Hashing to Elliptic Curves
draft-irtf-cfrg-hash-to-curve-01

Network Working Group                                           S. Scott
Internet-Draft                                              Cornell Tech
Intended status: Informational                               N. Sullivan
Expires: January 3, 2019                                      Cloudflare
                                                                 C. Wood
                                                              Apple Inc.
                                                           July 02, 2018

                       Hashing to Elliptic Curves
                  draft-irtf-cfrg-hash-to-curve-01

Abstract

   This document specifies a number of algorithms that may be used to
   hash arbitrary strings to Elliptic Curves.

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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Background  . . . . . . . . . . . . . . . . . . . . . . . . .   3
     2.1.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   4
       2.1.1.  Encoding  . . . . . . . . . . . . . . . . . . . . . .   5
       2.1.2.  Serialization . . . . . . . . . . . . . . . . . . . .   5
       2.1.3.  Random Oracle . . . . . . . . . . . . . . . . . . . .   5
   3.  Algorithm Recommendations . . . . . . . . . . . . . . . . . .   6
   4.  Utility Functions . . . . . . . . . . . . . . . . . . . . . .   7
   5.  Deterministic Encodings . . . . . . . . . . . . . . . . . . .   7
     5.1.  Interface . . . . . . . . . . . . . . . . . . . . . . . .   7
     5.2.  Encoding Variants . . . . . . . . . . . . . . . . . . . .   7
       5.2.1.  Icart Method  . . . . . . . . . . . . . . . . . . . .   7
       5.2.2.  Shallue-Woestijne-Ulas Method . . . . . . . . . . . .   9
       5.2.3.  Simplified SWU Method . . . . . . . . . . . . . . . .  10
       5.2.4.  Elligator2 Method . . . . . . . . . . . . . . . . . .  12
     5.3.  Cost Comparison . . . . . . . . . . . . . . . . . . . . .  13
   6.  Random Oracles  . . . . . . . . . . . . . . . . . . . . . . .  14
     6.1.  Interface . . . . . . . . . . . . . . . . . . . . . . . .  14
     6.2.  General Construction (FFSTV13)  . . . . . . . . . . . . .  14
   7.  Curve Transformations . . . . . . . . . . . . . . . . . . . .  14
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  15
   9.  Security Considerations . . . . . . . . . . . . . . . . . . .  15
   10. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  15
   11. Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  15
   12. Normative References  . . . . . . . . . . . . . . . . . . . .  15
   Appendix A.  Related Work . . . . . . . . . . . . . . . . . . . .  17
     A.1.  Probabilistic Encoding  . . . . . . . . . . . . . . . . .  17
     A.2.  Naive Encoding  . . . . . . . . . . . . . . . . . . . . .  17
     A.3.  Deterministic Encoding  . . . . . . . . . . . . . . . . .  18
     A.4.  Supersingular Curves  . . . . . . . . . . . . . . . . . .  18
     A.5.  Twisted Variants  . . . . . . . . . . . . . . . . . . . .  18
   Appendix B.  Try-and-Increment Method . . . . . . . . . . . . . .  19
   Appendix C.  Sample Code  . . . . . . . . . . . . . . . . . . . .  19
     C.1.  Icart Method  . . . . . . . . . . . . . . . . . . . . . .  19
     C.2.  Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . .  21
     C.3.  Simplified SWU Method . . . . . . . . . . . . . . . . . .  23
     C.4.  Elligator2 Method . . . . . . . . . . . . . . . . . . . .  23
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  24

1.  Introduction

   Many cryptographic protocols require a procedure which maps arbitrary
   input, e.g., passwords, to points on an elliptic curve (EC).
   Prominent examples include Simple Password Exponential Key Exchange

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   [Jablon96], Password Authenticated Key Exchange [BMP00], Identity-
   Based Encryption [BF01] and Boneh-Lynn-Shacham signatures [BLS01].

   Unfortunately for implementors, the precise mapping which is suitable
   for a given scheme is not necessarily included in the description of
   the protocol.  Compounding this problem is the need to pick a
   suitable curve for the specific protocol.

   This document aims to address this lapse by providing a thorough set
   of recommendations across a range of implementations, and curve
   types.  We provide implementation and performance details for each
   mechanism, along with references to the security rationale behind
   each recommendation and guidance for applications not yet covered.

   Each algorithm conforms to a common interface, i.e., it maps an
   element from a bitstring {0, 1}^* to a curve E.  For each variant, we
   describe the requirements for E to make it work.  Sample code for
   each variant is presented in the appendix.  Unless otherwise stated,
   all elliptic curve points are assumed to be represented as affine
   coordinates, i.e., (x, y) points on a curve.

1.1.  Requirements

   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 [RFC2119].

2.  Background

   Here we give a brief definition of elliptic curves, with an emphasis
   on defining important parameters and their relation to encoding.

   Let F be the finite field GF(p^k).  We say that F is a field of
   characteristic p.  For most applications, F is a prime field, in
   which case k=1 and we will simply write GF(p).

   Elliptic curves come in many variants, including, but not limited to:
   Weierstrass, Montgomery, and Edwards.  Each of these variants
   correspond to a different category of curve equation.  For example,
   the short Weierstrauss equation is of the form "y^2 = x^3 + Ax + B".
   Certain encoding functions may have requirements on the curve form
   and the parameters, such as A and B in the previous example.

   An elliptic curve E is specified by the equation, and a finite field
   F.  The curve E forms a group, whose elements correspond to those who
   satisfy the curve equation, with values taken from the field F.  As a
   group, E has order n, which is the number of points on the curve.
   When n is not prime, we write n = qh + r, where q is prime, and h is

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   said to be the cofactor.  It is frequently a requirement that all
   cryptographic operations take place in a prime order group.  In this
   case, we may wish an encoding to return elements of order q.  For a
   mapping outputting elements on E, we can multiply by the cofactor h
   to obtain an element in the subgroup.

   In practice, the input of a given cryptographic algorithm will be a
   bitstring of arbitrary length, denoted {0, 1}^*. Hence, a concern for
   virtually all protocols involving elliptic curves is how to convert
   this input into a curve point.

   Note that the number of points on an elliptic curve E is within
   2*sqrt(p) of p by Hasse's Theorem.  As a rule of thumb, for every x
   in GF(p), there is approximately a 1/2 chance that there exist a
   corresponding y value such that (x, y) is on the curve E.  Since the
   point (x, -y) is also on the curve, then this sums to approximately p
   points.

   Ultimately, an encoding function takes a bitstring {0, 1}^* to an
   element of E, of order n (or q), and represented by variables in
   GF(p).

   Summary of quantities:

   +--------+-------------------+--------------------------------------+
   | Symbol | Meaning           | Relevance                            |
   +--------+-------------------+--------------------------------------+
   |   p    | Order of finite   | Curve points need to be represented  |
   |        | field, F = GF(p)  | in terms of p. For prime powers, we  |
   |        |                   | write F = GF(p^k).                   |
   |        |                   |                                      |
   |   n    | Number of curve   | For map to E, needs to produce n     |
   |        | points, #E(F) = n | elements.                            |
   |        |                   |                                      |
   |   q    | Order of prime    | If n is not prime, may need mapping  |
   |        | subgroup of E, n  | to q.                                |
   |        | = qh + r          |                                      |
   |        |                   |                                      |
   |   h    | Cofactor of prime | For mapping to subgroup, need to     |
   |        | subgroup          | multiply by cofactor.                |
   +--------+-------------------+--------------------------------------+

2.1.  Terminology

   In the following, we categorize the terminology for mapping between
   bitstrings and elliptic curves.

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2.1.1.  Encoding

   The general term "encoding" is used to refer to the process of
   producing an elliptic curve point given as input a bitstring.  In
   some protocols, the original message may also be recovered through a
   decoding procedure.  An encoding may be deterministic or
   probabilistic, although the latter is problematic in potentially
   leaking plaintext information as a side-channel.

   In most cases, the curve E is over a finite field GF(p^k), with p >
   2.  Suppose as the input to the encoding function we wish to use a
   fixed-length bitstring of length L.  Comparing sizes of the sets, 2^L
   and n, an encoding function cannot be both deterministic and
   bijective.

   We can instead use an injective encoding from {0, 1}^L to E, with "L
   < log2(n)- 1", which is a bijection over a subset of points in E.
   This ensures that encoded plaintext messages can be recovered.

2.1.2.  Serialization

   A related issue is the conversion of an elliptic curve point to a
   bitstring.  We refer to this process as "serialization", since it is
   typically used for compactly storing and transporting points, or for
   producing canonicalized outputs.  Since a deserialization algorithm
   can often be used as a type of encoding algorithm, we also briefly
   document properties of these functions.

   A naive serialization algorithm maps a point (x, y) on E to a
   bitstring of length 2*log(p), given that x, y are both elements in
   GF(p).  However, since there are only n points in E (with n
   approximately equal to p), it is possible to serialize to a bitstring
   of length log(n).  For example, one common method is to store the
   x-coordinate and a single bit to determine whether the point is (x,
   y) or (x, -y), thus requiring log(p)+1 bits.  Thus exchanging
   computation (recovering the y coordinate) for storage.

2.1.3.  Random Oracle

   It is often the case that the output of the encoding function
   Section 2.1.1 should be distributed uniformly at random on the
   elliptic curve.  That is, there is no discernible relation existing
   between outputs that can be computed based on the inputs.  In
   practice, this requirement stems from needing a random oracle which
   outputs elliptic curve points: one way to construct this is by first
   taking a regular random oracle, operating entirely on bitstrings, and
   applying a suitable encoding function to the output.

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   This motivates the term "hashing to the curve", since cryptographic
   hash functions are typically modeled as random oracles.  However,
   this still leaves open the question of what constitutes a suitable
   encoding method, which is a primary concern of this document.

   A random oracle onto an elliptic curve can also be instantiated using
   direct constructions, however these tend to rely on many group
   operations and are less efficient than hash and encode methods.

3.  Algorithm Recommendations

   The following table lists algorithms recommended by use-case:

   +----------------+-----------------+--------------------------------+
   | Application    | Requirement     | Additional Details             |
   +----------------+-----------------+--------------------------------+
   | SPEKE          | Naive           | H(x)*G                         |
   | [Jablon96]     |                 |                                |
   |                |                 |                                |
   | PAKE  [BMP00]  | Random Oracle   | -                              |
   |                |                 |                                |
   | BLS [BLS01]    | Random Oracle   | -                              |
   |                |                 |                                |
   | IBE [BF01]     | Random Oracle   | Supersingular, pairing-        |
   |                |                 | friendly curve                 |
   |                |                 |                                |
   | PRF            | Injective       | F(k, m) = k*H(m)               |
   |                | encoding        |                                |
   +----------------+-----------------+--------------------------------+

   To find the suitable algorithm, lookup the requirement from above,
   with the chosen curve in the below:

         +------------+--------------------------+---------------+
         | Curve      | Inj. Encoding            | Random Oracle |
         +------------+--------------------------+---------------+
         | P-256      | Simple SWU Section 5.2.3 | FFSTV(SWU)    |
         |            |                          |               |
         | P-384      | Icart Section 5.2.1      | FFSTV(Icart)  |
         |            |                          |               |
         | Curve25519 | Elligator2 Section 5.2.4 | ...           |
         |            |                          |               |
         | Curve448   | Elligator2 Section 5.2.4 | ...           |
         +------------+--------------------------+---------------+

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4.  Utility Functions

   Algorithms in this document make use of utility functions described
   below.

   o  HashToBase(x): H(x)[0:log2(p) + 1], i.e., hash-truncate-reduce,
      where H is a cryptographic hash function, such as SHA256, and p is
      the prime order of base field Fp.

   o  CMOV(a, b, c): If c = 1, return a, else return b.

   Note: We assume that HashToBase maps its input to the base field
   uniformly.  In practice, there may be inherent biases in p, e.g., p =
   2^k - 1 will have non-negligible bias in higher bits.

5.  Deterministic Encodings

5.1.  Interface

   The generic interface for deterministic encoding functions to
   elliptic curves is as follows:

   map2curve(alpha)

   where alpha is a message to encode on a curve.

5.2.  Encoding Variants

5.2.1.  Icart Method

   The following map2curve_icart(alpha) implements the Icart method from
   [Icart09].  This algorithm works for any curve over F_{p^n}, where
   p^n = 2 mod 3 (or p = 2 mod 3 and for odd n), including:

   o  P384

   o  Curve1174

   o  Curve448

   Unsupported curves include: P224, P256, P521, and Curve25519 since,
   for each, p = 1 mod 3.

   Mathematically, given input alpha, and A and B from E, the Icart
   method works as follows:

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   u = HashToBase(alpha)
   x = (v^2 - b - (u^6 / 27))^(1/3) + (u^2 / 3)
   y = ux + v

   where v = ((3A - u^4) / 6u).

   The following procedure implements this algorithm in a straight-line
   fashion.  It requires knowledge of A and B, the constants from the
   curve Weierstrass form.  It outputs a point with affine coordinates.

  map2curve_icart(alpha)

  Input:

    alpha - value to be hashed, an octet string

  Output:

    (x, y) - a point in E

  Steps:

  1.   u = HashToBase(alpha)   // {0,1}^* -> Fp
  2.  u2 = u^2 (mod p)         // u^2
  3.  t2 = u2^2 (mod p)        // u^4
  4.  v1 = 3 * A (mod p)       // 3A
  5.  v1 = v1 - t2 (mod p)     // 3A - u^4
  6.  t1 = 6 * u (mod p)       // 6u
  7.  t3 = t1 ^ (-1) (mod p)   // modular inverse
  8.   v = v1 * t3 (mod p)     // (3A - u^4)/(6u)
  9.   x = v^2 (mod p)         // v^2
  10.  x = x - B (mod p)       // v^2 - b
  11. t1 = 27 ^ (-1) (mod p)   // 1/27
  12. t1 = t1 * u2 (mod p)     // u^4 / 27
  13. t1 = t1 * t2 (mod p)     // u^6 / 27
  14.  x = x - t1 (mod p)      // v^2 - b - u^6/27
  15. t1 = (2 * p) - 1 (mod p) // 2p - 1
  16. t1 = t1 / 3 (mod p)      // (2p - 1)/3
  17.  x = x^t1 (mod p)        // (v^2 - b - u^6/27) ^ (1/3)
  18. t2 = u2 / 3 (mod p)      // u^2 / 3
  19.  x = x + t2 (mod p)      // (v^2 - b - u^6/27) ^ (1/3) + (u^2 / 3)
  20.  y = u * x (mod p)       // ux
  21.  y = y + v (mod p)       // ux + v
  22. Output (x, y)

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5.2.2.  Shallue-Woestijne-Ulas Method

   The Shallue-Woestijne-Ulas (SWU) method, originated in part by
   Shallue and Woestijne [SW06] and later simplified and extended by
   Ulas [SWU07], deterministically encodes an artbirary string to a
   point on a curve.  This algorithm works for any curve over F_{p^n}.
   Given curve equation g(x) = x^3 + Ax + B, two separate HashToBase
   implementations, H0 and H1, this algorithm works as follows:

   1. t = H0(alpha)
   2. u = H1(alpha)
   3. X1 = u
   4. X2 = (-B / A)(1 + 1 / (t^4 * g(u)^2 + t^2 * g(u)))
   5. X3 = t^3 * g(u)^2  * g(X2)
   6. If g(X1) is square, output (X1, sqrt(g(X1)))
   7. If g(X2) is square, output (X2, sqrt(g(X2)))
   8. Output (X3(t, u), sqrt(g(X3)))

   The algorithm relies on the following equality:

   t^3 * g(u)^2  * g(X2(t, u)) = g(X1(t, u)) * g(X2(t, u)) * g(X3(t, u))

   The algorithm computes three candidate points, constructed such that
   at least one of them lies on the curve.

   The following procedure implements this algorithm.  It outputs a
   point with affine coordinates.  It requires knowledge of A and B, the
   constants from the curve Weierstrass form.

   map2curve_squ(alpha)

   Input:

     alpha - value to be hashed, an octet string
     H0 - HashToBase implementation
     H1 - HashToBase implementation

   Output:

     (x, y) - a point in E

   Steps:

   1.    t = H0(alpha)   // {0,1}^* -> Fp
   2.    u = H1(alpha)   // {0,1}^* -> Fp
   3.   t2 = t^2
   4.   t4 = t2^2
   5.   gu = u^3

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   6.   gu = gu + (A * u)
   7.   gu = gu + B      // gu = g(u)
   8.   x1 = u           // x1 = X1(t, u) = u
   9.   x2 = B * -1
   10.  x2 = x2 / A
   11.  gx1 = x1^3
   12.  gx1 = gx1 + (A * x1)
   13.  gx1 = gx1 + B    // gx1 = g(X1(t, u))
   14.  d1 = gu^2
   15.  d1 = d1 * t4
   16.  d2 = t2 * gu
   17.  d3 = d1 + d2
   18.  d3 = d3^(-1)
   19.  n1 = 1 + d3
   20.  x2 = x2 * n1     // x2 = X2(t, u)
   21. gx2 = x2^3
   22. gx2 = gx2 + (A * x2)
   23. gx2 = gx2 + B     // gx2 = g(X2(t, u))
   24.  x3 = t2 * gu
   25.  x3 = x3 * x2     // x3 = X3(t, u)
   26. gx3 = x3^3
   27. gx3 = gx3 + (A * x3)
   28. gx3 = gx3 + B     // gx3 = g(X3(t, u))
   29.  l1 = gx1^((p - 1) / 2)
   30.  l2 = gx2^((p - 1) / 2)
   31.  s1 = gx1^(1/2)
   32.  s2 = gx2^(1/2)
   33.  s3 = gx3^(1/2)
   34. if l1 == 1:
   35.   Output (x1, s1)
   36. if l2 == 1:
   37.   Output (x2, s2)
   38. Output (x3, s3)

5.2.3.  Simplified SWU Method

   The following map2curve_simple_swu(alpha) implements the simplfied
   Shallue-Woestijne-Ulas algorithm from [SimpleSWU].  This algorithm
   works for any curve over F_{p^n}, where p = 3 mod 4, including:

   o  P256

   o  ...

   Given curve equation g(x) = x^3 + Ax + B, this algorithm works as
   follows:

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   1. t = HashToBase(alpha)
   2. alpha = (-b / a) * (1 + (1 / (t^4 + t^2)))
   3. beta = -t^2 * alpha
   4. If g(alpha) is square, output (alpha, sqrt(g(alpha)))
   5. Output (beta, sqrt(g(beta)))

   The following procedure implements this algorithm.  It outputs a
   point with affine coordinates.  It requires knowledge of A and B, the
   constants from the curve Weierstrass form.

   map2curve_simple_swu(alpha)

   Input:

     alpha - value to be encoded, an octet string

   Output:

     (x, y) - a point in E

   Steps:

   1.     t = HashToBase(alpha)
   2. alpha = t^2 (mod p)
   3. alpha = alpha * -1 (mod p)
   4. right = alpha^2 + alpha (mod p)
   5. right = right^(-1) (mod p)
   6. right = right + 1 (mod p)
   7.  left = B * -1 (mod p)
   8.  left = left / A (mod p)
   9.    x2 = left * right (mod p)
   10.   x3 = alpha * x2 (mod p)
   11.   h2 = x2 ^ 3 (mod p)
   12.   i2 = x2 * A (mod p)
   13.   i2 = i2 + B (mod p)
   14.   h2 = h2 + i2 (mod p)
   15.   h3 = x3 ^ 3 (mod p)
   16.   i3 = x3 * A (mod p)
   17.   i3 = i3 + B (mod p)
   18.   h3 = h3 + i3 (mod p)
   19.   y1 = h2 ^ ((p + 1) // 4) (mod p)
   20.   y2 = h3 ^ ((p + 1) // 4) (mod p)
   21.    e = (y1 ^ 2 == h2)
   22.    x = CMOV(x2, x3, e)    // If e = 1, choose x2, else choose x3
   23.    y = CMOV(y1, y2, e)    // If e = 1, choose y1, else choose y2
   24. Output (x, y)

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5.2.4.  Elligator2 Method

   The following map2curve_elligator2(alpha) implements the Elligator2
   method from [Elligator2].  This algorithm works for any curve with a
   point of order 2 and j-invariant != 1728.  Given curve equation f(x)
   = y^2 = x(x^2 + Ax + B), i.e., a Montgomery form with the point of
   order 2 at (0,0), this algorithm works as shown below.  (Note that
   any curve with a point of order 2 is isomorphic to this
   representation.)

   1. r = HashToBase(alpha)
   2. If f(-A/(1+ur^2)) is square, then output f(-A/(1+ur^2))^(1/2)
   3. Else, output f(-Aur^2/(1+ur^2))^(1/2)

   Another way to express this algorithm is as follows:

   1. r = HashToBase(alpha)
   2. d = -A / (1 + ur^2)
   3. e = f(d)^((p-1)/2)
   4. u = ed - (1 - e)A/u

   Here, e is the Legendre symbol of y = (d^3 + Ad^2 + d), which will be
   1 if y is a quadratic residue (square) mod p, and -1 otherwise.
   (Note that raising y to ((p -1) / 2) is a common way to compute the
   Legendre symbol.)

   The following procedure implements this algorithm.

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   map2curve_elligator2(alpha)

   Input:

     alpha - value to be encoded, an octet string

     u - fixed non-square value in Fp.
     f() - Curve function

   Output:

     (x, y) - a point in E

   Steps:

   1.   r = HashToBase(alpha)
   2.   r = r^2 (mod p)
   3.  nu = r * u (mod p)
   4.   r = nu
   5.   r = r + 1 (mod p)
   6.   r = r^(-1) (mod p)
   7.   v = A * r (mod p)
   8.   v = v * -1 (mod p)   // -A / (1 + ur^2)
   9.  v2 = v^2 (mod p)
   10. v3 = v * v2 (mod p)
   11.  e = v3 * v (mod p)
   12. v2 = v2 * A (mod p)
   13.  e = v2 * e (mod p)
   14.  e = e^((p - 1) / 2)  // Legendre symbol
   15. nv = v * -1 (mod p)
   16.  v = CMOV(v, nv, e)   // If e = 1, choose v, else choose nv
   17. v2 = CMOV(0, A, e)    // If e = 1, choose 0, else choose A
   18.  u = v - v2 (mod p)
   19. Output (u, f(u))

   Elligator2 can be simplified with projective coordinates.

   ((TODO: write this variant))

5.3.  Cost Comparison

   The following table summarizes the cost of each map2curve variant.
   We express this cost in terms of additions (A), multiplications (M),
   squares (SQ), and square roots (SR).

   ((TODO: finish this section))

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               +----------------------+-------------------+
               | Algorithm            | Cost (Operations) |
               +----------------------+-------------------+
               | map2curve_icart      | TODO              |
               |                      |                   |
               | map2curve_swu        | TODO              |
               |                      |                   |
               | map2curve_simple_swu | TODO              |
               |                      |                   |
               | map2curve_elligator2 | TODO              |
               +----------------------+-------------------+

6.  Random Oracles

6.1.  Interface

   The generic interface for deterministic encoding functions to
   elliptic curves is as follows:

   hash2curve(alpha)

   where alpha is a message to encode on a curve.

6.2.  General Construction (FFSTV13)

   When applications need a Random Oracle (RO), they can be constructed
   from deterministic encoding functions.  In particular, let F :
   {0,1}^* -> E be a deterministic encoding function onto curve E, and
   let H0 and H1 be two hash functions modeled as random oracles that
   map input messages to the base field of E, i.e., Z_q.  Farashahi et
   al.  [FFSTV13] showed that the following mapping is indistinguishable
   from a RO:

   hash2curve(alpha) = F(H0(alpha)) + F(H1(alpha))

   This construction works for the Icart, SWU, and Simplfied SWU
   encodings.

   Here, H0 and H1 could be constructed as follows:

   H0(alpha) = HashToBase(0 || alpha)
   H1(alpha) = HashToBase(1 || alpha)

7.  Curve Transformations

   ((TODO: write this section))

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8.  IANA Considerations

   This document has no IANA actions.

9.  Security Considerations

   Each encoding function variant accepts arbitrary input and maps it to
   a pseudorandom point on the curve.  Points are close to
   indistinguishable from randomly chosen elements on the curve.  Not
   all encoding functions are full-domain hashes.  Elligator2, for
   example, only maps strings to "about half of all curve points,"
   whereas Icart's method only covers about 5/8 of the points.

10.  Acknowledgements

   The authors would like to thank Adam Langley for this detailed
   writeup up Elligator2 with Curve25519 [ElligatorAGL].  We also thank
   Sean Devlin and Thomas Icart for feedback on earlier versions of this
   document.

11.  Contributors

   o  Sharon Goldberg
      Boston University
      goldbe@cs.bu.edu

12.  Normative References

   [BF01]     "Identity-based encryption from the Weil pairing", n.d..

   [BLS01]    "Short signatures from the Weil pairing", n.d.,
              <https://iacr.org/archive/asiacrypt2001/22480516.pdf>.

   [BMP00]    "Provably secure password-authenticated key exchange using
              diffie-hellman", n.d..

   [ECOPRF]   "EC-OPRF - Oblivious Pseudorandom Functions using Elliptic
              Curves", n.d..

   [Elligator2]
              "Elligator -- Elliptic-curve points indistinguishable from
              uniform random strings", n.d., <https://dl.acm.org/
              ft_gateway.cfm?id=2516734&type=pdf>.

   [ElligatorAGL]
              "Implementing Elligator for Curve25519", n.d.,
              <https://www.imperialviolet.org/2013/12/25/
              elligator.html>.

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   [FFSTV13]  "Indifferentiable deterministic hashing to elliptic and
              hyperelliptic curves", n.d..

   [hacspec]  "hacspec", n.d., <https://github.com/HACS-workshop/
              hacspec>.

   [Icart09]  "How to Hash into Elliptic Curves", n.d.,
              <https://eprint.iacr.org/2009/226.pdf>.

   [Jablon96]
              "Strong password-only authenticated key exchange", n.d..

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997, <https://www.rfc-
              editor.org/info/rfc2119>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/info/rfc7748>.

   [RFC8017]  Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
              "PKCS #1: RSA Cryptography Specifications Version 2.2",
              RFC 8017, DOI 10.17487/RFC8017, November 2016,
              <https://www.rfc-editor.org/info/rfc8017>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017, <https://www.rfc-
              editor.org/info/rfc8032>.

   [SECG1]    "SEC 1 -- Elliptic Curve Cryptography", n.d.,
              <http://www.secg.org/sec1-v2.pdf>.

   [SimpleSWU]
              "Efficient Indifferentiable Hashing into Ordinary Elliptic
              Curves", n.d..

   [SW06]     "Construction of rational points on elliptic curves over
              finite fields", n.d..

   [SWU07]    "Rational points on certain hyperelliptic curves over
              finite fields", n.d., <https://arxiv.org/pdf/0706.1448>.

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Appendix A.  Related Work

   In this chapter, we give a background to some common methods to
   encode or hash to the curve, motivated by the similar exposition in
   [Icart09].  Understanding of this material is not required in order
   to choose a suitable encoding function - we defer this to Section 3 -
   the background covered here can work as a template for analyzing
   encoding functions not found in this document, and as a guide for
   further research into the topics covered.

A.1.  Probabilistic Encoding

   As mentioned in Section 2, as a rule of thumb, for every x in GF(p),
   there is approximately a 1/2 chance that there exist a corresponding
   y value such that (x, y) is on the curve E.

   This motivates the construction of the MapToGroup method described by
   Boneh et al.  [BLS01].  For an input message m, a counter i, and a
   standard hash function H : {0, 1}^* -> GF(p) x {0, 1}, one computes
   (x, b) = H(i || m), where i || m denotes concatenation of the two
   values.  Next, test to see whether there exists a corresponding y
   value such that (x, y) is on the curve, returning (x, y) if
   successful, where b determines whether to take +/- y.  If there does
   not exist such a y, then increment i and repeat.  A maximum counter
   value is set to I, and since each iteration succeeds with probability
   approximately 1/2, this process fails with probability 2^-I.  (See
   Appendix B for a more detailed description of this algorithm.)

   Although MapToGroup describes a method to hash to the curve, it can
   also be adapted to a simple encoding mechanism.  For a bitstring of
   length strictly less than log2(p), one can make use of the spare bits
   in order to encode the counter value.  Allocating more space for the
   counter increases the expansion, but reduces the failure probability.

   Since the running time of the MapToGroup algorithm depends on m, this
   algorithm is NOT safe for cases sensitive to timing side channel
   attacks.  Deterministic algorithms are needed in such cases where
   failures are undesirable.

A.2.  Naive Encoding

   A naive solution includes computing H(m)*G as map2curve(m), where H
   is a standard hash function H : {0, 1}^* -> GF(p), and G is a
   generator of the curve.  Although efficient, this solution is
   unsuitable for constructing a random oracle onto E, since the
   discrete logarithm with respect to G is known.  For example, given y1
   = map2curve(m1) and y2 = map2curve(m2) for any m1 and m2, it must be
   true that y2 = H(m2) / H(m1) * map2curve(m1).  This relationship

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   would not hold (with overwhelming probability) for truly random
   values y1 and y2.  This causes catastrophic failure in many cases.
   However, one exception is found in SPEKE [Jablon96], which constructs
   a base for a Diffie-Hellman key exchange by hashing the password to a
   curve point.  Notably the use of a hash function is purely for
   encoding an arbitrary length string to a curve point, and does not
   need to be a random oracle.

A.3.  Deterministic Encoding

   Shallue, Woestijne, and Ulas [SW06] first introduced a deterministic
   algorithm that maps elements in F_{q} to a curve in time O(log^4 q),
   where q = p^n for some prime p, and time O(log^3 q) when q = 3 mod 4.
   Icart introduced yet another deterministic algorithm which maps F_{q}
   to any EC where q = 2 mod 3 in time O(log^3 q) [Icart09].  Elligator
   (2) [Elligator2] is yet another deterministic algorithm for any odd-
   characteristic EC that has a point of order 2.  Elligator2 can be
   applied to Curve25519 and Curve448, which are both CFRG-recommended
   curves [RFC7748].

   However, an important caveat to all of the above deterministic
   encoding functions, is that none of them map injectively to the
   entire curve, but rather some fraction of the points.  This makes
   them unable to use to directly construct a random oracle on the
   curve.

   Brier et al.  [SimpleSWU] proposed a couple of solutions to this
   problem, The first applies solely to Icart's method described above,
   by computing F(H1(m)) + F(H2(m)) for two distinct hash functions H1,
   H2.  The second uses a generator G, and computes F(H1(m)) + H2(m)*G.
   Later, Farashahi et al.  [FFSTV13] showed the generality of the
   F(H1(m)) + F(H2(m)) method, as well as the applicability to
   hyperelliptic curves (not covered here).

A.4.  Supersingular Curves

   For supersingular curves, for every y in GF(p) (with p>3), there
   exists a value x such that (x, y) is on the curve E.  Hence we can
   construct a bijection F : GF(p) -> E (ignoring the point at
   infinity).  This is the case for [BF01], but is not common.

A.5.  Twisted Variants

   We can also consider curves which have twisted variants, E^d.  For
   such curves, for any x in GF(p), there exists y in GF(p) such that
   (x, y) is either a point on E or E^d.  Hence one can construct a
   bijection F : GF(p) x {0,1} -> E &#8746; E^d, where the extra bit is
   needed to choose the sign of the point.  This can be particularly

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   useful for constructions which only need the x-coordinate of the
   point.  For example, x-only scalar multiplication can be computed on
   Montgomery curves.  In this case, there is no need for an encoding
   function, since the output of F in GF(p) is sufficient to define a
   point on one of E or E^d.

Appendix B.  Try-and-Increment Method

   In cases where constant time execution is not required, the so-called
   try-and-increment method may be appropriate.  As discussion in
   Section Section 1, this variant works by hashing input m using a
   standard hash function ("Hash"), e.g., SHA256, and then checking to
   see if the resulting point E(m, f(m)), for curve function f, belongs
   on E.  This is detailed below.

   1. ctr = 0
   3. h = "INVALID"
   4. While h is "INVALID" or h is EC point at infinity:
      A.  CTR = I2OSP(ctr, 4)
      B.  ctr = ctr + 1
      C.  attempted_hash = Hash(m || CTR)
      D.  h = RS2ECP(attempted_hash)
      E.  If h is not "INVALID" and cofactor > 1, set h = h^cofactor
   5. Output h

   I2OSP is a function that converts a nonnegative integer to octet
   string as defined in Section 4.1 of [RFC8017], and RS2ECP is a
   function that converts of a random 2n-octet string to an EC point as
   specified in Section 5.1.3 of [RFC8032].

Appendix C.  Sample Code

   This section contains reference implementations for each map2curve
   variant built using [hacspec].

C.1.  Icart Method

   The following hacspec program implements map2curve_icart(alpha) for
   P-384.

from hacspec.speclib import *

prime = 2**384 - 2**128 - 2**96 + 2**32 - 1

felem_t = refine(nat, lambda x: x < prime)
affine_t = tuple2(felem_t, felem_t)

@typechecked

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def to_felem(x: nat_t) -> felem_t:
    return felem_t(nat(x % prime))

@typechecked
def fadd(x: felem_t, y: felem_t) -> felem_t:
    return to_felem(x + y)

@typechecked
def fsub(x: felem_t, y: felem_t) -> felem_t:
    return to_felem(x - y)

@typechecked
def fmul(x: felem_t, y: felem_t) -> felem_t:
    return to_felem(x * y)

@typechecked
def fsqr(x: felem_t) -> felem_t:
    return to_felem(x * x)

@typechecked
def fexp(x: felem_t, n: nat_t) -> felem_t:
    return to_felem(pow(x, n, prime))

@typechecked
def finv(x: felem_t) -> felem_t:
    return to_felem(pow(x, prime-2, prime))

a384 = to_felem(prime - 3)
b384 = to_felem(27580193559959705877849011840389048093056905856361568521428707301988689241309860865136260764883745107765439761230575)

@typechecked
def map2p384(u:felem_t) -> affine_t:
    v = fmul(fsub(fmul(to_felem(3), a384), fexp(u, 4)), finv(fmul(to_felem(6), u)))
    u2 = fmul(fexp(u, 6), finv(to_felem(27)))
    x = fsub(fsqr(v), b384)
    x = fsub(x, u2)
    x = fexp(x, (2 * prime - 1) // 3)
    x = fadd(x, fmul(fsqr(u), finv(to_felem(3))))
    y = fadd(fmul(u, x), v)
    return (x, y)

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C.2.  Shallue-Woestijne-Ulas Method

   The following hacspec program implements map2curve_swu(alpha) for
   P-256.

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from p256 import *
from hacspec.speclib import *

a256 = to_felem(prime - 3)
b256 = to_felem(41058363725152142129326129780047268409114441015993725554835256314039467401291)

@typechecked
def f_p256(x:felem_t) -> felem_t:
    return fadd(fexp(x, 3), fadd(fmul(to_felem(a256), x), to_felem(b256)))

@typechecked
def x1(t:felem_t, u:felem_t) -> felem_t:
    return u

@typechecked
def x2(t:felem_t, u:felem_t) -> felem_t:
    coefficient = fmul(to_felem(-b256), finv(to_felem(a256)))
    t2 = fsqr(t)
    t4 = fsqr(t2)
    gu = f_p256(u)
    gu2 = fsqr(gu)
    denom = fadd(fmul(t4, gu2), fmul(t2, gu))
    return fmul(coefficient, fadd(to_felem(1), finv(denom)))

@typechecked
def x3(t:felem_t, u:felem_t) -> felem_t:
    return fmul(fsqr(t), fmul(f_p256(u), x2(t, u)))

@typechecked
def map2p256(t:felem_t) -> felem_t:
    u = fadd(t, to_felem(1))
    x1v = x1(t, u)
    x2v = x2(t, u)
    x3v = x3(t, u)

    exp = to_felem((prime - 1) // 2)
    e1 = fexp(f_p256(x1v), exp)
    e2 = fexp(f_p256(x2v), exp)

    if e1 == 1:
        return x1v
    elif e2 == 1:
        return x2v
    else:
        return x3v

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C.3.  Simplified SWU Method

   The following hacspec program implements map2curve_simple_swu(alpha)
   for P-256.

from p256 import *
from hacspec.speclib import *

a256 = to_felem(prime - 3)
b256 = to_felem(41058363725152142129326129780047268409114441015993725554835256314039467401291)

def f_p256(x:felem_t) -> felem_t:
    return fadd(fexp(x, 3), fadd(fmul(to_felem(a256), x), to_felem(b256)))

def map2p256(t:felem_t) -> affine_t:
    alpha = to_felem(-(fsqr(t)))
    frac = finv((fadd(fsqr(alpha), alpha)))
    coefficient = fmul(to_felem(-b256), finv(to_felem(a256)))
    x2 = fmul(coefficient, fadd(to_felem(1), frac))

    x3 = fmul(alpha, x2)
    h2 = fadd(fexp(x2, 3), fadd(fmul(a256, x2), b256))
    h3 = fadd(fexp(x3, 3), fadd(fmul(a256, x3), b256))

    exp = fmul(fadd(to_felem(prime), to_felem(-1)), finv(to_felem(2)))
    e = fexp(h2, exp)

    exp = to_felem((prime + 1) // 4)
    if e == 1:
      return (x2, fexp(f_p256(x2), exp))
    else:
      return (x3, fexp(f_p256(x3), exp))

C.4.  Elligator2 Method

   The following hacspec program implements map2curve_elligator2(alpha)
   for Curve25519.

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from curve25519 import *
from hacspec.speclib import *

a25519 = to_felem(486662)
b25519 = to_felem(1)
u25519 = to_felem(2)

@typechecked
def f_25519(x:felem_t) -> felem_t:
    return fadd(fmul(x, fsqr(x)), fadd(fmul(a25519, fsqr(x)), x))

@typechecked
def map2curve25519(r:felem_t) -> felem_t:
    d = fsub(to_felem(p25519), fmul(a25519, finv(fadd(to_felem(1), fmul(u25519, fsqr(r))))))
    power = nat((p25519 - 1) // 2)
    e = fexp(f_25519(d), power)
    x = 0
    if e != 1:
        x = fsub(to_felem(-d), to_felem(a25519))
    else:
        x = d

    return x

Authors' Addresses

   Sam Scott
   Cornell Tech
   2 West Loop Rd
   New York, New York 10044
   United States of America

   Email: sam.scott@cornell.edu

   Nick Sullivan
   Cloudflare
   101 Townsend St
   San Francisco
   United States of America

   Email: nick@cloudflare.com

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   Christopher A. Wood
   Apple Inc.
   One Apple Park Way
   Cupertino, California 95014
   United States of America

   Email: cawood@apple.com

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