Secure Shell (SSH) Key Exchange Method Using Hybrid Streamlined NTRU Prime sntrup761 and X25519 with SHA-512: sntrup761x25519-sha512
draft-josefsson-ntruprime-ssh-00

Internet Engineering Task Force                             S. Josefsson
Internet-Draft                                                9 May 2023
Intended status: Informational                                          
Expires: 10 November 2023

  Secure Shell (SSH) Key Exchange Method Using Hybrid Streamlined NTRU
    Prime sntrup761 and X25519 with SHA-512: sntrup761x25519-sha512
                    draft-josefsson-ntruprime-ssh-00

Abstract

   We document a widely deployed hybrid key exchange methods in the
   Secure Shell (SSH) protocol that is based on Streamlined NTRU Prime
   sntrup761 and X25519 with SHA-512.

Status of This Memo

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   This Internet-Draft will expire on 10 November 2023.

Copyright Notice

   Copyright (c) 2023 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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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Language . . . . . . . . . . . . . . . . . . . .   2
   3.  Key Exchange Method: sntrup761x25519-sha512 . . . . . . . . .   3
   4.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   3
   5.  Security Considerations . . . . . . . . . . . . . . . . . . .   4
   6.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   4
   7.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   4
     7.1.  Normative References  . . . . . . . . . . . . . . . . . .   4
     7.2.  Informative References  . . . . . . . . . . . . . . . . .   5
   Appendix A.  Test vectors . . . . . . . . . . . . . . . . . . . .   6
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  10

1.  Introduction

   Secure Shell (SSH) [RFC4251] is a secure remote login protocol.  The
   key exchange protocol described in [RFC4253] supports an extensible
   set of methods.  [RFC5656] defines how elliptic curves are integrated
   into this extensible SSH framework, and [RFC8731] adds
   curve25519-sha256 to support the pre-quantum elliptic-curve Diffie-
   Hellman X25519 function [RFC7748].

   Streamlined NTRU Prime [NTRUPrime] [NTRUPrimePQCS] provides post-
   quantum small lattice-based key-encapsulation mechanisms.  The
   variant sntrup761 instance has been implemented widely.

   To hedge against attacks on either of sntrup761 or X25519 a hybrid
   construction may be used, with the intention that the hybrid would be
   secure if either of the involved algorithms are flawed.

   This document describes how to implement key exchange based on a
   hybrid between Streamlined NTRU Prime sntrup761 and X25519 with
   SHA-512 [RFC6234] in SSH.

   The algorithm is equivalent to the privately defined algorithm
   "sntrup761x25519-sha512@openssh.com", which is implemented and widely
   deployed in OpenSSH [OpenSSH] and TinySSH [TinySSH].

2.  Requirements Language

   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.

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3.  Key Exchange Method: sntrup761x25519-sha512

   The key-agreement is done by the X25519 Diffie-Hellman protocol as
   described in section 3 (Key Exchange Methods) of [RFC8731], and the
   key encapsulation method described in [NTRUPrimePQCS].

   The key exchange procedure re-use the Elliptic Curve Diffie-Hellman
   (ECDH) key exchange defined in section 4 (ECDH Key Exchange) and
   section 7.1 (ECDH Message Numbers) of [RFC5656], and the protocol
   flow and the SSH_MSG_KEX_ECDH_INIT and SSH_MSG_KEX_ECDH_REPLY
   messages are identical, except that we use different ephemeral public
   values Q_C and Q_S and shared secret K as described below.

   The SSH_MSG_KEX_ECDH_INIT's value Q_C that holds the client's
   ephemeral public key MUST be constructed by concatenating the 1158
   byte public key output from the key generator of sntrup761 with the
   32 byte K_A = X25519(a, 9) as described in [NTRUPrimePQCS] and
   [RFC8731].  The Q_C value is thus 1190 bytes.

   The SSH_MSG_KEX_ECDH_REPLY's value Q_S that holds the server's
   ephemeral public key MUST be constructed by concatenating the 1039
   byte ciphertext output from the key encapsulation mechanism of
   sntrup761 with the 32 byte K_B = X25519(b, 9) as described in
   [NTRUPrimePQCS] and [RFC8731].  The Q_S value is thus 1071 bytes.

   Clients and servers MUST abort if the length of the received public
   keys Q_C or Q_S are not the expected lengths.  An abort for these
   purposes is defined as a disconnect (SSH_MSG_DISCONNECT) of the
   session and SHOULD use the SSH_DISCONNECT_KEY_EXCHANGE_FAILED reason
   for the message [IANA-REASON].  No further validation is required
   beyond what is described in [RFC7748], [RFC8731] and [NTRUPrimePQCS].

   The SSH_MSG_KEX_ECDH_REPLY's signature value is computed as described
   in [RFC5656] with a shared secret K that MUST be the SHA-512 hash
   computed on the concatenation of the 32-byte key output from the key
   encapsulation mechanism of sntrup761 together with the 32 byte
   X25519(a, X25519(b, 9)) = X25519(b, X25519(a, 9)) output.  Note that
   unlike Curve25519-sha256 [RFC8731], the X25519 output is used raw
   instead of bignum-encoded, and the SHA-512 output value is string-
   encoded.

4.  Acknowledgements

   The sntrup761x25519-sha512 mechanism is identical to the
   sntrup761x25519-sha512@openssh.com mechanism designed by the OpenSSH
   project, inspired by the earlier
   sntrup4591761x25519-sha512@tinyssh.org mechanism designed by TinySSH.

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   This document was derived from [RFC8731].

5.  Security Considerations

   The security considerations of [RFC4251], [RFC5656], [RFC7748], and
   [RFC8731] are inherited.

   Streamlined NTRU Prime sntrup761 is aiming for the standard goal of
   IND-CCA2 security, is widely implemented with good performance on a
   wide range of architectures, and has been studied by researchers for
   several years.  However new cryptographic primitives should be
   introduced and trusted conservatively, and new research findings may
   be published at any time that may warrant implementation
   reconsiderations.

   The increase in communication size and computational requirements may
   be a concern for restricted computational devices, which would then
   not be able to take advantage of the improved security properties
   offer by this work.

   Since sntrup761x25519-sha512 is expected to offer no reduction of
   security compared to curve25519-sha256, we suggest that it is used
   and preferred whenever curve25519-sha256 is used today, when the
   extra communication size and computational requirements are
   acceptable.

6.  IANA Considerations

   IANA is requested to add "sntrup761x25519-sha512" to the "Key
   Exchange Method Names" registry for SSH [IANA-KEX] that was created
   in Section 4.10 of [RFC4250].

7.  References

7.1.  Normative References

   [NTRUPrimePQCS]
              Bernstein, D.J., Brumley, B. B., Chen,, M.,
              Chuengsatiansup, C., Lange, T., Marotzke, A., Peng, B.,
              Tuveri, N., Vredendaal, C. V., and B. Yang, "NTRU Prime:
              round 3", WWW https://ntruprime.cr.yp.to/nist/ntruprime-
              20201007.pdf, October 2020.

   [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>.

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   [RFC4250]  Lehtinen, S. and C. Lonvick, Ed., "The Secure Shell (SSH)
              Protocol Assigned Numbers", RFC 4250,
              DOI 10.17487/RFC4250, January 2006,
              <https://www.rfc-editor.org/info/rfc4250>.

   [RFC4251]  Ylonen, T. and C. Lonvick, Ed., "The Secure Shell (SSH)
              Protocol Architecture", RFC 4251, DOI 10.17487/RFC4251,
              January 2006, <https://www.rfc-editor.org/info/rfc4251>.

   [RFC4253]  Ylonen, T. and C. Lonvick, Ed., "The Secure Shell (SSH)
              Transport Layer Protocol", RFC 4253, DOI 10.17487/RFC4253,
              January 2006, <https://www.rfc-editor.org/info/rfc4253>.

   [RFC5656]  Stebila, D. and J. Green, "Elliptic Curve Algorithm
              Integration in the Secure Shell Transport Layer",
              RFC 5656, DOI 10.17487/RFC5656, December 2009,
              <https://www.rfc-editor.org/info/rfc5656>.

   [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>.

   [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>.

   [RFC8731]  Adamantiadis, A., Josefsson, S., and M. Baushke, "Secure
              Shell (SSH) Key Exchange Method Using Curve25519 and
              Curve448", RFC 8731, DOI 10.17487/RFC8731, February 2020,
              <https://www.rfc-editor.org/info/rfc8731>.

7.2.  Informative References

   [IANA-KEX] IANA, "Secure Shell (SSH) Protocol Parameters: Key
              Exchange Method Names",
              <https://www.iana.org/assignments/ssh-parameters/>.

   [IANA-REASON]
              IANA, "Secure Shell (SSH) Protocol Parameters:
              Disconnection Messages Reason Codes and Descriptions",
              <https://www.iana.org/assignments/ssh-parameters/>.

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   [NTRUPrime]
              Bernstein, D.J., Chuengsatiansup, C., Lange, T., and C.
              van Vredendaal, "NTRU Prime: reducing attack surface at
              low cost", WWW https://ntruprime.cr.yp.to/ntruprime-
              20170816.pdf, August 2017.

   [OpenSSH]  OpenSSH group of OpenBSD, "The OpenSSH Project",
              <https://www.openssh.com/>.

   [TinySSH]  TinySSH, "TinySSH - minimalistic SSH server which
              implements only a subset of SSHv2 features",
              <https://www.tinyssh.org/>.

Appendix A.  Test vectors

 SSH2_MSG_KEX_ECDH_INIT

 client public key sntrup761:
 0000: 5d b3 a9 d3 93 30 31 76 0e 8a f5 87 f7 b2 8c 4f  ]....01v.......O
 0016: 97 a1 74 0e 6b 6f cf 1a d9 d9 99 8a 32 a5 61 e5  ..t.ko......2.a.
 0032: 9e 4d 93 67 e2 66 18 f0 0a f5 54 f4 48 65 0c 60  .M.g.f....T.He.`
 0048: d1 12 92 c2 aa a9 e4 7c ea 32 a3 f5 86 cb c4 c3  .......|.2......
 0064: d5 c2 6f 34 5e 7f d3 57 51 d3 e3 d9 cc 1c e4 49  ..o4^..WQ......I
 0080: bb ea 3e 2e 58 5e ac ba 0a b8 22 00 7c 77 a4 e0  ..>.X^....".|w..
 0096: bd 16 5c 3a f7 b3 25 08 c1 81 fd 0d 9f 99 a3 be  ..\:..%.........
 0112: ae e3 38 84 13 ff f0 b4 0f cb ab 76 1e 95 3e 1e  ..8........v..>.
 0128: 7c 74 1e 58 46 f6 81 f0 f2 f2 56 5b f3 be ce c9  |t.XF.....V[....
 0144: c8 99 9f 03 88 81 db 17 75 1d fb f5 b1 e2 f3 5d  ........u......]
 0160: 32 ce 19 75 49 e7 e1 17 bf 35 0d 97 7c ac 0a cf  2..uI....5..|...
 0176: 6c 8a 0f fc 07 4b a7 8b c5 93 f7 47 7c b6 d5 bf  l....K.....G|...
 0192: 02 f0 96 80 e8 dc f3 87 c9 f0 b2 91 e7 37 70 82  .............7p.
 0208: 3e 47 b7 18 72 be 5a da b1 85 d3 6e 56 5d 8a a3  >G..r.Z....nV]..
 0224: 62 fa 3e d0 ea 6e b9 fa 69 ec 96 86 94 81 2e 88  b.>..n..i.......
 0240: 2b ba e5 af 70 1e ae ba 5f cb ea 82 e5 ba 67 0e  +...p..._.....g.
 0256: 4d f6 2a ec 13 a9 19 b4 08 9c b7 32 bb 40 de c3  M.*........2.@..
 0272: e9 33 e1 c4 0d 5b 72 00 06 c4 3b 7f 57 d4 85 76  .3...[r...;.W..v
 0288: 4c 4c 3d ab 8e 1b 00 00 ac d9 8c 05 b3 18 24 85  LL=...........$.
 0304: 77 28 74 71 0d 68 8b 02 2c 59 55 a7 4d a4 6e 37  w(tq.h..,YU.M.n7
 0320: 85 6c 77 68 f5 b7 a7 52 61 af 37 b4 09 07 34 68  .lwh...Ra.7...4h
 0336: b6 83 ca f2 03 25 47 f9 09 e6 da bd 82 07 7e d1  .....%G.......~.
 0352: 78 16 74 1a a5 4c 5b ac 78 d8 0f 1a 44 08 44 a7  x.t..L[.x...D.D.
 0368: ef 85 00 43 19 c3 3e b4 54 e6 3f f1 ac 83 03 ce  ...C..>.T.?.....
 0384: 7c bd ef 3c fd eb 47 6f f7 f9 e0 1f 13 9f cb 77  |..<..Go.......w
 0400: 52 40 9d 3a d7 8b ad bf cc f1 06 ec 93 32 48 be  R@.:.........2H.
 0416: 0a 53 99 5c dd 9e 96 3b 84 21 8f b2 b4 fd b8 97  .S.\...;.!......
 0432: 8b 7a 8f 71 aa e6 af 4e 22 53 18 f0 a2 30 a0 53  .z.q...N"S...0.S
 0448: 30 c9 d8 a9 d7 67 08 a5 ad 81 64 7b 3a 02 ae ff  0....g....d{:...
 0464: e7 fa 41 68 d0 54 e3 42 86 da f7 f0 98 31 38 e5  ..Ah.T.B.....18.

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 0480: 8c fa 86 5c 5c f9 82 f8 a2 09 91 91 96 72 12 e5  ...\\........r..
 0496: 8f 8b 8e 9b e8 5d bd 66 4b 6e ec a3 b3 03 c5 4e  .....].fKn.....N
 0512: 0f 7e a5 15 ef ab 01 8c 6d 02 52 77 bc 9a 02 f2  .~......m.Rw....
 0528: 2e bf 03 40 fe 5a 80 5a c0 78 1e 95 21 10 9d dd  ...@.Z.Z.x..!...
 0544: 37 87 00 ae 13 c5 9d 9c 81 87 37 3e 7d e0 40 bc  7.........7>}.@.
 0560: 83 76 69 4f 9f c4 08 fd aa a1 7e aa 88 0e 4c 56  .viO......~...LV
 0576: a0 47 c5 d6 94 fb 52 67 f3 36 de b2 7e bf d1 33  .G....Rg.6..~..3
 0592: 41 fd 05 20 66 60 f4 91 96 5f 19 33 2d 17 ec e0  A.. f`..._.3-...
 0608: 3e 93 7a 66 3b b0 de f4 ad 51 90 a4 a1 94 f3 37  >.zf;....Q.....7
 0624: 9a 77 11 02 67 45 6d 4d 19 80 33 58 56 2c b8 11  .w..gEmM..3XV,..
 0640: 51 7b bc ec 43 fe 3d 96 ac f7 f0 8b 8d c6 2c 02  Q{..C.=.......,.
 0656: 2f c0 67 21 56 49 ee bf 07 17 48 f9 30 0b 18 2c  /.g!VI....H.0..,
 0672: fa 7b 57 93 be f7 12 99 57 be 98 e7 55 84 da ed  .{W.....W...U...
 0688: 5c 94 71 fa 48 0f ed 97 ab e4 a5 d6 b6 26 3a e4  \.q.H........&:.
 0704: cb fe f9 ed 07 4b 42 bf e5 a1 d1 34 4d 7b 67 b9  .....KB....4M{g.
 0720: b7 06 7b d2 c7 ae 57 15 21 58 55 70 70 93 f1 87  ..{...W.!XUpp...
 0736: 31 bf 85 74 fe 36 0d 08 c8 07 a2 14 fc d5 96 8b  1..t.6..........
 0752: 59 62 97 30 43 75 c2 a9 4f ec f9 e9 33 a9 38 cb  Yb.0Cu..O...3.8.
 0768: ae ee 63 34 8c 65 54 e7 9d d4 23 a2 4f b9 00 ed  ..c4.eT...#.O...
 0784: b4 be 0b 1c df d4 97 c0 89 ab dd 5f 75 13 ce 37  ..........._u..7
 0800: f3 d2 26 55 72 39 61 f0 d2 11 e8 e7 5f 93 5b 79  ..&Ur9a....._.[y
 0816: e5 6c 28 f3 0a f9 5e 99 b8 a0 e6 4a 22 88 e5 28  .l(...^....J"..(
 0832: 82 0c 6f 72 1d dd 80 84 57 04 72 f4 26 56 71 f3  ..or....W.r.&Vq.
 0848: 92 23 ff 9e a9 fd 05 0b 51 99 72 32 98 a5 02 87  .#......Q.r2....
 0864: fe bb 99 18 5a b3 ec ab f9 26 7b 97 79 da 5f 19  ....Z....&{.y._.
 0880: 4e e7 7d a5 2d 53 40 2a 1f 1b 62 df 3b 11 82 e6  N.}.-S@*..b.;...
 0896: 90 7f 0f 56 0c 75 14 03 e7 6f aa f0 0e 0a 17 13  ...V.u...o......
 0912: 54 f5 ea d7 21 31 2c 7a c5 7f a3 ae 14 f3 05 42  T...!1,z.......B
 0928: e9 c9 6c 6d d1 0a cb 19 35 7f 01 8a 8c e2 a1 09  ..lm....5.......
 0944: b5 c6 e5 e8 2b 4f 1e a2 e9 ce 5b e4 76 f7 53 4f  ....+O....[.v.SO
 0960: 52 d4 75 22 4b aa 1e cd 42 0e be d7 dc 76 6f 94  R.u"K...B....vo.
 0976: 0a 37 47 ca 44 bd e6 9e c1 2a 0d 57 f3 c2 47 40  .7G.D....*.W..G@
 0992: 23 db a8 45 c7 9b 4a 96 13 6a 73 ad 6a a2 a8 e4  #..E..J..js.j...
 1008: df 92 34 76 f9 47 8d b9 21 63 46 c2 d7 f2 64 e6  ..4v.G..!cF...d.
 1024: 17 27 9f cf f3 ae cd 3a 7d ed 5e 46 7c 33 71 f6  .'.....:}.^F|3q.
 1040: 71 c8 92 dc ae e6 a0 c8 05 0c e0 37 fb ea 15 ed  q..........7....
 1056: b0 78 a5 bf b1 48 8b 46 64 1e c8 81 00 55 82 89  .x...H.Fd....U..
 1072: 25 f8 b1 8b 1c e4 96 54 f8 be 97 b1 d3 20 f3 a0  %......T..... ..
 1088: b5 c1 dd d5 27 d0 61 d9 96 2a 74 76 a8 33 10 78  ....'.a..*tv.3.x
 1104: ff b2 86 ee 4f 0b 78 73 dd 7f 7c b5 02 e9 12 35  ....O.xs..|....5
 1120: d3 9e ab 81 cd 9b 61 fb 2b 33 72 ee c6 bb 8a bc  ......a.+3r.....
 1136: bd 4f e5 9b c2 55 8f a0 b1 e7 1a 6a c1 e3 f1 5c  .O...U.....j...\
 1152: 83 8f f0 9c 5b 04                                ....[.

 client public key c25519:
 0000: be f9 23 79 d7 fd 4e 8a 10 55 9b dc e5 3e 62 13  ..#y..N..U...>b.
 0016: eb 9b 6a 6f ca de ed 90 04 db b1 30 f6 ff ef 4f  ..jo.......0...O

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 SSH2_MSG_KEX_ECDH_REPLY

 server cipher text:
 0000: 71 67 00 55 f8 ac 87 1a af 7c ef cf 1c b4 7d b9  qg.U.....|....}.
 0016: 4f b6 22 5e 4d 77 81 73 4f 1d b9 82 79 ff e9 34  O."^Mw.sO...y..4
 0032: 26 9f d2 2e 4e c6 a3 5f 79 9c 26 68 99 3a 0f 40  &...N.._y.&h.:.@
 0048: 33 2a 7d dd fa 7a e7 6b 1e e7 9d 50 b7 48 0f aa  3*}..z.k...P.H..
 0064: aa 97 ff e7 8c 6c ac 5d 10 df 2b e3 cc 93 ea dc  .....l.]..+.....
 0080: 18 17 b3 34 42 70 7a 27 85 58 2a ae c2 e6 b9 26  ...4Bpz'.X*....&
 0096: 93 fd 23 a9 ae ac 4a 35 8b 57 c1 5c 95 cb 23 fb  ..#...J5.W.\..#.
 0112: e5 93 0f 7c f5 63 6b 5b a1 53 b5 55 d0 75 16 21  ...|.ck[.S.U.u.!
 0128: 8a db 95 ff c8 58 ac f4 7e 46 69 0a 4c a9 c8 cc  .....X..~Fi.L...
 0144: eb e8 66 7c c4 fb fd 98 2c 0c 7f 41 8c 34 89 49  ..f|....,..A.4.I
 0160: a0 25 59 eb 63 a1 e6 8f 37 bf bc b3 ce 0a da 53  .%Y.c...7......S
 0176: 54 7f c2 41 52 eb 6c 9e 6e d0 ea af 6a 82 5d 17  T..AR.l.n...j.].
 0192: 6f 17 8d 06 8a 86 55 60 28 31 12 4a 0c de 6b be  o.....U`(1.J..k.
 0208: eb fd 38 13 6c 56 69 ad 0e 72 c8 bd b4 69 9d 32  ..8.lVi..r...i.2
 0224: b4 1c 8e 6f f4 25 e1 9b c5 6f 8b 02 77 52 ae 72  ...o.%...o..wR.r
 0240: eb 9b 03 c8 9f de 15 bd f6 5a e8 9d 83 81 7b 48  .........Z....{H
 0256: 7a 69 9a d0 91 41 aa 07 5a fa ad d6 e8 55 39 d9  zi...A..Z....U9.
 0272: d1 0f d2 18 dc a0 9d 1c f1 e4 1c 0d f8 88 85 6b  ...............k
 0288: 6d 11 24 3e 61 de 48 95 5f 2a d1 c9 ad 3f b8 41  m.$>a.H._*...?.A
 0304: 49 6d 9f 7c 3c bf 20 fe 37 7f 8c 8c 8f 72 ca f4  Im.|<. .7....r..
 0320: 19 e4 cc a1 d8 08 cb 69 ec da 2b 88 e8 98 e9 1e  .......i..+.....
 0336: 29 af 86 6f 19 a8 67 56 ef b4 33 e4 2b b8 fe 61  )..o..gV..3.+..a
 0352: ad 36 4c 42 f8 ec 04 38 09 62 02 66 b5 54 fc 69  .6LB...8.b.f.T.i
 0368: 46 29 05 27 d8 32 fd 37 4c d4 62 55 e1 ae e9 62  F).'.2.7L.bU...b
 0384: 66 a0 f4 cb 4b 01 af 6b ea 09 80 00 a2 2b ff 0e  f...K..k.....+..
 0400: 85 2c 92 b2 5c f9 f3 eb 44 a3 9a e8 55 bb e3 2f  .,..\...D...U../
 0416: 2d 20 5a 77 67 97 57 90 7f 4b b3 08 92 41 1a c0  - Zwg.W..K...A..
 0432: f6 1b e9 a4 06 29 ea 31 eb 81 f0 94 96 aa 26 95  .....).1......&.
 0448: 06 ed 4b f0 d3 9f aa 73 89 fa 6e f7 8f 4b f5 fa  ..K....s..n..K..
 0464: e4 5f 7c b6 08 e9 b2 18 77 99 9c ac 7b fb ec 41  ._|.....w...{..A
 0480: 41 1e 29 c2 d0 a5 de bc 59 2f 14 45 6d af b1 e0  A.).....Y/.Em...
 0496: 9c 77 73 0e ac 52 23 73 11 35 27 17 8c a3 ff 0e  .ws..R#s.5'.....
 0512: 52 5d b7 c8 06 c5 05 43 15 53 e8 fc 83 64 df 10  R].....C.S...d..
 0528: 8b 9c 74 5c 0e d9 54 5e 9a 49 cf 13 e4 1d 86 35  ..t\..T^.I.....5
 0544: 24 a3 27 75 d3 d6 b4 95 78 8f 0d 81 3b 80 6b 26  $.'u....x...;.k&
 0560: 25 9f 14 b1 65 73 e8 ce fa 95 6d b1 15 0c 76 3c  %...es....m...v<
 0576: b1 75 a9 96 78 c8 4b 91 06 a9 94 bc ec fa 44 eb  .u..x.K.......D.
 0592: 39 77 4d ee df ae eb 0e 90 61 eb ab 6a 17 1b 24  9wM......a..j..$
 0608: 3c 3a 6e c4 bb 6f 72 46 3d 9a b8 8c 6a e7 45 c7  <:n..orF=...j.E.
 0624: 0f 81 db 19 6e ce 65 74 ca db 73 ec 1e ce 5f d7  ....n.et..s..._.
 0640: 43 6b fe ff c0 e1 61 26 aa b7 6f e0 dc 7f d1 de  Ck....a&..o.....
 0656: 95 f0 28 fd 24 9c 73 1c cf ef 3e fe 21 a1 e5 4e  ..(.$.s...>.!..N
 0672: 77 da db 12 01 7a e4 2c b5 f3 9d 30 e6 49 99 d6  w....z.,...0.I..
 0688: 21 58 cc 5b 5b d5 ff ca ea df 9a fd d6 73 be cd  !X.[[........s..
 0704: ae 7c 0d ea 78 e4 dd 74 f9 93 53 21 70 b7 cd 16  .|..x..t..S!p...

Josefsson               Expires 10 November 2023                [Page 8]
Internet-Draft          NTRUPrime+X25519 for SSH                May 2023

 0720: ea c7 e9 5d 01 e0 e3 e6 53 46 7f fa a0 48 3e 5b  ...]....SF...H>[
 0736: af 64 46 ff 0f 0c b5 c9 92 48 e8 20 35 1d c8 ae  .dF......H. 5...
 0752: d8 c4 38 31 aa 2c b5 91 6b eb 86 ac 2b fa 86 f2  ..81.,..k...+...
 0768: d1 bd 7d 51 4c be f3 bf 4b d0 f0 78 0e 20 d3 30  ..}QL...K..x. .0
 0784: fc f8 00 53 2a 6a 9b d9 e4 0e 08 d1 ad 52 7a ca  ...S*j.......Rz.
 0800: f3 8b 0e a8 fb 45 3c 66 03 66 b4 54 a5 3d 8e df  .....E<f.f.T.=..
 0816: 4a 8f 66 f0 16 44 3b a9 f1 b3 db bb 7e d6 38 e5  J.f..D;.....~.8.
 0832: 5f 62 27 bb ba 34 0a 6f 9b 78 dd ae 54 ab 54 53  _b'..4.o.x..T.TS
 0848: 3a e1 d2 f1 d8 1e 8b 31 61 cd 69 8a 63 fb 7c 24  :......1a.i.c.|$
 0864: 75 5f e6 6d 64 3d e4 12 cb 2d b3 6f 0f 5a 19 28  u_.md=...-.o.Z.(
 0880: 1f d6 f6 9c ee 44 11 1a c5 84 d6 e3 a2 05 5d d4  .....D........].
 0896: 85 db f1 8f e4 17 df bc 4c 78 98 d1 70 3b 63 d6  ........Lx..p;c.
 0912: a4 91 db f1 9e 16 23 fa e0 54 f6 64 d1 0b d0 d6  ......#..T.d....
 0928: a6 fd f1 66 72 8c 65 d8 17 af c9 33 49 c8 e9 4d  ...fr.e....3I..M
 0944: 1c 0a 77 2b 96 86 f2 16 55 3a e3 f6 00 bb b6 5a  ..w+....U:.....Z
 0960: 86 f6 fc 3f d6 f9 a4 1d fd 29 1d 5b 65 dc b3 14  ...?.....).[e...
 0976: 96 10 3e c1 9a 90 23 e8 88 81 24 42 68 7a aa 25  ..>...#...$Bhz.%
 0992: ba f3 50 bd b9 ae be dc b3 ff 39 81 44 89 00 9d  ..P.......9.D...
 1008: 4e 26 d6 ef df 7c e0 53 d3 ed 34 07 3d f2 1e 42  N&...|.S..4.=..B
 1024: 28 af 1d 12 ce 98 c7 b0 7b 90 81 b5 ea f3 2c     (.......{.....,

 server public key c25519:
 0000: 18 6c 55 03 db 1c 38 e3 40 d7 09 24 77 46 14 b8  .lU...8.@..$wF..
 0016: 5e e4 7f 19 98 04 9b 90 1f f6 b9 7f b0 70 9e 32  ^............p.2

 shared secret
 0000: 9b 73 7d 41 d6 cf bb 12 56 c5 8c ad 0a 6a e2 c9  .s}A....V....j..
 0016: bf 84 a9 0a 72 91 eb 52 e4 c1 81 c8 d2 44 7b 56  ....r..R.....D{V

 client kem key:
 0000: 2c 0c 5a 36 e6 77 70 b4 d8 ab 38 9a 92 96 3a cd  ,.Z6.wp...8...:.
 0016: 10 82 38 36 40 be 2d 66 08 02 b8 17 cf eb b9 be  ..86@.-f........

 concatenation of KEM key and ECDH shared key:
 0000: 2c 0c 5a 36 e6 77 70 b4 d8 ab 38 9a 92 96 3a cd  ,.Z6.wp...8...:.
 0016: 10 82 38 36 40 be 2d 66 08 02 b8 17 cf eb b9 be  ..86@.-f........
 0032: 9b 73 7d 41 d6 cf bb 12 56 c5 8c ad 0a 6a e2 c9  .s}A....V....j..
 0048: bf 84 a9 0a 72 91 eb 52 e4 c1 81 c8 d2 44 7b 56  ....r..R.....D{V

 encoded shared secret:
 0000: 00 00 00 40 42 54 58 44 6f 22 75 63 04 de d7 5a  ...@BTXDo"uc...Z
 0016: 1f 23 fe f9 b1 8b 36 eb e0 e6 e2 60 c3 00 12 63  .#....6....`...c
 0032: b0 18 3f 42 49 07 e6 d8 22 b3 b7 6c 6c 38 37 b5  ..?BI..."..ll87.
 0048: b4 1f b0 d0 76 35 c7 57 e6 5e fb ef cb 5b c3 8a  ....v5.W.^...[..
 0064: 1a 15 a9 6d                                      ...m

                                Figure 1

Josefsson               Expires 10 November 2023                [Page 9]
Internet-Draft          NTRUPrime+X25519 for SSH                May 2023

Author's Address

   Simon Josefsson
   Email: simon@josefsson.org
   URI:   https://blog.josefsson.org/

Josefsson               Expires 10 November 2023               [Page 10]