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Versions: (draft-ladd-spake2) 00 01 02 03 04 05 06 07 08

Network Working Group                                            W. Ladd
Internet-Draft                                               UC Berkeley
Intended status: Informational                             B. Kaduk, Ed.
Expires: September 12, 2019                                       Akamai
                                                          March 11, 2019


                             SPAKE2, a PAKE
                       draft-irtf-cfrg-spake2-08

Abstract

   This document describes SPAKE2 and its augmented variant SPAKE2+,
   which are protocols for two parties that share a password to derive a
   strong shared key with no risk of disclosing the password.  This
   method is compatible with any prime order group, is computationally
   efficient, and SPAKE2 (but not SPAKE2+) has a security proof.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on September 12, 2019.

Copyright Notice

   Copyright (c) 2019 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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   include Simplified BSD License text as described in Section 4.e of




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   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Notation . . . . . . . . . . . . . . . . . . . .   2
   3.  Definition of SPAKE2  . . . . . . . . . . . . . . . . . . . .   2
   4.  Key Schedule and Key Confirmation . . . . . . . . . . . . . .   5
   5.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . . . .   6
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .   9
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   9
   8.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .   9
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   9
   Appendix A.  Algorithm used for Point Generation  . . . . . . . .  11
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  13

1.  Introduction

   This document describes SPAKE2, a means for two parties that share a
   password to derive a strong shared key with no risk of disclosing the
   password.  This password-based key exchange protocol is compatible
   with any group (requiring only a scheme to map a random input of
   fixed length per group to a random group element), is computationally
   efficient, and has a security proof.  Predetermined parameters for a
   selection of commonly used groups are also provided for use by other
   protocols.

2.  Requirements Notation

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in BCP
   14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

3.  Definition of SPAKE2

3.1.  Setup

   Let G be a group in which the computational Diffie-Hellman (CDH)
   problem is hard.  Suppose G has order p*h where p is a large prime; h
   will be called the cofactor.  Let I be the unit element in G, e.g.,
   the point at infinity if G is an elliptic curve group.  We denote the
   operations in the group additively.  We assume there is a
   representation of elements of G as byte strings: common choices would
   be SEC1 compressed [SEC1] for elliptic curve groups or big endian
   integers of a fixed (per-group) length for prime field DH.  We fix



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   two elements M and N in the prime-order subgroup of G as defined in
   the table in this document for common groups, as well as a generator
   P of the (large) prime-order subgroup of G.  P is specified in the
   document defining the group, and so we do not repeat it here.

   || denotes concatenation of strings.  We also let len(S) denote the
   length of a string in bytes, represented as an eight-byte little-
   endian number.  Finally, let nil represent an empty string, i.e.,
   len(nil) = 0.

   KDF is a key-derivation function that takes as input a salt,
   intermediate keying material (IKM), info string, and derived key
   length L to derive a cryptographic key of length L.  MAC is a Message
   Authentication Code algorithm that takes a secret key and message as
   input to produce an output.  Let Hash be a hash function from
   arbitrary strings to bit strings of a fixed length.  Common choices
   for H are SHA256 or SHA512 [RFC6234].  Let MHF be a memory-hard hash
   function designed to slow down brute-force attackers.  Scrypt
   [RFC7914] is a common example of this function.  The output length of
   MHF matches that of Hash.  Parameter selection for MHF is out of
   scope for this document.  Section 5 specifies variants of KDF, MAC,
   Hash, and MHF suitable for use with the protocols contained herein.

   Let A and B be two parties.  A and B may also have digital
   representations of the parties' identities such as Media Access
   Control addresses or other names (hostnames, usernames, etc).  A and
   B may share Additional Authenticated Data (AAD) of length at most
   2^16 - 1 bits that is separate from their identities which they may
   want to include in the protocol execution.  One example of AAD is a
   list of supported protocol versions if SPAKE2(+) were used in a
   higher-level protocol which negotiates use of a particular PAKE.
   Including this list would ensure that both parties agree upon the
   same set of supported protocols and therefore prevent downgrade
   attacks.  We also assume A and B share an integer w; typically w =
   MHF(pw) mod p, for a user-supplied password pw.  Standards such
   NIST.SP.800-56Ar3 suggest taking mod p of a hash value that is 64
   bits longer than that needed to represent p to remove statistical
   bias introduced by the modulation.  Protocols using this
   specification must define the method used to compute w: it may be
   necessary to carry out various forms of normalization of the password
   before hashing [RFC8265].  The hashing algorithm SHOULD be a MHF so
   as to slow down brute-force attackers.

   We present two protocols below.  Note that it is insecure to use the
   same password with both protocols; passwords MUST NOT be used for
   both SPAKE2 and SPAKE2+.





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3.2.  SPAKE2

   To begin, A picks x randomly and uniformly from the integers in
   [0,p), and calculates X=x*P and T=w*M+X, then transmits T to B.  Upon
   receipt of T, B computes T*h and aborts if the result is equal to I.
   (This ensures T is in the prime order subgroup of G.)

   B selects y randomly and uniformly from the integers in [0,p), and
   calculates Y=y*P, S=w*N+Y, then transmits S to A.  Upon receipt of S,
   A computes S*h and aborts if the result is equal to I.

   Both A and B calculate a group element K.  A calculates it as
   x*(S-wN), while B calculates it as y*(T-w*M).  A knows S because it
   has received it, and likewise B knows T.  A and B multiply protocol
   messages from each peer by h so as to avoid small subgroup attacks,
   but the result of the multiplication is not used for operations other
   than the comparison against I and the non-multiplied value is used in
   subsequent calculations.

   K is a shared value, though it MUST NOT be used as a shared secret.
   Both A and B must derive two shared secrets from K and the protocol
   transcript.  This prevents man-in-the-middle attackers from inserting
   themselves into the exchange.  The transcript TT is encoded as
   follows:

           TT = len(A) || A || len(B) || B || len(S) || S || len(T) || T
                || len(K) || K || len(w) || w

   If an identity is absent, it is omitted from the transcript entirely.
   For example, if both A and B are absent, then TT = len(S) || S ||
   len(T) || T || len(K) || K || len(w) || w.  Likewise, if only A is
   absent, TT = len(B) || B || len(S) || S || len(T) || T || len(K) ||
   K || len(w) || w.  This must only be done for applications in which
   identities are implicit.  Otherwise, the protocol risks Unknown Key
   Share attacks (discussion of Unknown Key Share attacks in a specific
   protocl is given in [I-D.ietf-mmusic-sdp-uks].

   Upon completion of this protocol, A and B compute shared secrets Ke,
   KcA, and KcB as specified in Section 4.  A MUST send B a key
   confirmation message so both parties agree upon these shared secrets.
   This confirmation message F is computed as a MAC over the protocol
   transcript TT using KcA, as follows: F = MAC(KcA, TT).  Similarly, B
   MUST send A a confirmation message using a MAC computed equivalently
   except with the use of KcB.  Key confirmation verification requires
   computing F and checking for equality against that which was
   received.





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3.3.  SPAKE2+

   This protocol appears in [TDH].  We use the same setup as for SPAKE2,
   except that we have two secrets, w0 and w1, derived by hashing the
   password pw with the identities of the two participants, A and B.
   Specifically, w0s || w1s = MHF(len(pw) || pw || len(A) || A ||
   len(B) || B), and then computing w0 = w0s mod p and w1 = w1s mod p.
   The length of each of w0s and w1s is equal to half of the MHF output,
   e.g., |w0s| = |w1s| = 128 bits for scrypt.  w0 and w1 MUST NOT equal
   I.  If they are, they MUST be iteratively regenerated by computing
   w0s || w1s = MHF(len(pw) || pw || len(A) || A || len(B) || B ||
   0x0000), where 0x0000 is 16-bit increasing counter.  This process
   must repeat until valid w0 and w1 are produced.  B stores L=w1*P and
   w0.

   When executing SPAKE2+, A selects x uniformly at random from the
   numbers in the range [0, p), and lets X=x*P+w0*M, then transmits X to
   B.  Upon receipt of X, A computes h*X and aborts if the result is
   equal to I.  B then selects y uniformly at random from the numbers in
   [0, p), then computes Y=y*P+w0*N, and transmits Y to A.  Upon receipt
   of Y, A computes Y*h and aborts if the result is equal to I.

   A computes Z as x*(Y-w0*N), and V as w1*(Y-w0*N).  B computes Z as
   y*(X- w0*M) and V as y*L.  Both share Z and V as common keys.  It is
   essential that both Z and V be used in combination with the
   transcript to derive the keying material.  The protocol transcript
   encoding is shown below.

            TT = len(A) || A || len(B) || B || len(X) || X || len(Y) || Y
             || len(Z) || Z || len(V) || V || len(w0) || w0

   As in Section 3.2, inclusion of A and B in the transcript is optional
   depending on whether or not the identities are implicit.

   Upon completion of this protocol, A and B follow the same key
   derivation and confirmation steps as outlined in Section 3.2.

4.  Key Schedule and Key Confirmation

   The protocol transcript TT, as defined in Sections Section 3.3 and
   Section 3.2, is unique and secret to A and B.  Both parties use TT to
   derive shared symmetric secrets Ke and Ka as Ke || Ka = Hash(TT).
   The length of each key is equal to half of the digest output,
   e.g., |Ke| = |Ka| = 128 bits for SHA-256.

   Both endpoints use Ka to derive subsequent MAC keys for key
   confirmation messages.  Specifically, let KcA and KcB be the MAC keys
   used by A and B, respectively.  A and B compute them as KcA || KcB =



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   KDF(nil, Ka, "ConfirmationKeys" || AAD), where AAD is the associated
   data each given to each endpoint, or nil if none was provided.  The
   length of each of KcA and KcB is equal to half of the KDF output,
   e.g., |KcA| = |KcB| = 128 bits for HKDF(SHA256).

   The resulting key schedule for this protocol, given transcript TT and
   additional associated data AAD, is as follows.

           TT   -> Hash(TT) = Ke || Ka
           AAD -> KDF(nil, Ka, "ConfirmationKeys" || AAD) = KcA || KcB

   A and B output Ke as the shared secret from the protocol.  Ka and its
   derived keys are not used for anything except key confirmation.

5.  Ciphersuites

   This section documents SPAKE2 and SPAKE2+ ciphersuite configurations.
   A ciphersuite indicates a group, cryptographic hash algorithm, and
   pair of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC.
   This ciphersuite indicates a SPAKE2 protocol instance over P-256 that
   uses SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash,
   KDF, and MAC functions, respectively.





























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   +--------------+-----------+-----------+---------------+------------+
   |      G       |    Hash   |    KDF    |      MAC      |    MHF     |
   +--------------+-----------+-----------+---------------+------------+
   |    P-256     |   SHA256  |    HKDF   |      HMAC     |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-256     |   SHA512  |    HKDF   |      HMAC     |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-384     |   SHA256  |    HKDF   |      HMAC     |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-384     |   SHA512  |    HKDF   |      HMAC     |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-512     |   SHA512  |    HKDF   |      HMAC     |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   | edwards25519 |   SHA256  |    HKDF   |      HMAC     |   scrypt   |
   |  [RFC7748]   | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |  edwards448  |   SHA512  |    HKDF   |      HMAC     |   scrypt   |
   |  [RFC7748]   | [RFC6234] | [RFC5869] |   [RFC2104]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-256     |   SHA256  |    HKDF   |  CMAC-AES-128 |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC4493]   | [RFC7914]  |
   |              |           |           |               |            |
   |    P-256     |   SHA512  |    HKDF   |  CMAC-AES-128 |   scrypt   |
   |              | [RFC6234] | [RFC5869] |   [RFC4493]   | [RFC7914]  |
   +--------------+-----------+-----------+---------------+------------+

                      Table 1: SPAKE2(+) Ciphersuites

   The following points represent permissible point generation seeds for
   the groups listed in the Table Table 1, using the algorithm presented
   in Appendix A.  These bytestrings are compressed points as in [SEC1]
   for curves from [SEC1].

   For P256:

   M =
   02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
   seed: 1.2.840.10045.3.1.7 point generation seed (M)

   N =
   03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
   seed: 1.2.840.10045.3.1.7 point generation seed (N)




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   For P384:

   M =
   030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
   36f15314739074d2eb8613fceec2853
   seed: 1.3.132.0.34 point generation seed (M)

   N =
   02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
   252c5490214cf9aa3f0baab4b665c10
   seed: 1.3.132.0.34 point generation seed (N)

   For P521:

   M =
   02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
   cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
   seed: 1.3.132.0.35 point generation seed (M)

   N =
   0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
   32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
   seed: 1.3.132.0.35 point generation seed (N)

   For edwards25519:

   M =
   d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
   seed: edwards25519 point generation seed (M)

   N =
   d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
   seed: edwards25519 point generation seed (N)

   For edwards448:

   M =
   b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
   942f5a92646109152292464f3e63d354701c7848d9fc3b8880
   seed: edwards448 point generation seed (M)

   N =
   6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
   f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
   seed: edwards448 point generation seed (N)






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6.  Security Considerations

   A security proof of SPAKE2 for prime order groups is found in [REF].
   Note that the choice of M and N is critical for the security proof.
   The generation method specified in this document is designed to
   eliminate concerns related to knowing discrete logs of M and N.

   SPAKE2+ appears in [TDH] along with a path to a proof that server
   compromise does not lead to password compromise under the DH
   assumption (though the corresponding model excludes precomputation
   attacks).

   Elements received from a peer MUST be checked for group membership:
   failure to properly validate group elements can lead to attacks.
   Beyond the cofactor multiplication checks to ensure that these
   elements are in the prime order subgroup of G, it is essential that
   endpoints verify received points are members of G.

   The choices of random numbers MUST BE uniform.  Randomly generated
   values (e.g., x and y) MUST NOT be reused; such reuse may permit
   dictionary attacks on the password.

   SPAKE2 does not support augmentation.  As a result, the server has to
   store a password equivalent.  This is considered a significant
   drawback, and so SPAKE2+ also appears in this document.

7.  IANA Considerations

   No IANA action is required.

8.  Acknowledgments

   Special thanks to Nathaniel McCallum and Greg Hudson for generation
   of test vectors.  Thanks to Mike Hamburg for advice on how to deal
   with cofactors.  Greg Hudson also suggested the addition of warnings
   on the reuse of x and y.  Thanks to Fedor Brunner, Adam Langley, and
   the members of the CFRG for comments and advice.  Chris Wood
   contributed substantial text and reformatting to address the
   excellent review comments from Kenny Paterson.  Trevor Perrin
   informed me of SPAKE2+.

9.  References

9.1.  Normative References







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   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,
              <https://www.rfc-editor.org/info/rfc2104>.

   [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>.

   [RFC4493]  Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
              AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
              2006, <https://www.rfc-editor.org/info/rfc4493>.

   [RFC5480]  Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
              "Elliptic Curve Cryptography Subject Public Key
              Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,
              <https://www.rfc-editor.org/info/rfc5480>.

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,
              <https://www.rfc-editor.org/info/rfc5869>.

   [RFC6234]  Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
              (SHA and SHA-based HMAC and HKDF)", RFC 6234,
              DOI 10.17487/RFC6234, May 2011,
              <https://www.rfc-editor.org/info/rfc6234>.

   [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>.

   [RFC7914]  Percival, C. and S. Josefsson, "The scrypt Password-Based
              Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
              August 2016, <https://www.rfc-editor.org/info/rfc7914>.

   [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>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [SEC1]     SEC, "STANDARDS FOR EFFICIENT CRYPTOGRAPHY, "SEC 1:
              Elliptic Curve Cryptography", version 2.0", May 2009.



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9.2.  Informative References

   [I-D.ietf-mmusic-sdp-uks]
              Thomson, M. and E. Rescorla, "Unknown Key Share Attacks on
              uses of TLS with the Session Description Protocol (SDP)",
              draft-ietf-mmusic-sdp-uks-03 (work in progress), January
              2019.

   [REF]      Abdalla, M. and D. Pointcheval, "Simple Password-Based
              Encrypted Key Exchange Protocols.", Feb 2005.

              Appears in A.  Menezes, editor.  Topics in Cryptography-
              CT-RSA 2005, Volume 3376 of Lecture Notes in Computer
              Science, pages 191-208, San Francisco, CA, US.  Springer-
              Verlag, Berlin, Germany.

   [RFC8265]  Saint-Andre, P. and A. Melnikov, "Preparation,
              Enforcement, and Comparison of Internationalized Strings
              Representing Usernames and Passwords", RFC 8265,
              DOI 10.17487/RFC8265, October 2017,
              <https://www.rfc-editor.org/info/rfc8265>.

   [TDH]      Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie
              Hellman Problem and Applications", 2008.

              EUROCRYPT 2008.  Volume 4965 of Lecture notes in Computer
              Science, pages 127-145.  Springer-Verlag, Berlin, Germany.

Appendix A.  Algorithm used for Point Generation

   This section describes the algorithm that was used to generate the
   points (M) and (N) in the table in Section 5.

   For each curve in the table below, we construct a string using the
   curve OID from [RFC5480] (as an ASCII string) or its name, combined
   with the needed constant, for instance "1.3.132.0.35 point generation
   seed (M)" for P-512.  This string is turned into a series of blocks
   by hashing with SHA256, and hashing that output again to generate the
   next 32 bytes, and so on.  This pattern is repeated for each group
   and value, with the string modified appropriately.

   A byte string of length equal to that of an encoded group element is
   constructed by concatenating as many blocks as are required, starting
   from the first block, and truncating to the desired length.  The byte
   string is then formatted as required for the group.  In the case of
   Weierstrass curves, we take the desired length as the length for
   representing a compressed point (section 2.3.4 of [SEC1]), and use
   the low-order bit of the first byte as the sign bit.  In order to



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   obtain the correct format, the value of the first byte is set to 0x02
   or 0x03 (clearing the first six bits and setting the seventh bit),
   leaving the sign bit as it was in the byte string constructed by
   concatenating hash blocks.  For the [RFC8032] curves a different
   procedure is used.  For edwards448 the 57-byte input has the least-
   significant 7 bits of the last byte set to zero, and for edwards25519
   the 32-byte input is not modified.  For both the [RFC8032] curves the
   (modified) input is then interpreted as the representation of the
   group element.  If this interpretation yields a valid group element
   with the correct order (p), the (modified) byte string is the output.
   Otherwise, the initial hash block is discarded and a new byte string
   constructed from the remaining hash blocks.  The procedure of
   constructing a byte string of the appropriate length, formatting it
   as required for the curve, and checking if it is a valid point of the
   correct order, is repeated until a valid element is found.

   The following python snippet generates the above points, assuming an
   elliptic curve implementation following the interface of
   Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
   Appendix A of [RFC8032]:































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  def iterated_hash(seed, n):
      h = seed
      for i in range(n):
          h = hashlib.sha256(h).digest()
      return h

  def bighash(seed, start, sz):
      n = -(-sz // 32)
      hashes = [iterated_hash(seed, i) for i in range(start, start + n)]
      return b''.join(hashes)[:sz]

  def canon_pointstr(ecname, s):
      if ecname == 'edwards25519':
          return s
      elif ecname == 'edwards448':
          return s[:-1] + bytes([s[-1] & 0x80])
      else:
          return bytes([(s[0] & 1) | 2]) + s[1:]

  def gen_point(seed, ecname, ec):
      for i in range(1, 1000):
          hval = bighash(seed, i, len(ec.encode()))
          pointstr = canon_pointstr(ecname, hval)
          try:
              p = ec.decode(pointstr)
              if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
                  return pointstr, i
          except Exception:
              pass

Authors' Addresses

   Watson Ladd
   UC Berkeley

   Email: watsonbladd@gmail.com


   Benjamin Kaduk (editor)
   Akamai Technologies

   Email: kaduk@mit.edu









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