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

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Internet-Draft                                                D. Brown
Intended status: Experimental                               BlackBerry
Expires: 2020-10-05                                         2020-04-03
          Elliptic curve 2y^2=x^3+x over field size 8^91+5


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

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Table of contents
  1.  Introduction
  2.  Requirements Language (RFC 2119)
  3.  Overview
  3.1. Not for single-curve ECC
  3.2.  Risks of new curve-specific attacks
  3.3.  Multi-curve ECC
  3.3.1.  Multi-curve ECC is a redundancy strategy
  3.3.2.  Whether to use multi-ECC  Benefits of multi-curve ECC  Costs of multi-curve ECC
  3.3.3.  Applying multi-curve ECC
  3.4.  Curve features
  3.4.1.  Field features
  3.4.3.  Equation features
  3.4.4.  Finite curve feature  Curve size and cofactor  Pollard rho security  Pohlig--Hellman security  Menezes--Okamoto--Vanstone security  Semaev--Araki--Satoh--Smart security  Edwards and Hessian form  Bleichenbacher security  Bernstein's "twist" security  Cheon security
  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
  5.1.4.  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

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  9.1.  Normative References
  9.2.  Informative References
  Appendix A.  Test vectors
  Appendix B.  Minimizing trapdoors and backdoors
  Appendix C.  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  Pseudocode for test vectors
  C.3.  Pseudocode for a command-line demo of Diffie--Hellman
  C.4  Pseudocode for public-key validation and twist insecurity
  C.5.  Elligator i
  D. Primality proofs and certificates
  D.1.  Pratt certificate for the field size 8^91+5
  D.2.  Pratt certificate for subgroup order

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

  Multi-curve ECC mitigates 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).

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.  Overview

  This sections how curve 2y^2=x^3+x/GF(8^91+5) improves ECC.

3.1. Not for single-curve ECC

  Curve 2y^2=x^3+x/GF(8^91+5) SHOULD NOT be used in single-curve ECC,
  because it is is riskier than other IETF-approved curves, such as
  NIST P-256 and Curve25519, for at least two reasons:

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    - it is newer: common sense says newer is riskier, all else equal,

    - 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 latter
  consensus by listing 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.

3.2.  Risks of new curve-specific attacks

  A risk for ECC is new curve-specific attacks --- "new" meaning
  hypothetical and not yet published, so either future or hidden.

  Prime-field curves were affected by two curve-specific attacks on
  the discrete logarithm: the MOV attacks the SASS attack, both from
  before 2001.  For non-prime-field curves, more recent curve-specific
  attacks have been discovered.  The rarity of the attacks is evidence
  that the probability of new curve-specific attacks is low, but is
  not proof.

  Sensible curves include 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.

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

  - Brainpool's curve are derived using hash functions to
    number-up-my-sleeve numbers, perhaps aiming to mitigate both
    intential subversion and accidental rare weakness.

  A reasonable inference from these curves is that risk of new
  curve-specific attacks warranted the mitigations used.   The
  risk may be less now that further time has passed, yet the
  mitigations may still be warranted.

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  The curve 2y^2=x^3+x/GF(8^91+5) includes 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, much like a
    nothing-up-my-sleeve number,

  - a set of special efficiencies, such as a curve endomorphism,
    Montgomery form, and fast field operation, much like a "rigid"
    favors efficiency 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).

  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 the
  new curve-specific attack (if a new attack ever appears).  This aim
  is only plausible if the set of curves used is diverse.

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

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

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

3.3.  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).

3.3.1.  Multi-curve ECC is a redundancy strategy

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  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,

3.3.2.  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 well be the case
      for at least financial, medical, or personally-identifying data,

    - 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).  Benefits of multi-curve ECC

  The benefit of multi-curve ECC over single-curve ECC, its extra
  security, is difficult to quantify.

  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 inuition 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.

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  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.  Costs of multi-curve ECC

  The cost of multi-curve ECC can be cost compared to single-curve
  ECC.  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.

3.3.3.  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 derive one symmetric key from the result of multiple key
  establishments.  In essense, 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

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  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.  One would still need to a mechanism to support
  sending more than one ECDH public key (usually ephemeral), with an
  indication of the curve for each ECDH key.

  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.)

3.4.  Curve features
  This subsection describes some general features of the curve


  presuming a familiarity with elliptic curve cryptography (ECC).

  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 Section 3,4, and 5.

3.4.1.  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.  

  Prime fields in ECC tend be more efficient in software than in
  hardware.  The most recent known curve-specific attacks on
  prime-field ECC are from 2000.

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

3.4.3.  Equation features

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

  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
    longer used, perhaps because quaternionic multiplication is

  This 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 they have
    more than two automorphisms.  Over complex numbers, the moduli
    space of elliptic curves is an orbifold, with two non-smooth
    points, at j=0 and j=1728, which is yet another reason these
    j-invariants are special.

3.4.4.  Finite curve feature

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  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+)[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 the finite curve.  Curve size and cofactor

  The curve (of points rational over GF(8^91+5) has size (order) 72q
  for a large prime q.  (See Appendix for a Pratt primality certifcate
  for q.)

  In other words, the curve has cofactor 72.

    Note: The curve size 72q can be found using the CM method and
    Cornacchia's algorithm, instead of the more costly
    Schoo--Elkies--Atkin algorithm(s).  For this curve, this method
    amounts to finding integers (a,b) such that a^2 + b^2 = p, and
    then putting 72q = a^2 + (b-1)^2.  Pollard rho security

  The prime q is 267-bit number, so the Pollard rho algorithm takes
  (proportional to) sqrt(q) ~ 2^133 elliptic curve operations.  So, it
  seems to provide well over 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.

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  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,  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 factor 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.)  Menezes--Okamoto--Vanstone security

  The curve has a large embedding degree, so it resists the
  Menezes--Okamoto--Vanstone attack.  The curve 2y^2=x^3+x /
  GF(8^91+5) is not supersingular.

    Note: For about half of all primes q, then curve 2y^2=x^3+x is
    supersingular over GF(q).  Supersingular curves have q+1 points,
    and are vulnerable to the MOV attack, which reduces the elliptic
    curve discrete logarithm to the finite field discrete logarithm
    over GF(q^2).  This is one of the sense in which the curve
    2y^2=x^3+x / GF(8^91+5) is close to being insecure.  To be clear,
    this curve was chosen after, and with full knowledge of, the MOV

    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.

  The large embedding degree also means that it has no efficient
  pairing operation, so it cannot be used for pairing-based
  cryptography.  Semaev--Araki--Satoh--Smart security
  The fact that the curve size 72q is not p, means that the curve
  resists the Semaev--Araki--Satoh--Smart attack.  Edwards and Hessian form

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  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.

    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.  Bleichenbacher security

  The prime q is not particularly close to a power of two.

  This means that for faulty implementations of digital signatures may
  be more vulnerable to Bleichenbacher's attack, which would exploits
  the non-uniformity in secret numbers obtained by reducing uniformly
  random bit strings modulo q.

  Therefore, q-uniformization of the secret numbers is critical for
  signature applications of 2y^2=x^3+x/GF(8^91+5).  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 Legendre symbol.

  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 *

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  An adversary can exploit this, by finding such invalid x that
  corresond 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.  Cheon security

  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, with d not re-used across ECDH sessions, so
      that 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.  

  The Brown--Gallant--Cheon q-1 algorithm, finds d given [d^e]G if
  e|q-1, with 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.  These algorithm rely on factors e of q-1 or
  q+1, so the factorization of these numbers affects the security
  against the algorithm.  Cheon security refers to the ability to
  render these algoirthms unusable.

  It is possible seek out curves such that q-1 and q+1 have no small
  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:
  To be completed.

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4.  Encoding points
  Elliptic curve cryptography uses points for public keys and raw
  shared secrets.

  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, both of whose coordinates
  may be deemed as infinity.

  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: for practicality, an implementation will often internally
    represent the x-coordinate as a ratio [X:Z] of field elements.
    Each field element has multiple representations, but [x:1] can
    viewed as normal representation of x.  (Infinity can be then
    represented by [1:0], though one must be careful.)

  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 many typical ECC schemes, 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 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, then the 34-byte encoding is
  probably not usable unless the following pre-encoding procedure is

  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 ord(G)).

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 done if 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 x or -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.  In some situations, such invalid b can  |
    |  lead to a severe attack.  In these situations, the   |
    |  decoded point (x,y) MUST be validated, as described  |
    |  below in Section 4.                                  |
    |                                                       |

  In cases where a value for at least one of y, -y, iy, or -iy is
  needed such as Diffie--Hellman key agreement using some other
  coordinate system (such as one might need when converting to Edwards
  coordinates), the candidate value can be obtained by computing a
  square root:
    y = ((x^3+x)/2)^(1/2).

  In some cases, 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 replace x by p-x, and adjusts the discrete
  logarithm appropriately.  These steps can be done by the encoder,
  with the decoder doing nothing.

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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
    a little milder than the average 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 several strategies.

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 groun, an impelmentatio 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                                                        |

    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.

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    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.

5.1.4.  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
  implemeneted with a Montgomery ladder, such that the static secret
  key is used at most ten million times hashed-ECDH transactions.
  Even if exposed to invalid points on the twist, the secruity risk is
  nearly negligible.

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 moduli smaller than 8^91+5.

  More slowly, but perhaps more simply, one can compute the Legendre
  symbol using powering in the field: (z/p) = z^((p-1)/2) =
  z^(2^272+2).  This will have value 0,1 or p-1 (which is equivalent
  to -1).

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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 include a check that x is

  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 n = ord(G).  The costly method
  to check this is checking that [n]P = O.  An alternative method is
  to try to solve for Q in the equation [12]Q=P, which involves
  methods such a division polynomials.

  To be completed.

6.  OPTIONAL encodings
  The following two encodings are not usually required to obtain
  interoperability in the typical ECC applications, but can sometimes
  be useful.

6.1.  Encoding scalars
  Scalar (integer point multipliers) sometimes needed to be encoding
  as byte strings, at least internally to an implementation.

  Basically, little-endian byte encoding of integers is recommended.

  In Diffie--Hellman only implementations, the scalars s and p-s
  really have not significant distinction, so all scalars can be
  represented with 34 bytes.


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

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   - 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.

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) for use in
      password-authenticated key exchange.

    - 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 3.2 does not suffice to
    encode strings, becausse only about half of all 34-byte strings
    are decodable.  

    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].  The
    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.

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  To encode a 34-byte string b,

    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, but risk is difficult to

  This section lists the most plausible threats against
  2y^2=x^3+x/GF(8^91+5), comparing their risk to a typical generic
  curve in ECC, or in some cases, to specific well-established
  curves in ECC.

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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.  Perhaps a smaller field, such as that used in
    Curve25519, has a 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.

  - 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 Kolmogorov 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 much slower implementation than
    8^91+5, which is a significant runtime cost, and perhaps also a
    security risk (due to implementation bugs).

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    The prime 8^95-9 is, just like 8^91+5, very close to a power of
    two.  So should have comparable efficiency for basic field
    arithmetic operations, such as addition, multiplication and
    reduction.  The field 8^95-9 is a little larger, but can still be
    implemented using five 64-bit words.  Being larger, 8^95-9, it 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 bug.

  - 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.

    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
    arugably 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.

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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].  

  To be completed:

  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.

  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).

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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.

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,

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  - 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.

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 cryptographic forum such as the IACR conferences like
  Crypto and Eurocrypt.  So, it has only been reviewed by very few

  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

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  - 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.

  - 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.

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  - 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

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. <https://ia.cr/2018/749>

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

  [B2] D. Brown.  RKHD ElGamal signing and 1-way sums, IACR ePrint,
     2018. <http://ia.cr/2018/186>

  [KKM] A. Koblitz, N. Koblitz and A. Menezes.  Elliptic Curve
     Cryptography: The Serpentine Course of a Paradigm Shift, IACR
     ePrint, 2008.  <https://ia.cr/2008/390>

  [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. <https://ia.cr/2014/571>

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

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  [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.

  [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.  <http://ia.cr/2013/549>

  [Teske] E. Teske.  An Elliptic Curve Trapdoor System, IACR ePrint,
     2003.  <http://ia.cr/2003/058>

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

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Appendix A.  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 big-endian.

    Note:  Public keys are encoded as 34-byte strings are little, so
    one reverses the order of the bytes found in the test vectors.
    The pseudocode in Appendix C.2 should make this clear.

  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, which would serve as a very
  insecure private key.  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.

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

  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.

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  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 (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 catch any more bugs, but simply 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 reassigning y with the
  equation y = yG.  (So, the y on the left is new, the y on the right
  is old, they are not the same.)  Then 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.  The left side is the new
  value of y.  Finally, Bob's replaces y by y^900 mod order(G), just
  as Alice did.

  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.

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

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

  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

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  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).

    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 (secure) elliptic curves using 21 or fewer
     characters.  (But if the specification new interpretations, then
     new interpretation might also be used to further compress the
     specification of 2y^2=x^3+x/GF(8^91+5) to below 21 characters.)

  The curve is actually isomorphic to a curve specifiable in 20


  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?].

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


  This has a longer overall specification, but the field part is
  shorter field specification.  Perhaps an isomorphic curve can be
  found (one with three terms), so that total length is 20 or fewer

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

  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.

  Low Kolmogorov complexity is not directly correlated with security
  of the curve.

   Note: Indeed, as shown further below, the very insecure examples
   exist with lower complexity, by choosing a defective curve

  The benefit of low Kolmogorov complexity is an idea, which general
  to cryptography, sometimes called nothing-up-my-sleeve, or
  subversion-resistance, or similar.  For elliptic curves, the benefit
  may be stated as the two 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.

  Singular cubics, though not formally elliptic curves, are arguably
  among the same class of object, and can be described similarly,
  using equations and so.  For smooth singular curves (irreducible
  cubics) a group can be define, 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.
  Worse, a low Kolmogorov complexity curve can be described that
  suffers from three attacks simultaneously: y^2=x^3+1/GF(2^127-1).
  To be completed.

  Of course, the weak cubics are vulnerable to extremely well-known
  attacks, so when estimating the bits of information in the
  Kolmogorov complexity of curves that resist known attacks, we can
  ignore such examples.  The point of these examples, however, is to
  demonstrate that there exists known attacks that affect curves of
  low Kolmogorov complexity, and therefore secret attacks might have
  the same property.

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  So, it is sensible to disclaim any resistance to secret attacks of
  such a nature.  For this reason, 2y^2=x^3+x/GF(8^91+5) should be
  used with other elliptic curves.


Appendix C.  Pseudocode

  This section uses a C-like pseudocode to demonstrate both the
  well-known algorithms one can use implement this curve, and some
  details particular to this curve.

    Note: Some implementers, such as C programmers, may prefer such
    pseudocode over the wordy and formulaic specifications given
    earlier in this draft.  Besides the principles and algorithms are
    well-known, so I have opted to put the pseudocode in a more
    runnable form than traditional language-agnostic pseudocode.

    Note: The pseudocode is not standard C (e.g., it uses non-standard
    C type __int128), not portable, not thoroughly hardened against
    side channels or any other implementation attacks.

    Note: The pseudocode is highly constricted to minimize line and
    character counts, with Python-like indentation and Lisp-like
    clumping of closing delimiters.  Tools may exist that can put
    transform the pseudocode into more conventional C indentation.
    The pseudocode borrows various yet further C brevities: some
    idiomatic and conventional, some altogether peculiar.  Anything
    too indecipherable deserves explanation in a future revision of
    this draft.

    Note: this pseudocode has not yet received any independent review.

C.1.  Scalar multiplication of 34-byte strings

  The pseudocode for scalar multiplication provides an interface for
  scalar multiplication.  A function takes as input 3 pointer to
  unsigned character strings; it also returns a Boolean value,
  indicating success or failure.

  The pseudocode is to be consider to form a single file, pseudo.c,
  which is then include into other 3 pieces pseudocode: one to
  generate test vectors, one to demo a command-line Diffie--Hellman,
  one to demo public-key validation and twist insecurity of the curve.

  The file pseudo.c has two sections, one for field arithmetic, and
  one form scalar multiplication using Montgomery's ladder.

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    Note: I have been able to improve the speed of Montgomery's ladder
    by ~10% using Bernstein's 2-D ladder.  I have also been to improve
    the speed by ~20% using Gallant--Lambert--Vanstone and Edwards
    coordinates.  These improvements are not likely to carry through
    to a proper optimization regime, since I never used any assembly
    optimizations.  Also these improvements involve more complex
    algorithms, which may suffer higher risk of implementation

  To be completed.

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

  The field arithmetic pseudocode, is the first part of the file
  pseudo.c, implements all the necessary field operations to implement
  a Montgomery for elliptic curve 2y^2=x^3+x.  This means that it does
  not include a square computation: instead it has a Legendre symbol

    Note: The Legendre symbol is used for public-key validation.  The
    pseudocode implements field inversion and the Legendre symbol
    using exponentiation, with the aim of being simple and
    constant-time.  Alternative algorithms for these tasks are known
    to experts.

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  #define RZ return z
  #define B 34
  #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)\
    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)), !j;}
  This pseudocode makes uses of some extra C-like pseudocode features:

  - #define is used to create macros, which expand within the source
    code (as in C pre-processing).

  - type ii is 128-bit integer

  - multiplying a type i by a type ii variable yields a type ii
    variable.  If both inputs can fit into a type i variable, then
    the result has no overflow or reduction: it is exact as a product
    of integers.

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  - type ff is array of five type ii values.  It is used to represent
    a field in a radix expansion, except the limbs (digits) can be
    128-bits instead of 64-bits.  The variable zz has type ff and is
    used to intermediately store the product of two field element
    variables x and y (of type f).

  - function mod takes an ff variable and produce f variable
    representing the same field element.  A pseudocode example may be
    defined further below.

  TO DO: Add some notes (answer these questions):

  - How small the limbs of the inputs to function mul and squ must be
    to ensure no overflow occurs?

  - How small are the limbs of the output of functions mul and squ?

  TO DO: add notes answering these questions:

  - How small must be the input limbs to avoid overflow?

  - How small are the output limbs (to know how to safely use of
    output in further calculations).

  Note: The partial reduction technique used in the multiplication
  pseudocode is sometimes known as lazy reduction.  It aims to do
  just enough calculation to avoid overflow errors, and thus it may be
  regarded as attempt at optimization.

  To be completed.
  The input variable is x and the output variable is b.  The declared
  types and functions are as follows:

  - type c: curve representative, length-34 array of non-negative
    8-bit integers ("characters"),

  - type f: field element, a length-5 array of 64-bit integers
    (negatives allowed), representing a field element as an integer in
    base 2^55,

  - type i: 64-bit integers (e.g. entries of f),
  - function mal: multiply a field element by a small integer (result
    stored in 1st argument),

  - function fix: fully reduce an integer modulo 8^91+5,

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  - function cmp: compare two field element (after fixing), returning
    -1, 0 or 1.

    Note: The two for-loops in the pseudocode are just radix
    conversion, from base 2^55 to base 2^8.  Because both bases are
    powers of two, this amount to moving bits around.  The entries of
    array b are compute modulo 256.  The second loop copies the bits
    that the first loop misses (the bottom bits of each entry of f).

    Note: Encoding is lossy, several different (x,y) may encode to the
    same byte string b.  Usually, if (x,y) generated as a part of
    Diffie--Hellman key exchange, this lossiness has no effect.

    Note: Encoding should not be confused with encryption.  Encoding
    is merely a conversion or representation process, whose inverse is
    called decoding.

  - the expression (i)b[j] means that 8-bit integer b[j] is converted
    to a 64-bit integer (so is no longer treated modulo 256).  (In C,
    this is operation is called casting.)

    Note: the decode function 'feed' only has 1 for-loop, which is the
    approximate inverse of the first of the 2 for-loops in the encode
    function 'bite'.  The reason the 'bite' needs the 2nd for-loop is
    due to the lossy conversion from integers to bytes, whereas in the
    other direction the conversion is not lossy.  The second loop
    recovers the lost information.


C.1.2.  Montgomery ladder scalar multiplication

  The pseudocode below, 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

  Again, the pseudocode is a continuation of the pseudocode for field
  arithmetic, and all previous definitions are assumed.

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  #define X z[0]
  #define Z z[1]
  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
  _ 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

  The pseudocode function mulch -- which multiplies byte string
  (character) representations of point b by the byte string
  representation of integer d -- omits public key validation of the
  input point b if the base of scalar multiplication is the chosen
  fixed base, or if the input integer d and output point db have the
  same location.

  The reason for the latter omission of public key validation is the
  integer d is overwritten presumably the caller of mulch intended to
  use d only once, so that d is likely to be an ephemeral secret,
  largely obviating the need to validate b.

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  In other words, the caller of mulch can control whether public key
  validation is done by choosing the locations of db, b, b
  appropriately.  (An alternative would be for mulch to include a flag
  to indicate whether b needs to be validated.  Instead, the
  pseudocode tries to make mulch do the sensible choice for
  Diffie--Hellman if the caller forgets whether public key validation
  is necessary.)

  The pseudocode files tv.c, dhe.c and pkv.c, define in the sections
  below, demonstrate the use of mulch, and its features regarding
  public key validation.

  In case, mulch returns a Boolean-valued integer indicating whether b
  was valid.  If validation was requested by the interface, and b is
  invalid, then mulch return false (0), and the memory location db
  should remain unaltered.

    Note: the pseudocode makes types c and C volatile, with the aim
    that the C compiler will preserve attempts to zeroize values of
    this type.  Such zeroization steps in the pseudocode do add
    clutter to the code, but have usually been delimited by
    parentheses or braces to indicate their implementation-specific

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.)

  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. 

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  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  Pseudocode for test vectors

  The following pseudocode, describing the contents of a file tv.c,
  includes the previously defined file pseudo.c, and stdio.h, and then
  generates some test vectors.

  #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 j=1e5/2,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)/1e5);//else//debug

  To be completed: Explain this properly, if possible.

  The test vectors should output this:

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

  The following code, representing a file dhe.c, is a bilingual: being
  valid C and bash script.

  As a bash script, it will compile the C code as dhe, then run it
  twice, once as Alice and once as Bob, piping the ephemeral public
  keys, and writing the resulting Diffie--Hellman agreed secret keys
  into pipes.  The agreed secret keys are fed into SHA-256 to
  demonstrate their equality, but also to show the typical way to use
  DH agree keys (to hash them rather than use them directly).

  This pseudocode assumes a Linux-like system.  

  #include "pseudo.c" /* dhe.c (also a bash script) 
  : demos ephemeral DH, also creates, clobbers files dhba dha dhb
  : -- Dan Brown, BlackBerry, '19 */
  #include <stdio.h>
  _ get(c p,_*f){if(f)while(!fread((_*)p,B,1,f));}
  _ put(c p,_*f){if(f)fwrite((_*)p,B,1,f),fflush(f); bite(p,O);}
  int main (_){C s="/dev/urandom",p="EPHEMERAL s => OK if p INVALID";
    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 -O3 dhe.c -o 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)# '*/

C.4  Pseudocode for public-key validation and twist insecurity

  The following pseudocode, describing a file pkv.c, demonstrates the
  public-key validation features of mulch from pseudo.c, by
  deliberately supplying invalid points to mulch.  It also
  demonstrates how to turn PKV on and off using the mulch interface.

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  It also demonstrates the need for PKV despite using the Montgomery
  by finding points of low order on the twist of the curve, and
  showing that such points can leak bits of the secret multiplier.

  It further demonstrates the order of the curve, and complex
  multiplication by i, and the fact the 34-byte representation of
  points is unaffected by multiplication by i. 

  #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:


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 pseudocode

  Pseudocode (to be verified):

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  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);

  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)) ;

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  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);}
D. 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.

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

  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

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  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.  

D.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).

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    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,

D.2.  Pratt certificate for subgroup order

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

   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, to Gaelle Martin-Cocher for encouraging submission
  of this I-D.  Thanks to David Jacobson for sending Pratt primality

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Author's Address
  Dan Brown
  4701 Tahoe Blvd.
  BlackBerry, 5th Floor
  Mississauga, ON

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