*2^110 + z[3]*2^165 + z[4]*2^220

  In other words, the radix (base) is 2^55.  Say that z has m-bit
  limbs if each |z[i]| < 2^m.

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  The field arithmetic function input order follows the C assignment
  order, as input z=x*y, so usually the first input is the location
  for the result of the operation.  The return value is usually just a
  pointer to the result's location, the first input, indicated by the
  preprocessor macro RZ.   The functions, inv, cmp, and leg, also
  return an integer, which is not a field element, but usually a
  Boolean (or for function leg, a value in {-1,0,1}.)

  The utility functions are fix and cmp.  They are meant to take
  inputs with 58-bit limbs, and produce an output with 55-bit
  non-negative limbs, with the highest limb, a 53-bit value.  The
  purpose of fix is to provide a single array representation of each
  field element.  The function cmp fixes both its inputs, and then
  returns a sigend comparison indicator (in {-1,0,1}).

  The multiplicative functions are mul, squ, inv and leg.  They are
  meant to take inputs with 58-bit limbs, and produce either an output
  with 57-bit limbs, or a small integer output.  They try to do this
  as follows:

    1. Some of the input limbs are multiplied by 20, then multiplied
       in pairs to 128-bit limbs, and then summed in groups of five
       (with at least one of the pairs having both elements not
       multiplied by 20).  The multiplications by 20 should not cause
       64-bit overflow 20*2^58 < 32*2^58=2^63, while the sums of
       128-bit numbers should not cause overflow, because
       (1+4*20)*2^58*2^58 = 81*2^116 < 2^7*2^116 = 2^123.

    2. The five 128-bit limbs are partially reduced to five 57-bit
       limbs.   Each the five smaller limbs is obtained by summing two
       55-bit limbs, extracted from sections of the 128-bit limbs, and
       then summing one or two much smaller values summing to less
       than a 55-bit limb.  So, the final limbs in the multiplication
       are a sum of at most three 55-bit sub-limbs, making each final
       limb at most a 57-bit limb.

  The additive functions are add, sub and mal.  They are meant to take
  inputs with 57-bit limbs, and product an output with 58-bit limbs.

  The utility and multiplicative function can be used repeatedly,
  because they do not lengthen the limbs.

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  The additive functions potentially increase the limb length, because
  they do not perform any reduction on the output.    The additive
  functions should not be applied repeatedly.  For example, if the
  output of addtive additive function is fed directly as the input to
  an additive function, then the final output might have 59-bit
  limbs.  In this case, if 2nd output might not be evaluated corrected
  if given as input to one of the multipilcative functions, an error
  due to overflow of 64-bit arithmetic might occur.

  The lack of reduction in the additive functions trades generality
  for efficiency.  The elliptic curve arithmetic code aims to never
  send the output of an additive function directly into the input of
  another additive function.

    Note: Zeroizing temporary field values is attempted by subtracting
    them from themselves.  Some compilers might remove these
    zeroization steps.

    Note: The defined types f and F are essentially the equivalent.
    The main difference is that type F is an array, so it can be used
    to allocate new memory (on the stack) for a field value.

C.1.2.  Montgomery ladder scalar multiplication

  The second part of the file "pseudo.c" implements Montgomery's
  well-known ladder algorithm for elliptic curve scalar point
  multiplication, as it applies to the curve 2y^2=x^3+x.

  The sample code, as part of the same file, is a continuation of the
  sample code for field arithmetic.  All previous definitions are

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  #define X z[0]
  #define Z z[1]
  enum {B=34}; typedef void _;typedef volatile unsigned char *c,C[B];
  typedef F*e,E[2];typedef E*v,V[2];
  f feed(f x,c z){i j=((mal(x,0,x)),B);
    for(;j--;) x[j/7]+=((i)z[j])<<((8*j)%55); return fix(x);}
  c bite(c z,f x){F t;i j=((fix(mal(x,cmp(mal(t,-1,x),x),x))), B),k=5;
    for(;j--;) z[j]=x[j/7]>>((8*j)%55); {(sub(t,t,t));}
    for(;--k;) z[7*k-1]+=x[k]<<(8-k); {(sub(x,x,x));} RZ;}
  i lift(e z,f x,i t){F y;return mal(X,1,x),mal(Z,1,I),t||
  i drop(f x,e z){return inv(Z)&&mul(x,X,Z)&&(sub(X,X,X)&&sub(Z,Z,Z));}
  _ let(e z,e y){i j=2;for(;j--;)mal(z[j],1,y[j]);}
  _ smv(v z,v y){i j=4;for(;j--;)add(((e)z)[j],((e)z)[j],((e)y)[j]);}
  v mav(v z,i a){i j=4;for(;j--;)mal(((e)z)[j],a,((e)z)[j]);RZ;}
  _ due(e z){F a,b,c,d;
  _ ade(e z,e u,f w){F a,b,c,d;f ad=a,bc=b;
  _ duv(v a,e z){ade(a[1],a[0],z[0]);due(a[0]);}
  v adv(v z,i b){V t;
   e mule(e z,c d){V a;E o={{1}};i
    for(;n--;) c=1&d[n/8]>>n%8,duv(adv(a,c!=b),z),b=c;
    let(z,*adv(a,b)); (due(*mav(a,0))); RZ;}
  C G={23,1};
  i mulch(c db,c d,c b){F x;E p; return

  This part of the sample code represents points and scalar
  multipliers as character strings of 34 bytes.

    Note: Types c and C are used for these 34-byte encodings.
    Following the previous pattern for f and F, type C is an array,
    used for allocating new memory (on the stack) for these arrays.

  The conversion functions feed and bite convert
  between a 34-byte string and a field value (recall, stored as five
  element array, base 2^55).

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  The conversion functions lift and drop convert between field
  elements and the projective line point, so that x <-> (X:1).   The
  function lift can also test if x is the x-coordinate of the a point
  (x,y) on the curve 2y^2=x^3+x.

    Note: Projective line points are stored in defined types e and E
    (for extended field element).

    Note: The Montgomery ladder can implemented by working with a
    pair of extended field elements.

  The raw scalar multiplication function "mule" takes a projective
  point (with defined type e), multiplies it by a scalar (encoded as
  byte string with defined type c), and then replaces the projective
  point by the multiple.

  The main loop of mule is written a double-and-always-add, acting on
  pair projective line points.  Basically it acts on the x-coordinates
  of the points nB and (n+1)B, for n changing.

  Because the Montogomery ladder algorithm is being used, the "adv"
  called by mule function does nothing but swap the two values.  With
  an appropriate isogeny, this can be viewed as addition operation.

  The function "duv" called by mule, does the hard work of finding
  (2n)B and (2n+1)B from nB and (n+1)B.  It does so, using doubling in
  the function "due" and differntial addition, in the function "ade".

  The functions "due" and "ade" are non-trivial, and use field
  arithmetic.  They are fairly specific to 2y^2=x^3+x.  They try to
  avoid repeated application of additive field operations.

  The function smv, mav and let are more utilitarian.  They are used
  for initialization, swapping, and zeroization.

C.1.3.  Bernstein's 2-dimensional Montgomery ladder

  Bernstein's 2-dimensional ladder is a variant of Montgomery's ladder
  that computes aP+bQ, for any two points P and Q, more quickly than
  computing aP and bQ separately.

  Curve 2y^2=x^3+x has an efficient endomorphism, which allows a point
  Q = [i+1]P to compute efficiently.  Gallant, Lambert and Vanstone
  introduced a method (now called the GLV method), to compute dP more
  efficiently, given such an efficient endomorphism.  They write d = a
  + eb where e is the integer multiplier corresponding to the
  efficient endomorphism, and a and b are integers smaller than d.
  (For example, 17 bytes each instead of 34 bytes.)

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  The GLV method can be combined with Bernstein's 2D ladder algorithm
  to be applied to compute dP = (a+be)P = aP + beP = aP + bQ, where

  This algorithm is not implemented by any pseudocode in the version
  the draft.  (Previous versions had it.)

  See [B1] for further explanation and example pseudocode.

  I have estimate a ~10% speedup of this method compared to the plain
  Montgomery ladder.  However, the code is more complicated, and
  potentially more vulnerable to implementation-based attacks.

C.1.4.  GLV in Edwards coordinates (Hisil--Carter--Dawson--Wong)

  To be completed.

  It is also possible to convert to Edwards coordinates, and then use
  the Hisil--Carter--Dawson--Wong (HCDW) elliptic curve arithmetic.

  The HCDW arithmetic can be combined with the GLV techniques to
  obtain a scalar multiplication potentially more efficient than
  Bernstein's 2-dimensional Montgomery.  The downside is that it may
  require key-dependent array look-ups, which can be a security risk.

  I have implemented this, finding ~20% speed-up over my
  implementation of the Montgomery ladder.  However, this speed-up may
  disappear upon further optimization (e.g. assembly), or further
  security hardening (safe table lookup code).

C.2.  Sample code for test vectors

  The following sample code describes the contents of a file "tv.c",
  with the purpose of generating the test vectors in Appendix B.

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  //gcc tv.c -o tv -O3 -flto -finline-limit=200;strip tv;time ./tv
  #include <stdio.h>
  #include "pseudo.c"
  #define M mulch
  void hx(c x){i j=B;for(;j--;)printf("%02x",x[j]);printf("\n");}
  int main (void){i n=1e5,j=n/2,wait=/*your mileage may vary*/7000;
    C x="TEST 2y^2=x^3+x/GF(8^91+5)",y="yet another test",z;
    M(z,x,G); hx(x),hx(G),hx(z);
    fprintf(stderr,"%30s(wait=~%ds, ymmv)","",j/wait);
     j=0*printf("Mulch fail rate ~%f :(\n",(2*j)/n);//else//debug
    fprintf(stderr,"\r%30s                  \r",""),hx(x),hx(z);

  It includes the previously defined file pseudo.c, and the standard
  header file stdio.h.

  The first for-loop in main aims to terminate in the event of the bug
  such that the output of mulch is an invalid value, not on the curve

  Of the 100,000 scalar multiplication in this for-loop, the aim is
  that 50,000 include public-key validation.  All 100,000 include a
  field-inversion, to encode points uniquely as 34-byte strings.

  The second and three for-loops aims to test the compatibilty with
  Diffie--Hellman, by showing the 900 applications of scalar
  multipliers x and y are the same, whether x or y is applied first.

  The 1st line comment suggest possible compilation commands, with
  some optimization options.  The run-time depends on the system, and
  should be slower on older and weaker systems.

  Anecdotally, on a ~3 year-old personal computer, it runs in time as
  low as 5.7 seconds, but these were under totally uncontrolled
  conditions (with no objective benchmarking).  (Experience has shown
  that on a ~10 year-old personal computer, it could be ~5 times

C.3.  Sample code for a command-line demo of Diffie--Hellman

  The next sample code is intended to demonstrate ephemeral (elliptic
  curve) Diffie--Hellman: (EC)DHE in TLS terminology.

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  The code can be considered as a file "dhe.c".  It has both C and bash
  code, intermixed within comments and strings. It is bilingual: a
  valid bash script and valid C source code.  The file "dhe.c" can be
  made executable (using chmod, for example), so it can be run as a
  bash script.

  #include "pseudo.c" /* dhe.c (also a bash script)
  : demos ephemeral DH, also creates, clobbers files dhba dha dhb
  : -- Dan Brown, BlackBerry, '20 */
  #include <stdio.h>
  _ get(c p,_*f){f&&fread ((_*)p,B,1,f)||mulch(p,p,G);}
  _ put(c p,_*f){f&&fwrite((_*)p,B,1,f)&&fflush(f); bite(p,O);}
  int main (_){C p="not validated",s="/dev/urandom" "\0"__TIME__;
    get(s,fopen((_*)s,"r")), mulch(p,s,G), put(p,stdout);
    get(p,stdin),            mulch(s,s,p), put(s,stderr);} /*'
  [ dhe.c -nt dhe ]&&gcc -O2 dhe.c -o dhe&&strip dhe&&echo "$(<dhe.c)"
  mkfifo dh{a,b,ba} 2>/dev/null || ([ ! -p dhba ] && :> dhba)
  ./dhe <dhba 2>dha | ./dhe >dhba 2>dhb &
  sha256sum dha & sha256sum dhb  # these should be equal
  (for f in dh{a,b,ba} ; do [ -f $f ] && \rm -f $f; done)# '*/

  Run as a bash script, file "dhe.c" will check if it needs compile
  its own C code, into an executable named "dhe".  Then the bash
  script file "dhe.c" runs the compiled executable "dhe" twice.  One
  run is Alice's, and the other Bob's.

  Each run of "dhe" generates an ephemeral secret key, by reading the
  file "/dev/urandom".  Each run then writes to "stdout", the
  ephemeral public key.  Each run then reads the peer's ephemeral
  public key from "stdin".  Each run then writes to "stderr" the
  shared Diffie--Hellman secret.  (Public-key validation is mostly
  unnecessary, because the ephemeral is only used once, so it is
  skipped by using the same pointer location for the ephemeral secret
  and final shared secret.)

  The script "dhe.c" connects the input and output of these two using
  pipes.  One pipe is generated by the shell command line using the
  shell operator "|".  The other pipe is a pipe name "dhab", created
  with "mkfifo".  The script captures the shared secrets from each run
  by redirecting "stderr" (as file descriptor 2), to files "dha" and
  "dhb", which will be made named pipes if possible.

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  The scripts fees each shared secret keys into SHA-256.  This
  demonstrates their equality.  It also illusrates a typical way to
  use Diffie--Hellman, by deriving symmetric keys using a hash
  function.  In multi-curve ECC, hashing a concatenation of such
  shared secrets (one for each curve used), could be done instead.

C.4.  Sample code for public-key validation and curve basics

  The next sample code demonstrates the public-key validation issues
  specific to 2y^2=x^3+x/GF(8^91+5).  It also demonstrates the order
  of the curve.  It also demonstrates complex multiplication by i, and
  the fact the 34-byte representation of points is unaffected by
  multiplication by i.

  The code can be considered to describe a file "pkv.c".  It uses the
  "mulch" function by including "pseudo.c".

  #include <stdio.h>
  #include "pseudo.c"
  #define M mulch // works with +/- x, so P ~ -P ~ iP ~ -iP
  void hx(c x){i j=B;for(;j--;)printf("%02x",x[j]);printf("\n");}
  int main (void){i j;// sanity check, PKV, twist insecurity demo
    C y="TEST 2y^2=x^3+x/GF(8^91+5)",z="zzzzzzzzzzzzzzzzzzzz",
    q = "\xa9\x38\x04\xb8\xa7\xb8\x32\xb9\x69\x85\x41\xe9\x2a"
    "\xc7\x71\x1c\xc7\x71\x1c\x07", // q=order(G)
    i = "\x36\x5a\xa5\x56\xd6\x4f\xb9\xc4\xd7\x48\x74\x76\xa0"
    "\x63\xfc\x0a\x26\x0c\x1b\x04", // i^2=-1 mod q
    w5= "\xb4\x69\xf6\x72\x2a\xd0\x58\xc8\x40\xe5\xb6\x7a\xfc"
    "\x66\x66\x66\x66\x66\x66\x66"; // w5=(2p+2-72q)/5
    for(j=0;j<=3;j++)M(z,(C){j},G),hx(z); // {0,1,2,3}G, but reject 0G
    M(z,q,G),hx(z); // reject qG; but qG=O, under hood:
    {F x;E p;lift(p,feed(x,G),1);mule(p,q);hx(bite(z,p[1]));}
    for(j=0;j<0*25;j++){F x;E p;lift(p,feed(x,(C){j,1}),1);mule(p,q);
    printf("%3d ",j),hx(bite(z,p[1]));}// see j=23 for choice of G
    for(j=3;j--;)q[0]-=1,M(z,q,G),hx(z);// (q-{1,2,3})G ~ {1,2,3}G
    M(z,i,G),hx(z); i[0]+=1,M(z,i,G),M(z,i,z),hx(z);// iG~G,(i+1)^2G~2G
    M(w5,w5,(C){5}),hx(w5);// twist, ord(w5)=5, M(z,z,p) skipped PKV(p)
    M(G,(C){1},w5),hx(G);// reject w5 (G unch.); but w5 leaks z mod 5:

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  The sample code demonstrates the need for public-key validation
  even when using the Montgomery ladder for scalar multiplication.  It
  does this by finding points of low order on the twist of the curve.
  This invavlid points can leak bits of the secret multiplier.  This
  is because the curve 2y^2=x^3+x/GF(8^91+5) is not fully "twist
  secure".  (Its twist security is typical of that of a random curve.)

C.5.  Elligator i

  To be deleted (or completed).

  This pseudocode would show how to implement to the Elligator i map
  from byte strings to points.

  This is INCOMPATIBLE with previous samples of code above, and is
  taken from an earlier version of experimental code.

  Pseudocode (to be verified):

  typedef f xy[2] ;
  #define X p[0]
  #define Y p[1]
  lift(xy p, f r) {
    f t ; i b ;
    squ(t,r);        // r^2
    mul(t,I,t);      // ir^2
    sub(t,(f){1},t); // 1-ir^2
    inv(t,t);        // 1/(1-ir^2)
    mal(t,3,t);      // 3/(1-ir^2)
    mul(t,I,t);      // 3i/(1-ir^2)
    sub(X,I,t);      // i-3i/(1-ir^2)
    b = get_y(t,X);
    mal(t,1-b,I);    // (1-b)i
    add(X,X,t);      // EITHER  x  OR  x + i
    mal(Y,2*b-1,Y);  // (-1)^(1-b)""
    fix(X);  fix(Y);

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  drop(f r, xy p)
    f t ; i b,h ;
    fix(X); fix(Y);
    sub(t,X,t);   // EITHER x or x-i
    sub(t,I,t);   // i-x
    inv(t,t);     // 1/(i-x)
    mal(t,3,t);   // 3/(i-x)
    add(t,I,t);   // i+ 3/(i-x)
    mal(t,-1,t);  // -i-3/(i-x)) = (1-3i/(i-x))/i
    b = root(r,t) ;
    h = (r[4]<(1LL<<52)) ;

  elligator(xy p,c b) {f r; feed(r,b); lift(p,r);}

  crocodile(c b,xy p) {f r; drop(r,p); bite(b,r);}

Appendix D.  Minimizing trapdoors and backdoors

  The main advantage of curve 2y^2=x^3+x/GF(8^91+5) over almost all
  other elliptic curves is its Kolmogorov complexity is almost minimal
  among curves of sufficient resistance to the Pollard rho attack on
  the discrete logarithm problem.

  See [AB] and [B1] for some details.

D.1.  Decimal exponential complexity

  The curve can be described with 21 characters:

     2  y  ^  2  =  x  ^  3  +  x  /  G  F  (  8  ^  9  1  +  5  )
     1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21

  Those familiar with ECC will recognize that these 21 characters
  suffice to specify the curve up to the level of detail needed to
  describe the cost of the Pollard rho algorithm, as well as many
  other security properties (especially resistance to other known
  attacks on the discrete logarithm problem, such as Pohlig--Hellman
  and Menezes--Okamoto--Vanstone).

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    Note: The letters GF mean Galois Field, and are quite traditional
    mathematics, and every elliptic curve in cryptographic needs to
    use some notation for the finite field.

  We may therefore describe the curve's Kolmogorov complexity as 21

     Note: The idea of low Kolmogorov complexity is hard to specify
     exactly.  Nonetheless, a claim of nearly minimal Kolmogorov
     complexity is quite falsifiable.  The falsifier need merely
     specify several other (secure) elliptic curves using 21 or fewer
     characters.  (But if the other curves use a different
     specificaion language, then a fair comparison should re-specify
     2y^2=x^3+x/GF(8^91+5) in this specification language.)

D.1.1.  A shorter isomorophic curve

  The curve is isomorphic to a curve specifiable in 20 characters:


  Generally, isomorphic curves have essentially equivalently hard
  discrete logarithm problems, so one could argue that curve
  2y^2=x^3+x/GF(8^91+5) could be rated as having Kolmogorov complexity
  at most 20 characters.

  Isomorphic curves, however, may differ slightly in security, due to
  issues of efficiency, and implementability.  The 21-character
  specification uses an equation in Montgomery form, which creates an
  incentive to use the Montgomery ladder algorithm, which is both safe
  and efficient [Bernstein?].

D.1.2.  Other short curves

  Allowing for non-prime fields, then the binary-field curve known as
  sect283k1 has a 22-character description:


  This curve was formerly one of the fifteen curves recommended by
  NIST.  Today, a binary curve is curve is considered risky, due to
  advances in elliptic curve discrete logarithm problem over extension
  fields, such as recent asymptotic advances on discrete logarithms in
  low-characteristic fields [HPST] and [Nagao].  According to [Teske],
  some characteristic-two elliptic curves could be equipped with a
  secretly embedded backdoor (but sect283k1's short description should
  help mitigate that risk).

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  This has a longer overall specification than curve
  2y^2=x^3+x/GF(8^91+5), but the field part is shorter field
  specification.  Perhaps an isomorphic curve can be found (one with
  three terms), so that total length is 21 or fewer characters.

  A non-prime field tends to be slower in software.  A non-prime field
  is therefore perhaps riskier due to some recent research on
  attacking non-prime field discrete logarithms and elliptic curves,

  To be completed.

D.1.3.  Converting DEC characters to bits

  The units of characters as measuring Kolmogorov complexity is not
  calibrated as bits of information.  Doing so formally would be very
  difficult, but the following approach might be reasonable.

  Set the criteria for the elliptic curve.  For example, e.g. prime
  field, size, resistance (of say 2^128 bit operations) to known
  attacks on the discrete logarithm problem (Pollard rho, MOV, etc.).
  Then list all the possible ECC curve specification with Kolmogorov
  complexity of 21 characters or less.  Take the base two logarithm of
  this number.  This is then an calibrated estimate of the number of
  bits needed to specify the curve.  It should be viewed as a lower
  bound, in case some curves were missed.

  To be completed.

D.1.4.  Common acceptance of decimal exponential notation

  The decimal exponentiation notation used in to measure decimal
  exponential complexity is quite commonly accepted, almost standard,
  in mathematical computer programming.

  For example, as evidence of this commmon acceptance, here is a
  slightly edited session of the program "bc" (versions of which are
  standardized in POSIX).

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   $ BC_LINE_LENGTH=71 bc
   bc 1.06.95
   Copyright ... Free Software Foundation, Inc.
     p=8^91+5 ; p; obase=16; p


     define v(b,e,m){
       auto a; for(a=1;e>0;e/=2){
       if(e%2==1) {a=(a*b)%m;}
     x = (1*256) + (23*1)
     y = (((p+1)/2)*v(2*(x^3+x),(p+3)/8,p))%p
     (2*y^2)%p == (x^3+x)%p
     (2*y^2 -(x^3+x))%(8^91+5)

     Note: Input lines have been indented at least two extra spaces,
     and can be pasted into a "bc" session.  (Pasting the output lines
     causes a few spurious results.)

  The sample code demonstrates that "bc" directly accepts the
  notations "8^91+5" and "x^3+x": parts parts of the curve
  specification "2y^2=x^3+x/GF(8^91+5)", which goes to show how much
  of the notation used in this specifcation is commonly accepted.

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     Note: Defined function "v" implements modular exponentiation,
     with returning v(b,e,m) returning (b^e mod m).  Then, "v" is used
     to show that p=8^91+5 is a Fermat pseudoprime to base 571
     (evidence that p is prime).  The value x defined is the
     x-coordinate of the recommend base point G.  Then, another
     computation with "v" shows that 2(x^3+x) has Legendre symbol 1,
     which implies (assuming p is prime) that there exists y with
     2y^2=x^3+x, namely y = (1/2)sqrt(2(x^3+x)).  The value of y is
     computed, again using "v" (but also a little luck).  The curve
     equation is then tested twice with two different expressions,
     somewhat similar to the mathematical curve specification

D.2.  General benefits of low Kolmogorov complexity to ECC

  The benefit of low Kolmogorov complexity to cryptography is well
  known, but very informal.  The general benefit is believed to a form
  of subversion-resistance, where the attacker is the designer of the

  Often, fixed numbers in cryptographic algorithms with low Kolmogorov
  complexity are called "nothing-up-my-sleeve" numbers.  (Bernstein et
  al. uses terms in "rigid", for a very similar idea, but with an
  emphasis on efficiency instead of compressibility.)

  For elliptic curves, the informal benefit may be stated as the
  following gains.

    - Low Kolmogorov complexity defends against insertion of a keyed
      trapdoor, meaning the curve can broken using a secret trapdoor,
      by an algorithm (eventually discovered by the public at large).
      For example, the Dual EC DRBG is known to capable of having such
      a trapdoor.  Such a trapdoor would information-theoretically
      imply an amount of information, comparable the size of the
      secret, to be embedded in the curve specification.  If the
      calibrated estimate for the number of bits is sufficiently
      accurate, then such a key cannot be large.

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    - Low Kolmogorov complexity defends against a secret attack
      (presumably difficult to discover), which affects a subset of
      curves such that (a) whether or not a specific curve is affected
      is a somewhat pseudorandom function of its natural
      specification, and (b) the probably of a curve being affected
      (when drawn uniformly from some sensible of curve
      specification), is low.  For an example of real-world attacks
      meeting the conditions (a) and (b) consider the MOV attack.
      Exhaustively finding curve meeting these two conditions is
      likely to prevent low Kolmogorov complexity, essentially by the
      low probability of the attack, and the independence of attack's
      success from the natural Kolmogorov complexity.

    - Even more hypothetically, there may yet exist undisclosed
      classes of weak curves, or attacks, for which
      2y^2=x^3+x/GF(8^91+5) is lucky enough to avoid.  This would be a
      fluke.  A real-world example is prime-order, or low cofactor
      curves, which are are among all curves, but which better resist
      the Pohlig--Hellman attack.

  Of course, low Kolmogorov complexity is not a panacea.  The worst
  failure would be attacks that increase in strength as Kolmogorov
  complexity gets lower.  Two examples illustrate this strongly.

D.2.1.  Precedents of low Komogorov complexity in ECC

   To be completed.

   Basically, the curves sect283k1, Curve25519, and Brainpool curves
   can be argued as mitigating the risk of manipulated designed-in
   weakness, by virtue of the low Kolmogorov complexity.

   To be completed.

D.3.  Risks of low Kolmogorov complexity

  Low Kolmogorov complexity is not a panacea for cryptography.

  Indeed, it may even add its own risks, if some weakness are
  positively correleated with low Kolmogorov complexity, making some
  attacks stronger.

  In other words, choosing low Kolmogorov complexity might just
  accidentally weaken the cryptography.  Or worse, if attackers find
  and hold secret such weaknesses, then attackers can intentionally
  include the weakness, by using low Kolmogorov serving as a cover,
  thereby subverting the algorithm.

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  Evidence of positive correlations between curve weakness and low
  Kolmogorov complexity might help assess this risk.

  In general cryptography (not ECC), the shortest cryptography
  algorithms may be the least secure, such as the identity function as
  an encryption function.

  Within ECC, however, some minimum threshold of complexity must be
  met for interoperability.  But curve size is positively correlated
  with security (via Pollard rho) and negatively correlated with
  complexity (at least for fields, larger fields needs larger
  specifications).  Therefore, there is a somewhat negative correlation
  between Pollard rho security of ECC and Kolmogorov complexity of the
  field size.

  Beyond field size in ECC, there is some negative correlations in the
  curve equation.

  Singular cubics have equations that look very simlar to those
  commonly used elliptic curves.  For smooth singular curves
  (irreducible cubics) a group can be defined, using more or less the
  same arithmetic as for a elliptic curve.  For example
  y^2=x^3/GF(8^91+5) is such a cubic.  The resulting group has an easy
  discrete logarithm problem, because it can be mapped to the field.

  Supersingular elliptic curves can also be specified with low
  Kolmogorov complexity, and these are vulnerable to MOV attack,
  another negative correlation.

  Combining the above, a low Kolmogorov complexity elliptic curve,
  y^2=x^3+1/GF(2^127-1), with 21-character decimal exponential
  complexity, suffers from three well-known attacks:

    1. The MOV (Menezes--Okamato--Vanstone) attack.

    2. The Pohlig--Hellman attack (since it has 2^127 points).

    3. The Pollard rho attack (taking 2^63 steps, instead of the 2^126
    of exhaustive).

  Had all three attacks been unknown, an implementer seeking low
  Kolmogorov complexity, might have been drawn to curve
  y^2=x^3+1/GF(2^127-1).  (This document's curve 2y^2=x^3+x/GF(8^91+5)
  uses 1 more character and is much slower since, the field size has
  twice as many bits.)

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  Had an attacker known one of three attacks, the attacker could found
  y^2=x^3+1/GF(2^127-1), proposed it, touted its low Kolmogorov
  complexity, and maybe successfully subverted the system security.

    Note: The curve y^2=x^3+1/GF(2^127-1) not only has low decimal
    exponential complexity, it also has high efficiency: fast field
    arithmetic and fairly fast curve arithmetic (for its bit lengths).
    So high efficiency can also be positively correlated with

  It can be argued, that pseudorandomized curves, such as NIST P-256
  and Brainpool curves, are an effective way mitigate such attacks
  positively correlated with low complexity.  More precisely, strong
  pseudorandomization somewhat mitigates the attacker's subversion
  ability, by reducing an easy look up of the weakest curve to an
  exhaustive search by trial and error, intuitively implying a
  probable high Kolmogorov complexity (proportional the rarity of the

  It can be further argued that all major known weak classes of curves
  in ECC are positively correlated with low complexity, in that the
  weakest curves have very low complexity.  No major known weak
  classes of curves imply an increase in Kolmogorov complexity, except
  perhaps Teske's class of curves.

  In defense of low complexity, it can be argued that the strongest
  way to resist secret attacks is to find the attacks.

  For these reasons, this specification suggests to use curve
  2y^2=x^3+x/GF(8^91+5) in  multi-curve elliptic curve cryptography,
  in combination with at least one pseudo-randomized curve.

  To be completed.

D.4.  Alternative measures of Kolmogorov complexity

  Decimal exponential complexity arguably favors decimal and the
  exponentiation operators, rather than the arbitrary notion of

  Allowing more arbitrary compression schemes introduces another
  possible level of complexity, the compression scheme itself,
  somewhat defeating the purpose of nothing-up-sleeve number.  An
  attacker might be able to choose a compression scheme among
  many that somehow favors a weak curve.

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  Despite this potential extra complexity, one can still seek a
  measure more objective than decimal complexity.  To this end, in
  [B3], I adapted the Godel's approach for general recursive
  functions, which breaks down all computation into succession,
  composition, repetition, and minimization.

  The adaption is a miniature programming language called Roll to
  describe number-related functions, including constant functions.  A
  Roll program for the constant function that always return 8^91+5 is:

    8^91+5 subs 8^91+1 in +4
    8^91+1 subs 2^273 in +1
    2^273 subs 273 in 2^
    273 subs 17 in *16+1
    17  subs  1 in *16+1
    *16+1 roll +16 up 1
    +16 subs +8 in +8
    +8  subs +4 in +4
    +4  subs +2 in +2
    2^ roll *2 up  1
    1 subs  in +2
    *2 roll +2 up  0
    +2 subs +1 in +1
    0 subs   in +1

  A Roll program has complexity measured in its length in number of
  words (space-separated substrings).  This program has 68 words.
  Constants (e.g. field sizes) can be compared using roll complexity,
  the shortest known length of their implementations in Roll.

  In [B3], several other ECC field sizes are given programs.  The only
  prime field size implemented with 68 or fewer words was 2^521-1.
  (The non-prime field size (2^127-1)^2 has 58-word "roll" program.)
  Further programming effort might produce shorter programs.

    Note: Roll programs have a syntax implying some redundancy.
    Further work may yet establish a reasonable normalization for roll
    programs, resulting in a more calibrated complexity measure in
    bits, making the units closed to a universal kind of Kolmogorov

Appendix E. Primality proofs and certificates

  Recent work of Albrecht and others [AMPS] has shown the combination
  of an adversarially chosen prime, and users using improper
  probabilistic primality tests can make user vulnerable to an attack.

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  The adversarial primes in their attack are typically the result of
  an exhaustive search.  These bad primes would therefore typically
  contain an amount of information corresponding to the length of
  their search, putting a predictable lower bound on their Kolmogorov

  The two primes involved for 2y^2=x^3+x/GF(8^91+5) should perhaps
  already resist [AMPS] because of the following compact
  representation of these primes:

    p = 8^91+5
    q = #(2y^2=x^3+x/GF(8^91+5))/72

  This attack [AMPS] can also be resisted by:

   - properly implementing probabilistic primality test, or
   - implementing provable primality tests.

  Provable primality tests can be very slow, but can be separated into
  two steps:

  -- a slow certificate generation, and

  -- a fast certificate verification.

  The certificate is a set of data, representing an intermediate step
  in the provable primality test, after which the completion of the
  test is quite efficient.

  Pratt primality certificate generation for any prime p, involves
  factorizing p-1, which can be very slow, and then recursively
  generating a Pratt primality certificate for each prime factor of
  p-1.  Essentially, each prime has a unique Pratt primality

  Pratt primality certificate verification of (p-1), involves search
  for g such that 1 = (g^(p-1) mod p) and 1 < (g^((p-1)/q) mod p) for
  each q dividing p-1, and then recursively verifying each Pratt
  primality certificate for each prime factor q of p-1.

  In this document, we specify a Pratt primality certificate as a
  sequence of (candidate) primes each being 1 plus a product of
  previous primes in the list, with certificate stating this product.

  Although Pratt primality certificate verification is quite
  efficient, an ECC implementation can opt to trust 8^91+5 by virtue
  of verifying the certificate once, perhaps before deployment or
  compile time.

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E.1.  Pratt certificate for the field size 8^91+5

  Define 52 positive integers, a,b,c,...,z,A,...,Z as follows:

   a=2 b=1+a c=1+aa d=1+ab e=1+ac f=1+aab g=1+aaaa h=1+abb i=1+ae
   j=1+aaac k=1+abd l=1+aaf m=1+abf n=1+aacc o=1+abg p=1+al q=1+aaag
   r=1+abcc s=1+abbbb t=1+aak u=1+abbbc v=1+ack w=1+aas x=1+aabbi
   y=1+aco z=1+abu A=1+at B=1+aaaadh C=1+acu D=1+aaav E=1+aeff F=1+aA
   G=1+aB H=1+aD I=1+acx J=1+aaacej K=1+abqr L=1+aabJ M=1+aaaaaabdt
   N=1+abdpw O=1+aaaabmC P=1+aabeK Q=1+abcfgE R=1+abP S=1+aaaaaaabcM
   T=1+aIO U=1+aaaaaduGS V=1+aaaabbnuHT W=1+abffLNQR X=1+afFW
   Y=1+aaaaauX Z=1+aabzUVY.

    Note: variable concatenation is used to indicate multiplication.
    For example, f = 1+aab = 1+2*2*(1+2) = 13.

    Note: One must verify that Z=8^91+5.

    Note: The Pratt primality certificate involves finding a generator
    g for each the prime (after the initial prime).  It is possible to
    list these in the certificate, which can speed up verification by
    a small factor.

     (2,b), (2,c), (3,d), (2,e), (2,f), (3,g), (2,h), (5,i), (6,j),
     (3,k), (2,l), (3,m), (2,n), (5,o), (2,p), (3,q), (6,r), (2,s),
     (2,t), (6,u), (7,v), (2,w), (2,x), (14,y),(3,z), (5,A), (3,B),
     (7,C), (3,D), (7,E), (5,F), (2,G), (2,H), (2,I), (3,J), (2,K),
     (2,L),(10,M), (5,N), (10,O),(2,P), (10,Q),(6,R), (7,S), (5,T),
     (3,U), (5,V), (2,W), (2,X), (3,Y), (7,Z).

    Note: The decimal values for a,b,c,...,Y are given by: a=2, b=3,
    c=5, d=7, e=11, f=13, g=17, h=19, i=23, j=41, k=43, l=53, m=79,
    n=101, o=103, p=107, q=137, r=151, s=163, t=173, u=271, v=431,
    w=653, x=829, y=1031, z=1627, A=2063, B=2129, C=2711, D=3449,
    E=3719, F=4127, G=4259, H=6899, I=8291, J=18041, K=124123,
    L=216493, M=232513, N=2934583, O=10280113, P=16384237, Q=24656971,
    R=98305423, S=446424961, T=170464833767, U=115417966565804897,
    V=4635260015873357770993, W=1561512307516024940642967698779,

E.2.  Pratt certificate for subgroup order

  Define 56 variables a,b,...,z,A,B,...,Z,!,@,#,$, with new

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   a=2 b=1+a c=1+a2 d=1+ab e=1+ac f=1+a2b g=1+a4 h=1+ab2 i=1+ae
   j=1+a2d k=1+a3c l=1+abd m=1+a2f n=1+acd o=1+a3b2 p=1+ak q=1+a5b
   r=1+a2c2 s=1+am t=1+ab2d u=1+abi v=1+ap w=1+a2l x=1+abce y=1+a5e
   z=1+a2t A=1+a3bc2 B=1+a7c C=1+agh D=1+a2bn E=1+a7b2 F=1+abck
   G=1+a5bf H=1+aB I=1+aceg J=1+a3bc3 K=1+abA L=1+abD M=1+abcx N=1+acG
   O=1+aqs P=1+aqy Q=1+abrv R=1+ad2eK S=1+a3bCL T=1+a2bewM U=1+aijsJ
   V=1+auEP W=1+agIR X=1+a2bV Y=1+a2cW Z=1+ab3oHOT !=1+a3SUX @=1+abNY!
   #=1+a4kzF@ $=1+a3QZ#

    Note: numeral after variable names represent powers.  For example,
    f = 1 + a2b = 1 + 2^2 * 3 = 13.

  The last variable, $, is the order of the base point, and the order
  of the curve is 72$.

    Note: Punctuation used for variable names !,@,#,$, would not scale
    for larger primes.  For larger primes, a similar format might work
    by using a prefix-free set of multi-letter variable names.
    E.g. replace, Z,!,@,#,$ by Za,Zb,Zc,Zd,Ze:


  Thanks to John Goyo and various other BlackBerry employees for past
  technical review, and to Gaelle Martin-Cocher and Takashi Suzuki for
  encouraging work on I-D.  Thanks to David Jacobson for sending Pratt
  primality certificates.

Author's Address

  Dan Brown
  4701 Tahoe Blvd., 5th Floor
  Mississauga, ON

Brown             ECC with 2y^2=x^3+x/GF(8^91+5)             [Page 64]