Secure Shell (SSH) Key Exchange Method Using Hybrid Streamlined NTRU Prime sntrup761 and X25519 with SHA-512: sntrup761x25519-sha512
draft-josefsson-ntruprime-ssh-02

Internet Engineering Task Force                                M. Friedl
Internet-Draft                                                   OpenSSH
Intended status: Informational                                 J. Mojzis
Expires: 22 March 2024                                           TinySSH
                                                            S. Josefsson
                                                       19 September 2023

  Secure Shell (SSH) Key Exchange Method Using Hybrid Streamlined NTRU
    Prime sntrup761 and X25519 with SHA-512: sntrup761x25519-sha512
                    draft-josefsson-ntruprime-ssh-02

Abstract

   This document describe 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 22 March 2024.

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Table of Contents

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

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.

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].

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   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].  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, see section 11.1 (Disconnection Message) of
   [RFC4253].  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 the following changes.  Instead of encoding the
   shared secret K as 'mpint', it MUST be encoded as 'string'.  The
   shared secret K value MUST be the 64-byte output octet string of the
   SHA-512 hash computed with the input as the 32-byte octet string key
   output from the key encapsulation mechanism of sntrup761 concatenated
   with the 32-byte octet string of X25519(a, X25519(b, 9)) = X25519(b,
   X25519(a, 9)).

4.  Acknowledgements

   Jan Mojzis added "sntrup4591761x25519-sha512@tinyssh.org" to TinySSH
   [TinySSH] in 2018 and Markus Friedl implemented it for OpenSSH
   [OpenSSH] during 2019.  During 2020 Damien Miller replaced
   sntrup4591761 with sntrup761 in OpenSSH, to create
   "sntrup761x25519-sha512@openssh.com".  TinySSH added support for it
   during 2021.  It became the default key exchange algorithm in OpenSSH
   during 2022.  That is identical to the "sntrup761x25519-sha512"
   mechanism described in this document.

   This document was derived from [RFC8731].

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   We wish to thank the following people who contributed to this
   document: Roman Danyliw, Loganaden Velvindron, Panos Kampanakis, Mark
   Baushke.

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 method described here to combine Curve25519
   with sntrup761 (i.e., SHA-512 hashing the concatenated outputs) is
   also available for the same kind of cryptographic scrutiny.

   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
   offered by this work.

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

   As discussed in the security considerations of Curve25519-sha256
   [RFC8731], the X25519 shared secret K is used bignum-encoded in that
   document, and this raise a potential for a hash-processing time side-
   channel that could leak one bit of the secret due to different length
   of the bignum sign pad.  This document resolve that problem by using
   string-encoding instead of bignum-encoding.

6.  IANA Considerations

   IANA is requested to add a new “Method Name” of
   "sntrup761x25519-sha512" to the "Key Exchange Method Names" registry
   for Secure Shell (SSH) Protocol Parameters [IANA-KEX] with a
   “reference” field to this RFC and the “OK to implement” field of
   “MAY”.

7.  References

7.1.  Normative References

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

   [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, Submission to the NIST PQC Standardization Round
              3 Process", WWW 
              https://csrc.nist.gov/CSRC/media/Projects/post-quantum-
              cryptography/documents/round-3/submissions/NTRU-Prime-
              Round3.zip, 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>.

   [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>.

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   [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

   [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, "OpenSSH", <https://www.openssh.com/>.

   [TinySSH]  TinySSH, "TinySSH", <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.

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 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.
 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                                ....[.

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 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

 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.....

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 0656: 95 f0 28 fd 24 9c 73 1c cf ef 3e fe 21 a1 e5 4e  ..(.$.s...>.!..N
 0672: 77 da db 12 01 7a e4 2c b5 f3 9d 30 e6 49 99 d6  w....z.,...0.I..
 0688: 21 58 cc 5b 5b d5 ff ca ea df 9a fd d6 73 be cd  !X.[[........s..
 0704: ae 7c 0d ea 78 e4 dd 74 f9 93 53 21 70 b7 cd 16  .|..x..t..S!p...
 0720: ea c7 e9 5d 01 e0 e3 e6 53 46 7f fa a0 48 3e 5b  ...]....SF...H>[
 0736: af 64 46 ff 0f 0c b5 c9 92 48 e8 20 35 1d c8 ae  .dF......H. 5...
 0752: d8 c4 38 31 aa 2c b5 91 6b eb 86 ac 2b fa 86 f2  ..81.,..k...+...
 0768: d1 bd 7d 51 4c be f3 bf 4b d0 f0 78 0e 20 d3 30  ..}QL...K..x. .0
 0784: fc f8 00 53 2a 6a 9b d9 e4 0e 08 d1 ad 52 7a ca  ...S*j.......Rz.
 0800: f3 8b 0e a8 fb 45 3c 66 03 66 b4 54 a5 3d 8e df  .....E<f.f.T.=..
 0816: 4a 8f 66 f0 16 44 3b a9 f1 b3 db bb 7e d6 38 e5  J.f..D;.....~.8.
 0832: 5f 62 27 bb ba 34 0a 6f 9b 78 dd ae 54 ab 54 53  _b'..4.o.x..T.TS
 0848: 3a e1 d2 f1 d8 1e 8b 31 61 cd 69 8a 63 fb 7c 24  :......1a.i.c.|$
 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.^...[..

Friedl, et al.            Expires 22 March 2024                 [Page 9]
Internet-Draft          NTRUPrime+X25519 for SSH          September 2023

 0064: 1a 15 a9 6d                                      ...m

                                Figure 1

Authors' Addresses

   Markus Friedl
   OpenSSH
   Email: markus@openbsd.org

   Jan Mojzis
   TinySSH
   Email: jan.mojzis@gmail.com

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

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