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Internet-Draft                                                D. Brown
Intended status: Experimental                               BlackBerry
Expires: 2018 Apr 14                                       2017 Oct 11


          Elliptic curve 2y^2=x^3+x over field size 8^91+5
         <draft-brown-ec-2y2-x3-x-mod-8-to-91-plus-5-00.txt>

Abstract


  This document specifies: the field of size 8^91+5, the elliptic
  curve 2y^2=x^3+x (over this field), encoding a point (on the curve)
  into 34 bytes, public key validation, encoding a private key into 34
  bytes, and encoding 34 bytes into a point.  Test vectors and
  pseudocode are to be provided.

Status of This Memo

  This Internet-Draft is submitted in full conformance with the
  provisions of BCP 78 and BCP 79.  Internet-Drafts are working
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  Internet-Drafts are draft documents valid for a maximum of six
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  Copyright (c) 2017 IETF Trust and the persons identified as the
  document authors.  All rights reserved.

  This document is subject to BCP 78 and the IETF Trust's Legal
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  translate it into languages other than English.



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Table of Contents

  1.  Introduction
  1.1.  Background
  1.2.  Motivation
  2.  Requirements Language (RFC 2119)
  3.  Encoding a point into 34 bytes
  3.1.  Encoding a point into bytes
  3.2.  Decoding bytes into a point
  4.  Point validation
  4.1.  When a point MUST be validated
  4.2.  How to validate a point (given only x)
  5.  OPTIONAL encodings
  5.1.  Encoding scalar multipliers as 34 bytes
  5.2.  Encoding 34 bytes into a point (sketch)
  6.  Cryptographic schemes
  6.1.  Diffie--Hellman key agreement
  6.2.  Signatures
  6.3  Menezes--Qu--Vanstone key agreement
  7.  IANA Considerations
  8.  Security considerations
  8.1.  Field choice
  8.2.  Curve choice
  8.3.  Encoding choices
  8.4.  General subversion concerns
  9.  References
  9.1.  Normative References
  9.2.  Informative References
  Appendix A.  Test vectors
  Appendix B.  Motivation: minimizing the room for backdoors
  Appendix C.  Pseudocode
  C.1.  Byte encoding
  C.2.  Byte decoding
  C.3.  Fermat inversion
  C.4.  Branchless Legendre symbol computation
  C.5.  Field multiplication and squaring
  C.6.  Field element partial reduction
  C.7.  Field element final reduction
  C.8.  Scalar point multiplication
  C.9.  Diffie--Hellman pseudocode
  C.10.  Elligator i

1.  Introduction

  This document specifies some conventions for using the elliptic
  curve 2y^2=x^3+x over the field of size 8^91+5 in cryptography.

  This draft focuses on applications to Diffie--Hellman exchange.


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1.1.  Background

  This document presumes that its reader already has familiarity with
  elliptic curve cryptography.

  The symbol '^', as used in '2y^2=x^3+x' and '8^91+5' means
  exponentiation, also known as powering.  In particular, it does not
  mean bit-wise exclusive-or (as in the C programming language
  operator).  For example, y^3=yyy (or y*y*y, if * is used for
  multiplication.)

  In particular, p=8^91+5 is a (positive) prime number.  Its encoding
  into bytes, using little-endian ordering (least significant bytes
  first), requires 35 bytes, and has the form {5,0,0,...,2}, with the
  first byte equal to 5, the last 2, and the 33 intermediate bytes are
  each 0.  A byte encoding of p is not needed for this document, and
  is only shown here for illustrative purposes.  Its hexadecimal
  representation (i.e. big-endian, base 16), is 20...05, with 67 zeros
  between 2 and 5.

1.2.  Motivation

  The motivations for curve 2y^2=x^3+x over field 8^91+5 are discussed
  in Appendix B.

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

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 a point into 34 bytes

  Elliptic curve cryptography uses points for public keys and raw
  shared secrets.  A point can be defined as either pair (x,y), where
  x and y are field elements, or a special point O located at
  infinity.  Field elements for this curve are integers modulo 8^91+5.

    Note: for practicality, an implementation will usually represent
    the x-coordinate as a ratio (X:Z) of field elements.  This
    specification ignores that detail, assuming x has been normalized
    to (x:1).



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  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 finite point (x,y).

    Note: In many typical ECC schemes, the 34-byte encoding works
    well, despite being neither injective nor surjective.

3.1.  Encoding a point into bytes

  In short: a finite point (x,y) by the little-endian byte
  representation of x or -x, whichever fits into 34 bytes.

  In detail: a point (x,y) is encoded into 34 bytes b[0], b[1], ...,
  b[33], 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.  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) = b[0] + b[1]*256 + ... + b[33]*256^33.

  Pseudocode can be found in Appendix C.



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3.2.  Decoding bytes into a point

  In short: the bytes are little-endian decoded into an integer which
  becomes the x-coordinate.  The y-coordinate is implicit (in
  Diffie--Hellman).

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

  (TO DO: if y is needed explicitly, then one of y matching x must be
  solved; in that case, y-needing application, after a point (x,y) is
  encoded to b, it should be replaced by (x',y'), where (x',y') is the
  decoding of b.  In the rare case that x and x' do not match, then
  (x,y) should be re-generated or rejected.)

  In greater detail: if the 34 bytes are b[0], b[1], ..., b[33], each
  with an integer value between 0 and 255 inclusive, then

   x = b[0] + b[1]256 + ... + b[i]256^i + ... + b[33]256^33

4.  Point validation

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

4.1.  When a point MUST be validated

  Public keys from other parties MUST undergo validation if they are
  combined with private keys as part of multiple Diffie--Hellman
  computations:



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    +---------------------------------------------------------------+
    |                                                               |
    |  STATIC                               _  ___                  |
    |  SECRET        ---\         |\/| | | (_`  |                   |
    |  SCALAR        ---/  POINT  |  | \_/ ._)  |  BE VALIDATED.    |
    |  MULTIPLIER                                                   |
    |                                                               |
    +---------------------------------------------------------------+

  Additionally, public keys SHOULD undergo validation if they are
  received from an unauthenticated source, even if scalar is ephemeral
  or public.

  TO DO: certain exceptional exemptions to the point validation
  REQUIREMENT may be added in future versions of this draft.  In
  particular, when one party has received key confirmation from the
  other side, some of the harm of an using invalid point is
  diminished.  The difference in risk is similar to the difference
  between online and offline attacks on passwords.  In this setting,
  (after key confirmation is received) alternatives to point
  validation might suffice, such as somehow limiting the rate or
  amount of use of the static scalar secret.  So, point validation
  would be reduced to a RECOMMENDATION, but at least one of point
  validation or rate-limiting would still be REQUIREMENT.  For this
  draft, point validation remains a REQUIREMENT, even if the key
  confirmation is received.  (Not every protocol makes a clear
  distinction between ephemeral and static keys, and it seems that
  only one side of Diffie--Hellman key exchange can receive key
  confirmation before using the key.)

4.2.  How to validate a point (given only x)

  Upon decoding the 34 bytes into x, the next step is to compute
  z=2(x^3+x). Then one checks if z has a nonzero square root.  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.



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  More slowly, but perhaps more simply, one 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, 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.

  The curve 2y^2=x^3+x is not twist-secure.  So, using the Montgomery
  ladder for scalar multiplication is not enough to thwart invalid
  public key attacks.  In other words, public key validation MUST be
  combined with the Montgomery ladder, unless the scalar multiplier
  involved is public or a single-DH-use secret (i.e. computing kG and
  kP, counts as a single DH use of k).

    Note: a given point need only be validated once, if the
    implementation can track validation state.

  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 12P != O, then the point P has
  large order.

  OPTIONAL: In even rarer situations, it may be necessary to ensure
  that the point also has prime order.  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 scalar multipliers as 34 bytes

  To be completed.

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

  The main application is to signatures.

  Another application is for test vectors (to be completed).




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5.2.  Encoding 34 bytes into a point (sketch)

  In special applications, beyond mere Diffie--Hellman key exchange or
  digital signatures, it may be desired to encode arbitrary bytes as
  points.

  Example reasons are anonymity, or hiding the presence of a key
  exchange.

    Note: the point encoding described earlier does a different job.
    It encodes every point.  The task here is to encode every byte
    string.

  This method is slower than the representations above, and yields
  biased elliptic curve points, but has the advantage that the
  byte-strings are unbiased.

  The idea is a minor variation of the Elligator 2 construction
  [Elligator].  Unfortunately, Elligator 2 itself fails for curves
  with j-invariant 1728, which includes 2y^2=x^3+x.  In case of
  confusion, this map here can be called Elligator i.

  Fix a square root i of -1 in the field.

  Given any random field element r, compute

    x=i- 3i/(1-ir^2)

  If there is no y solving 2y^2=x^3+x for this x, then replace x by
  x+i and try to solve for y once again.

  If the first x fails, then the second x succeeds.

  So, now r determines a unique x.  To determine y, solve it per the
  equation, getting two roots.  Label the 2 roots y0 and y1 according
  to a deterministic rule.  Then choose y0 if the first x works, else
  choose y2.  This ensures that the map from r^2 to (x,y) is
  injective.

  Finally, to encode a byte string b, just let it represent a field
  element r.  Note that -r will be require more than 34 bytes.  So the
  map from b to (x,y) is now injective.

  This map is reversible.

  To be completed.



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6.  Cryptographic schemes

  To be completed, or even removed!

  List all possible cryptographic schemes in which this curve could be
  used is outside the scope of this short document.  Only a few
  highlights are mentioned.

6.1.  Diffie--Hellman key agreement

  To be completed.

  Question: should DH use cofactor multiplication?  For now, let's say
  no.

  Non-cofactor multiplication risks leaking the private key mod 72, or
  at least mod 12, or perhaps even worse (if the field arithmetic has
  additional leaks).

  But cofactor multiplication reduces the private key size similarly.
  Also, if we start from a 34-byte private key scalar, then we achieve
  a similar effect to cofactor multiplication.

6.2.  Signatures

  For signatures, such as ECDSA, the verifier must fully decompress
  the 34-byte representation.  The verifier must do this twice, once
  with the signer's public key, and once with one component of the
  signature.

  To do this, the verifier can take, and make the most natural choice
  of the two possible y.  The signer, anticipating the verifier, then
  must ensure that the signature will verify correctly under the
  verifier's choices for the y values.  The signer incurs only a small
  extra cost for ensuring this.

  To be completed.

  Given that this curve is experimental and non-radically distinct
  from previous curves, signers and may opt to consider an
  experimental and non-radically distinct signature scheme with the
  curve 2y^2=x^3+x.

  The RKHD ElGamal signatures is an example of such a signature
  scheme.



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  In short, fix a base point G.  The signing key is d, the verifying
  key is Q=dG.  A pair (R,s), R is a point, and s is an integer, is a
  (valid) signature of message with integer hash h, if

       sG = rR + hQ

  where r is obtained from R by re-interpreting its byte as an
  integer.

  To sign a message with hash h, the signer computes a
  message-unique secret k, computes R=kG, computers r as above, and
  computes

       s = rk + hd mod n

  where n is the order of G.

  The signer may compute k as the hash of s and h, or through some
  other method which ensures that k depends (pseudorandomly) on h.

  The signer MUST choose k such that no linear relation between the k
  for different h can be discovered by the adversary.  The signer
  SHOULD use some kind of pseudorandom function to achieve this.

    Note: this ElGamal signature variant corresponds to type 4 ElGamal
    signature in the Handbook of Applied Cryptography.

6.3  Menezes--Qu--Vanstone key agreement

  To be completed.

7.  IANA Considerations

  This document requires no actions by IANA, yet.

8.  Security considerations

  No cryptographic algorithms is without risks. Consequently, risks
  are comparative.  This section will not fully list the risks of all
  other forms of elliptic curve cryptography.  Instead it will list
  the most plausible risks of this curve, and only to a limited degree
  contrast these to a few other standardized curves.

8.1.  Field choice

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



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  - 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.

    Many other standard curves, such as the NIST P-256 and
    Curve25519, also use special prime field sizes, so have a similar
    risk.  Yet other standard curves, such as the Brainpool, use
    pseudorandom field sizes, so have less risk to this threat.

  - 8^91+5, 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 cost is wasteful, and partially a form of denial of service.

  - 8^91+5, is smaller than 8^95-9, yet uses no fewer symbols.  Since
    larger field sizes lead to strong Pollard rho resistance, it can
    be argued that this field size does not optimize security against
    (specification) simplicity.  (The main reason this document
    prefers 8^91+5 over 8^95-9 is its simpler field inversion.)
    Similarly, 8^91+5 is smaller than the six-symbol primes 9^99+4 and
    9^87+4, but these are not close to powers of two, which means that
    modular multiplication and reduction for them is not likely to be
    as efficient as for 8^91+5.

  - 8^91+5, is smaller than 2^283 (used by sect283k1 in Zigbee), and
    many other five-symbol and four-symbol powers of primes (such as
    9^97).  So, it less to provide less resistance to Pollard rho.
    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 prime
    field 8^91+5 (and other large characteristic fields) over small
    characteristic fields, is the generally better software speed of
    large characteristic fields: which arises because most software is
    implemented on a general purpose hardware processor that has fast
    multiplication circuits.  (This speed advantage probably does not
    apply for hardware.)



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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, with complex multiplication leading to an efficient
  endomorphism.  Many other standard curves, NIST P-256, Curve25519,
  Brainpool, do not have any efficient endomorphisms.  Yet some
  standard curves do, NIST K-283 and secp256k1 (in 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.

  A third risk is a biased ephemeral private key generation in a
  digital signature scheme.  Most standard curve lack this risk
  because the field is close to a power of two, and the cofactor is a
  power of two.

  A fourth risk is a Cheon-type attack.  Few standard curves address
  this risk.

  A fifth risk is a small-subgroup confinement attack, which can also
  leak a few bits of the private key.

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, such as the one
  described by [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.)



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  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 user 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

  - 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 over 8^91+5 was (seemingly) first
  described to the public in 2017 [AB].  So, it has a very low age.

  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 been review by very few eyes, most
  of which had little incentive to study it seriously.

  Under the metric of aegis, as in age * eyes, it 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 some more well-studied curves.



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

  - 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 score is 10000.

  - P-256: has a special field size, and maybe an estimated aegis
    score 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 score
    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, which is essentially one of Miller's curves, except
    that a=1 is a quadratic residue.  Miller's curve has not been
    studied directly, but probably much more so than this than the
    curve in this document.  Miller also hinted that it was not
    prudent to use a special curve y^2=x^3-ax: such a comment may have
    encourage some cryptanalysts, but discouraged cryptographers,
    perhaps balancing out the effect on the eyes factor the aegis
    score.  An estimate aegis score 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,
          <http://www.rfc-editor.org/info/bcp14>.

9.2.  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/>

  [KKM] A. Koblitz, N. Koblitz and A. Menezes.  Elliptic Curve
        Cryptography: The Serpentine Course of a Paradigm Shift, IACR
        ePrint, 2008.  <http://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. <http://ia.cr/2014/571>

  [Elligator] To do: fill in this reference.

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

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

  [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.


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  [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: Projective coordinates leak.  Pointcheval, Smart,
  Malone-Lee?

  [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.
         <http://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>

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

Appendix A.  Test vectors

  To be completed.

Appendix B.  Motivation: minimizing the room for backdoors

  To be completed.

  See [AB] for some details.

  The field and curve are described with very few symbols, while
  retaining many basic security and speed features.


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  A prime field was chosen due to 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.

    Note: this curve is isomorphic to the non-Montgomery curve
    y^2=x^3-x, which requires just 9 symbols in its description, 1
    fewer than required by 2y^2=x^3+x.

Appendix C.  Pseudocode

  This section uses a C-like pseudocode to describe some of the
  algorithms useful for implementing this curve.

  Real-world implementations adapting this pseudocode had better
  harden this pseudocode against real-world implementation issues.
  Better yet, real-world code could start from scratch, using the
  pseudocode only for comparison.

    Note: the pseudocode relies on some C idioms (hacks?), which might
    make the pseudocode unclear to those unfamiliar with these idioms.

    Note: this pseudocode was adapted from a few different
    experimental prototypes of the author, (which might not be
    consistent).  The pseudocode has not yet received any independent
    review.

    Note: this pseudocode uses a terse non-conventional coding style,
    partly as an exercise in arbitrary source code compression (code
    golf), but also in the mathematics tradition of using many
    single-letter variable names, which enables seeing an entire
    formula in a single view and emphasizes the essential mathematical
    operations rather than the variable's purpose.

    Note: the pseudocode does not use the C operator ^ for bitwise XOR
    of integers, which (luckily) avoid possible confusion with the use
    of ^ as exponentiation operator in the rest of this document.



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C.1.  Byte encoding

  Pseudocode for byte representation encoding process is

  bite(c b,f x) {
   i j=34,k=5; f t;
   mal(t,-1,x);
   mal(x,cmp(t,x),x);
   fix(x);
   for(;j--;) b[j]=x[j/7]>>((8*j)%55);
   for(;--k;) b[7*k-1]+=x[k]<<(8-k);
   }

  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,

  - 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.



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C.2.  Byte decoding

  Pseudocode for decoding is:

  feed(f x,c b) {
   i j=34;
   mal(x,0,x);
   for(;j--;) x[j/7]+=((i)b[j])<<((8*j)%55);
   fix(x);
   }

  with similar conventions as used in the pseudocode function bite
  (defined in the section on encoding), and some extra conventions:

  - 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.3.  Fermat inversion

  Projective coordinates help avoid costly inversion steps during
  scalar multiplication.

  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].



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  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.

  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).

  Pseudocode for Fermat inversion is:

  i inv(f y,f x) {
    i j=272;f z;
    squ(z,x);
    mul(y,x,z);
    for(;j--;) squ(z,z);
    mul(y,z,y);
    return !!cmp(y,(f){});
  }

  Other inversion techniques, such as the binary extended GCD, may be
  faster, but generally run in variable-time.

  When field elements are sometimes secret keys, using a variable-time
  algorithm risk leaking these secrets, and defeating security.

C.4.  Branchless Legendre symbol computation

  Pseudocode for branchlessly computing if a field element x has a
  square root:

  i has_root(f x) {
    i j=270;f y,z;
    squ(y,x);squ(z,y);
    for(;j--;)squ(z,z);
    mul(y,y,z);
    return 0==cmp(y,(f){1});
  }



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    Note: Legendre symbol is usually most appropriately applied to
    public keys, which mostly obviates the need for side-channel
    resistance.  In this case, the implementer can use quadratic
    reciprocity for greater speed.

C.5.  Field multiplication and squaring

  To be completed.

    Note (on security): 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 multiple 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 supplies inputs likely induce
    overflows [bug attacks], [IT].  The following pseudocode
    should therefore be considered only for illustrative purposes.
    The implementer is responsible for ensuring that inputs cannot
    cause overflows or bugs.

  The pseudocode below for multiplying and squaring: uses unrolled
  loops for efficiency, uses refactoring for source code compression,
  relies on a compiler optimizer to detect common sub-expressions (in
  squaring).

  #define TRI(m,_)\
    zz[0]=m(0,0)_(1,4)_(2,3)_(3,2)_(4,1);\
    zz[1]=m(0,1)_(1,0)_(2,4)_(3,3)_(4,2);\
    zz[2]=m(0,2)_(1,1)_(2,0)_(3,4)_(4,3);\
    zz[3]=m(0,3)_(1,2)_(2,1)_(3,0)_(4,4);\
    zz[4]=m(0,4)_(1,3)_(2,2)_(3,1)_(4,0);
  #define CYC(M) ff zz; TRI(+M,-20*M); mod(z,zz);
  #define MUL(j,k) x[j]*(ii)y[k]
  #define SQR(j,k) x[j]*(ii)x[k]
  #define SQU(j,k) SQR(j>k?j:k,j<k?j:k)
  mul(f z,f x,f y) {CYC(MUL);}
  squ(f a,f x) {CYC{SQU};}

  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



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  - 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.

  - 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?

C.6.  Field element partial reduction

  To be completed.

  The function mod used by pseudocode function mul and squ above is
  defined below.

  #define QUO(x)(x>>55)
  #define MOD(x)(x&((((i)1)<<5)-1))
  #define Q(j) QUO(QUO(zz[j]))
  #define P(j) MOD(QUO(zz[j]))
  #define R(j)     MOD(zz[j])
  mod(f z,ff zz){
   z[0]=R(0)-P(4)*20-Q(3)*20;
   z[1]=R(1)-P(0)-Q(4)*20;
   z[2]=R(2)-P(1)-Q(0);
   z[3]=R(3)-P(2)-Q(1);
   z[4]=R(4)-P(3)-Q(2);
   z[1]+=QUO(z[0]);
   z[0]=MOD(z[0]);
   }

  TO DO: add notes answering these questions:

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



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

C.7.  Field element final reduction

  To be completed.

  The partial reduction technique is sometimes known as lazy
  reduction.  It is an optimization technique.  It aims to do only
  enough calculation to avoid overflow errors.

  For interoperability, field elements need to be fully reduced,
  because partial reduction means the elements still have multiple
  different representations.

  Pseudocode that aims for final reduction is the following:

  #define FIX(j,r,k) {q=x[j]>>r;\
   x[j]-=q<<r; x[(j+1)%5]+=q*k;}
  fix(f x) {
   i j,q,t=2;
   for(;t--;) for(j=0;j<5;j++) FIX(j,(j<4?55:53),(j<4?1:-5));
   q=x[0]<0;
   x[0]+=q*5; x[4]+=q>>53;
   }

C.8.  Scalar point multiplication

  Work in progress.

  A recommended method of scalar point multiplication is the
  Montgomery ladder.  However, the curve 2y^2=x^3+x has an efficient
  endomorphism.  So, this can be used to speed-up scalar point
  multiplication, as suggested by Gallant, Lambert and Vanstone.

  Combining both GLV and Montgomery is also possible, such as
  suggested as by Bernstein.

    Note: The following pseudocode is not entirely consistent with
    previous pseudocode examples.

    Note and Warning: The following pseudocode uses secret indices to
    access (small) arrays.  This has a risk of cache-timing attacks.



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  typedef f p[2];
  typedef struct rung {i x0; i x1; i y; i z;} k[137];
  monty_2d (f ps,k sk,f px) {
    i j,h; f z; p w[3],x[3],y[2]={{{},{1}}},z[2];
    fix(px);mal(y[0][0],1,px);
    endomorphism_1_plus_i(z[0],px);
    endo_i(y[1],y[0]); endo_i(z[1],z[0]);
    copy(x[1],y[0]); copy(x[2],z[0]);
    double_xz(x[0],y[0]);
    for(j=0;j<137;j+=){
     double_xz(w[0],     x[sk[j].x0 /* cache attack here? */ ]);
     diff_add (w[1],x[1],x[sk[j].x1],y[sk[j].y]);
     diff_add (2[2],x[2],x[0],       z[sk[j].z]);
     for(h=0;h<3;h++) {copy(x[h],w[h]);}
    }
    inv(ps,x[1][1]);
    mul(ps,x[1][0],ps);
    fix(ps);
  }

    Note: The pseudocode uses some other functions not defined here,
    but whose meaning can be inferred by ECC experts.

    Note: The pseudocode uses a specialized format for the scalar.
    Normal scalars would have to be re-coded into this format, and
    re-coding has non-negligible run-time.  Perhaps in
    Diffie--Hellman, re-coding is not necessary if one can ensure that
    uniformly selection of coded scalars is not a security risk.

  TO DO:
  -  Define the functions used by monty_2d.
  -  Prove that these function avoid overflow.
  -  Define functions to re-code scalars for monty_2d.

C.9.  Diffie--Hellman pseudocode

  To be completed.

  This pseudocode would show how to use to scalar multiplication,
  combined with point validation, and so on.


C.10.  Elligator i

  To be completed.



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

  Pseudocode (to be verified):

  typedef f xy[2] ;
  #define X p[0]
  #define Y p[1]
  lift(xy p, f r) {
    f t ; i b ;
    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);
  }



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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);}

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.

Author's Address

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


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