Internet-Draft                                                D. Brown
Intended status: Experimental                               BlackBerry
Expires: 2020-04-05                                         2019-10-03

Elliptic curve 2y^2=x^3+x over field size 8^91+5
<draft-brown-ec-2y2-x3-x-mod-8-to-91-plus-5-04.txt>

Abstract

In elliptic curve cryptography, 2y^2=x^3+x/GF(8^91+5) hedges a
remote risk of potential weakness in other curves, if used in
multi-curve Diffie--Hellman, for example.  This curve features:
isomorphism to Miller curves from 1985; low Kolmogorov complexity
(little room for secretly embedded trapdoors of Gordon, Young--Yung,
or Teske); likeness to a Bitcoin curve; 34-byte keys; prime field;
5*64-bit field arithmetic; easy reduction, inversion, Legendre
symbol, and square root; Montgomery ladder or Edwards unified curve
arithmetic (Hisil--Carter--Dawson--Wong); multiplication by i
(Gallant--Lambert--Vanstone); and string-as-point encoding.

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1.  Introduction
1.1.  Background
1.1.1.  Notation
1.1.2.  Basic features
1.1.3.  Multi-curve ECC
1.2.  Speculative security motivation
2.  Requirements Language (RFC 2119)
3.  Encoding points
3.1.  Point encoding process
3.1.1.  Summary
3.1.2.  Details
3.2.  Point decoding process
3.2.1.  Summary
3.2.2.  Detail
4.  Point validation
4.1.  When to validate
4.1.1.  Mandatory validation
4.1.2.  Simplified validation
4.1.4.  Minimal validation
4.2.  Point validation process
5.  OPTIONAL encodings
5.1.  Encoding scalars
5.2.  Encoding strings as points
6.  IANA Considerations
7.  Security considerations
7.1.  Field choice
7.2.  Curve choice
7.3.  Encoding choices
7.4.  General subversion concerns
8.  References
8.1.  Normative References
8.2.  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.1.4.  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

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D.2.  Pratt certificate for subgroup order

1.  Introduction

This document specifies a type of elliptic curve cryptography (ECC)
using the curve

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

This curve is useful as part of a multi-curve ECC system that
combines a diverse set of curves for extra security.

The extra security in using multiple curves is a strongest-link,
multi-layer, fail-safe, defense-in-depth against potential (but not
yet known) attacks against one or more of the curves.

Note: Using multiple curves adds a nonzero cost to an ECC system.
On a current personal computer, this extra cost includes up to 1
millisecond of runtime and sending an extra 34 bytes, per ECC
transaction.  In low-end devices, the time may be higher due to
slower processors, making the cost might be unaffordable.  Even in
high-end devices, the benefit-to-cost comparison is quite
questionable: is the little extra security (against a potential
but unlikely and unknown threat) even worth the cost of extra
runtime and traffic?  The answer may depend on the data being
protected.  If the answer is deemed to be "yes", then multi-curve
ECC is useful, and curve 2y^2=x^3+x/GF(8^91+5) can contribute to
this security.

Comparing single curves when used in isolation, which is current ECC
tradition, curve 2y^2=x^3+x/GF(8^91+5) is arguably riskier than the
more well-established curves (such as NIST P-256, Curve25519, and
even Brainpool).

In traditional single-curve ECC systems, the curve
2y^2=x^3+x/GF(8^91+5) SHOULD NOT be used, due to its risk begin
greater than more well-established curves.

Multi-curve ECC is not noticeably more secure than ECC if all of the
multiple curves are the actually the same curve.  Therefore, a
diversity of dissimilar curves is needed to achieve extra security,
with each curve hedging against a failure in dissimilar curves.

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The curve 2y^2=x^3+x/GF(8^91+5) has features marking it as
dissimilar from some of well-established curves: especially lower
Kolmogorov complextiy and complex multiplication by i.

1.1.  Background

elliptic curve cryptography (ECC).

1.1.1.  Notation

The symbol '^', as used in '2y^2=x^3+x' and '8^91+5' means
exponentiation, also known as powering.  For example, y^3=yyy, or
y*y*y, if * is used for multiplication, and 8^91 = 8*8*...*8, with
91 eights in the product on the right.

Note: This document does not use '^' the way that C (and similar
programming languages) does (as a bit-wise exclusive-or).

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

200000000000000000000000000000000000000000000000000000000000000000005

with with 67 zeros between 2 and 5.

Note: For a lack of a better term, standard ECC terminology uses
the a slight misnomer, "scalar multiplication" for the computation
dP = P + ... + P for the P a point on the curve, d an integer, +
the elliptic curve group addition and law, and the right hand side
implying d terms.  This suggests calling the integer d a scalar.
This is a misnomer, because, in other areas of mathematics,
scalars are used to multiply vectors, but elliptic curve scalar
multiplication is not really vector multiplication, and risks the
suggestion of confusing d(x,y) with (dx,dy).  (That said, an
elliptic curve group, like any abelian group, is a module over the
ring of integers.  Since a module is to a ring the analogue of a
vector space to a field, the terminology is arguably justifiable.)

1.1.2.  Basic features

The underlying field (for defining the curve) is a prime, p=8^91+5.
It is very close to a power of two, which is sometimes known as a
Crandall prime, making reduction modulo p relatively efficient.

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The prime p being slightly larger (not smaller) than a power of two,
means that common algorithms for computing inverses, Legendre
symbols, and square roots are relatively simple (and slightly more
efficient).

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

by^2=x^3+ax^2+x,

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 by the
map

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

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.

Note: Over the complex numbers, j-invariant 0 and 1728 are
special, being the only two non-smooth orbifold points the moduli
space of elliptic curves, which also means that the curves have
extra symmetry.

The curve 2y^2=x^3+x is not supersingular (as defined over the prime
p=8^91+5).

The curve has order 72q for a large prime q, meaning it has cofactor
72, so it is not vulnerable Pohlig--Hellman attack, and it not
vulnerable to the Semaev--Araki--Satoh--Smart attack.

The cofactor 72 is divisible by 4 (and also 3), meaning it is
isomorphic to a curve with an Edwards equation (and also to cure
with a Hessian equation), which may permit yet more efficient
implementation (and yet further combination with the GLV method).

The curve has a large embedding degree, so it has no efficient
pairing operation.  It is therefore also not vulnerable to the
Menezes--Okamoto--Vanstone attack.

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The best known algorithm to solve the discrete logarithm in the
group are Pollard rho algorithms and its variants, (with minor
enhancements due to Gallant, Lambert, Vanstone, which take advantage
of the extra map for complex multiplication), which takes
approximately sqrt(q) point additions to compute a discrete
logarithm (with success rate 1/2).

1.1.3.  Multi-curve ECC

This document does not specify how to do multi-curve elliptic curve
cryptography, but some ideas are sketch without much detail.

Multi-curve Diffie--Hellman key agreement could perhaps compute 10
shared secrets (hashed) with 10 very different curves, and then XOR
them together to get one secret.  Presumably, as long as one of the
Diffie--Hellman secrets is secure, the XOR of 10 is secure.  All the
Diffie--Hellman private keys (scalars) should be independent and so
on.

For signatures, one might just apply multiple signatures, with
different curves (and perhaps different signature algorithms).

This document does not specifically recommend which other curves
should be combined with 2y^2=x^3+x/GF(8^91+5), but suggests the
at least following:

- Use one or more well-established curves, such as NIST P-256 or
Curve25519.

- Use one or more curves without complex multiplication, such as
NIST P-256 or Curve25519.

- Use one or more pseudo-randomized curves, such as NIST P-256 or
Brainpool or something else.

- Use one or more curves whose security features complement those
2y^2=x^3+x/GF(8^91+5) in any other way.

- Use at least three or more curves, since is likely two rather
different curves are reasonably less risky than
2y^2=x^3+x/GF(8^91+5).

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1.2.  Speculative security motivation

The section explain why to use 2y^2=x^3+x/GF(8^91+5) in a set of
three of more curves, rather than a set of three or more curves
without the curve 2y^2=x^3+x/GF(8^91+5).

The main motivation for the specific curve is that its description
of the curve is very short (for an otherwise secure elliptic curve),
thereby reducing the room for a possible secretly embedded trapdoor,
as in [Teske].

A lesser motivation for the curve is its special features.  A very
remote potential catastrophe in ECC would be attack on most curves.
In this disaster scenario, perhaps only a few curves survive, saved
by some special feature.  Complex multiplication by i is perhaps one
of those features.  Surviving such a disaster would be a fluke, but
diversity is perhaps the best possible hedge against this event.
More probable than such a disaster would be an attack that exploits
precisely the special features of curve 2y^2=x^3+x/GF(8^91+5) which
makes it different from better established curves.  So, it is only
really sensible to use the curve in combination with other very
different curves.

More detailed motivations for curve 2y^2=x^3+x over field 8^91+5 are
discussed in Appendix B (and in [AB] and [B1]).

2.  Requirements Language (RFC 2119)

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [BCP14].

3.  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].

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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
strings.

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
encoding.

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.

3.1.  Point encoding process

3.1.1.  Summary

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

3.1.2.  Details

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

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.
Let

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).

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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
bytes.

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
practical:

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

3.2.  Point decoding process

3.2.1.  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.

3.2.2.  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].

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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.

4.  Point validation

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

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.

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4.1.  When to validate

This section specifies several strategies.

4.1.1.  Mandatory validation

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

4.1.2.  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
multiplication.

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
key.

+---------------------------------------------------------------+
|   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.

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.

4.1.4.  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.

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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.

Note: the twist of 2y^2=x^3+x/GF(8^91+5) curve has order:

2^2 * 5 * 1526119141 * 788069478421 * 182758084524062861993 *
3452464930451677330036005252040328546941

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.

4.2.  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
((x^3+x)/p)=-1.)

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).

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
valid.

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

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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.

5.  OPTIONAL encodings

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

5.1.  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.

Applications:

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

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

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5.2.  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

- 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.

To encode a 34-byte string b,

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1. Let b represent a field element r, using little-endian base
256.

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
j+1.

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

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
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.

6.  IANA Considerations

This document requires no actions by IANA, yet.

7.  Security considerations

No cryptographic algorithm is without risk.

Theoretically, therefore, cryptographic risk analysis should be
comparative: so that the least risky cryptographic algorithm can be
chosen.  Practically, however, it is difficult to compare an
algorithm to all others.

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For practicality, this section lists the most plausible risks of
2y^2=x^3+x/GF(8^91+5), comparing these against a general background
of any curve in ECC.  To a lesser degree, this section contrasts
these risks to a few other well-established and standardized
specific curves.

7.1.  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
properties.

- 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).

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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).

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
faster.)

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

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Arguably, in traditional mathematical notation, the symbol '^' is
not actually written, with operation being marked by the use of
superscripts.  In this view, using an ASCII character count
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.

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
coordinates.

- 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
[Bleichenbacher].

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
report.

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

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

7.2.  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

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
non-negligible.

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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.

7.3.  Encoding choices

To be completed.

7.4.  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
discussion.)

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
either

- 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,
or

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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.

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
eyes.

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

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

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

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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.

- 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.

8.  References

8.1.  Normative References

[BCP14] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997,
<http://www.rfc-editor.org/info/bcp14>.

8.2.  Informative References

To be completed.

[AB] A. Allen and D. Brown.  ECC mod 8^91+5, presentation to CFRG,
2017.
<https://datatracker.ietf.org/doc/slides-99-cfrg-ecc-mod-8915/>

[AMPS] Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, and
Juraj Somorovsky.  Prime and Prejudice: Primality Testing Under
2018. <https://ia.cr/2018/749>

[B1] D. Brown.  ECC mod 8^91+5, IACR ePrint, 2018.
<https://ia.cr/2018/121>

[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
number.

[SEC2] Standards for Efficient Cryptography.  SEC 2: Recommended
Elliptic Curve Domain Parameters, version 2.0, 2010.
<http://www.secg.org/sec2-v2.pdf>

[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
parameters.

[HPST] Y. Huang, C. Petit, N. Shinohara and T. Takagi.  On
Generalized First Fall Degree Assumptions, IACR ePrint 2015.
<https://ia.cr/2015/358>

[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>

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

Appendix A.  Test vectors

The following are some test vectors.

000000000000000029352b31395e382846472f782b335e783d325e79322054534554
00000000000000000000000000000000000000000000000000000000000000000117
fc023b3f03b469dca32446db80d9b388d753cc77aa4c3ee7e2bb86e99e7bed38f509
8c2b0d58eb27185715a48d6071657273dfbb861e515ac8bac9bfe58f2baa85908221
8c2b0d58eb27185715a48d6071657273dfbb861e515ac8bac9bfe58f2baa85908221

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
multiplications.

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

The purpose of the large number of iterations is to catch a bug that
has probability larger than 1/100000 of arising on pseudorandom
inputs.  The iterations do nothing to find rarer bugs (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.

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The curve can be described with 21 characters:

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

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

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
characters.

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
characters:

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

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:

y^2+xy=x^3+1/GF(2^283)

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

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
equation.

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
attacks.

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
computation.

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

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.

- 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?

- 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.

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
2y^2=x^3+x.

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

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<CODE BEGINS>
#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
inv(Z)&&mul(x,X,Z)&&(sub(X,X,X)&&sub(Z,Z,Z));}
_ let(e z,e y){i j=2;for(;j--;)mal(z[j],1,y[j]);}
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;
let(t[0],z[1]),let(t[1],z[0]);smv(mav(z,!b),mav(t,b));mav(t,0);RZ;}
e mule(e z,c d){V a;E o={{1}};i
b=0,c,n=(let(a[0],o),let(a[1],z),8*B);
C G={23,1};
i mulch(c db,c d,c b){F x;E p; return
lift(p,feed(x,b),(db==d||b==G))&&drop(x,mule(p,d))&&bite(db,x);}
<CODE ENDS>

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
purpose.

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
e=i+1.

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.

<CODE BEGINS>
#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);
for(;j--;)if(fprintf(stderr,"\r%7d\r",j),!(M(z,x,z)&&M(x,z,G)))
j=0*printf("Mulch fail rate ~%f :(\n",(2*j)/1e5);//else//debug
hx(x),hx(z);
M(y,y,G);M(y,y,y);
for(M(z,G,G),j=900;j--;)M(z,x,z);for(j=900;j--;)M(z,y,z);hx(z);
for(M(z,G,G),j=900;j--;)M(z,y,z);for(j=900;j--;)M(z,x,z);hx(z);}
<CODE ENDS>

To be completed: Explain this properly, if possible.

The test vectors should output this:

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000000000000000029352b31395e382846472f782b335e783d325e79322054534554
00000000000000000000000000000000000000000000000000000000000000000117
fc023b3f03b469dca32446db80d9b388d753cc77aa4c3ee7e2bb86e99e7bed38f509
8c2b0d58eb27185715a48d6071657273dfbb861e515ac8bac9bfe58f2baa85908221
8c2b0d58eb27185715a48d6071657273dfbb861e515ac8bac9bfe58f2baa85908221

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.

<CODE BEGINS>
#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>
_ 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)# '*/ <CODE ENDS> 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. Brown 2y^2=x^3+x over 8^91+5 [Page 39]  Internet-Draft 2019-10-03 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. <CODE BEGINS> #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" "\xd1\xce\x4a\x7a\x1c\xc7\x71\x1c\xc7\x71\x1c\xc7\x71\x1c" "\xc7\x71\x1c\xc7\x71\x1c\x07", // q=order(G) i = "\x36\x5a\xa5\x56\xd6\x4f\xb9\xc4\xd7\x48\x74\x76\xa0" "\xc4\xcb\x4e\xa5\x18\xaf\xf6\x8f\x74\x48\x4e\xce\x1e\x64" "\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" "\x3b\xc4\xca\xeb\x65\x66\x66\x66\x66\x66\x66\x66\x66\x66" "\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: for(j=10;j--;)M(z,y,G),z[0]+=j,M(z,z,w5),hx(z);} <CODE ENDS> 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 above. Pseudocode (to be verified): Brown 2y^2=x^3+x over 8^91+5 [Page 40]  Internet-Draft 2019-10-03 <CODE BEGINS> typedef f xy[2] ; #define X p[0] #define Y p[1] lift(xy p, f r) { f t ; i b ; fix(r); 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 get_y(Y,X); 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); get_y(t,X); b=eq(t,Y); mal(t,1-b,I); 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) ; fix(r); h = (r[4]<(1LL<<52)) ; mal(r,2*h-1,r); fix(r); } Brown 2y^2=x^3+x over 8^91+5 [Page 41]  Internet-Draft 2019-10-03 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);} <CODE ENDS> D. Primality proofs and certificates Recent work of Albrecht and others [AMPS] has shown the combination of adversarially chosen prime and improper probabilistic primality tests can result in attacks. The adversarial primes are generally result of an exhaustive search, and 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 compact representation of these primes: p = 8^91+5 q = #(2y^2=x^3+x/GF(8^91+5))/72 The [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 certificate. 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. Brown 2y^2=x^3+x over 8^91+5 [Page 42]  Internet-Draft 2019-10-03 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). Brown 2y^2=x^3+x over 8^91+5 [Page 43]  Internet-Draft 2019-10-03 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, X=167553393621084508180871720014384259, Y=1453023029482044854944519555964740294049. D.2. Pratt certificate for subgroup order Define 56 variables a,b,...,z,A,B,...,Z,!,@,#,$, with new
values:

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:

Acknowledgments

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
certificates.

Brown                2y^2=x^3+x over 8^91+5                 [Page 44]


Internet-Draft                                             2019-10-03

Dan Brown
4701 Tahoe Blvd.
BlackBerry, 5th Floor
Mississauga, ON
danibrown@blackberry.com

Brown                2y^2=x^3+x over 8^91+5                 [Page 45]
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