Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-11

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Network Working Group                                 S. Smyshlyaev, Ed.
Internet-Draft                                                 CryptoPro
Intended status: Informational                              V. Nozdrunov
Expires: December 20, 2019                                   V. Shishkin
                                                                   TC 26
                                                          E. Smyshlyaeva
                                                               CryptoPro
                                                           June 18, 2019

                     Multilinear Galois Mode (MGM)
                        draft-smyshlyaev-mgm-11

Abstract

   Multilinear Galois Mode (MGM) is an authenticated encryption with
   associated data block cipher mode based on EtM principle.  MGM is
   defined for use with 64-bit and 128-bit block ciphers.

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   the Trust Legal Provisions and are provided without warranty as
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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Conventions Used in This Document . . . . . . . . . . . . . .   2
   3.  Basic Terms and Definitions . . . . . . . . . . . . . . . . .   2
   4.  Specification . . . . . . . . . . . . . . . . . . . . . . . .   4
     4.1.  MGM Encryption and Authentication Procedure . . . . . . .   4
     4.2.  MGM Decryption and Authentication Check Procedure . . . .   6
   5.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .   7
   6.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     6.1.  Normative References  . . . . . . . . . . . . . . . . . .   8
     6.2.  Informative References  . . . . . . . . . . . . . . . . .   9
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .   9
   Appendix B.  Contributors . . . . . . . . . . . . . . . . . . . .  12
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  13

1.  Introduction

   Multilinear Galois Mode (MGM) is an authenticated encryption with
   associated data block cipher mode based on EtM principle.  MGM is
   defined for use with 64-bit and 128-bit block.  The MGM design
   principles can easily be applied to other block sizes.

2.  Conventions Used in This Document

   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].

3.  Basic Terms and Definitions

   This document uses the following terms and definitions for the sets
   and operations on the elements of these sets:

   V*      the set of all bit strings of a finite length (hereinafter
           referred to as strings), including the empty string;
           substrings and string components are enumerated from right to
           left starting from zero;

   V_s     the set of all bit strings of length s, where s is a non-
           negative integer;

   |X|     the bit length of the bit string X (if X is an empty string,
           then |X| = 0);

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   X || Y  concatenation of strings X and Y both belonging to V*, i.e.,
           a string from V_{|X|+|Y|}, where the left substring from
           V_{|X|} is equal to X, and the right substring from V_{|Y|}
           is equal to Y;

   a^s     the string in V_s that consists of s 'a' bits: a^s = (a, a,
           ... , a), 'a' in V_1;

   (xor)   exclusive-or of the two bit strings of the same length,

   Z_{2^s} ring of residues modulo 2^s;

   MSB_i: V_s -> V_i   the transformation that maps the string X =
           (x_{s-1}, ... , x_0) in V_s into the string MSB_i(X) =
           (x_{s-1}, ... , x_{s-i}) in V_i, i <= s, (most significant
           bits);

   Int_s: V_s -> Z_{2^s}    the transformation that maps a string X =
           (x_{s-1}, ... , x_0) in V_s into the integer Int_s(X) =
           2^{s-1} * x_{s-1} + ... + 2 * x_1 + x_0 (the interpretation
           of the bit string as an integer);

   Vec_s: Z_{2^s} -> V_s  the transformation inverse to the mapping
           Int_s (the interpretation of an integer as a bit string);

   E_K: V_n -> V_n  the block cipher permutation under the key K in V_k;

   k       the bit length of the block cipher key;

   n       the block size of the block cipher (in bits);

   len: V_s -> V_{n/2}  the transformation that maps a string X in V_s,
           0 <= s <= 2^{n/2} - 1, into the string len(X) =
           Vec_{n/2}(|X|) in V_{n/2}, where n is the block size of the
           used block cipher;

   [+]     the addition operation in Z_{2^{n/2}}, where n is the block
           size of the used block cipher;

   (x)     multiplication in GF(2^n), where n is the block size of the
           used block cipher; if n = 64, then the field polynomial is
           equal to f = x^64 + x^4 + x^3 + x + 1; if n = 128, then the
           field polynomial is equal to f = x^128 + x^7 + x^2 + x + 1;

   incr_l: V_n -> V_n  the transformation that maps a string L || R,
           where L, R in V_{n/2}, into the string incr_l(L || R ) =
           Vec_{n/2}(Int_{n/2}(L) [+] 1) || R;

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   incr_r: V_n -> V_n  the transformation that maps a string L || R,
           where L, R in V_{n/2}, into the string incr_r(L || R ) = L ||
           Vec_{n/2}(Int_{n/2}(R) [+] 1).

4.  Specification

   An additional parameter that defines the functioning of MGM mode is
   the bit length S of the authentication tag, 32 <= S <= 128.  The
   value of S MUST be fixed for a particular protocol.  The choice of
   the value S involves a trade-off between message expansion and the
   forgery probability.

4.1.  MGM Encryption and Authentication Procedure

   The MGM encryption and authentication procedure takes the following
   parameters as inputs:

   1.  Encryption key K in V_k.

   2.  Initial counter nonce ICN in V_{n-1}.

   3.  Plaintext P, 0 <= |P| < 2^{n/2}. If |P| > 0, then P = P_1 ||
       ... || P*_q, P_i in V_n, for i = 1, ... , q - 1, P*_q in V_u, 1
       <= u <= n.  If |P| = 0, then by definition P*_q is empty, and the
       q and u parameters are set as follows: q = 0, u = n.

   4.  Associated authenticated data A, 0 <= |A| < 2^{n/2}. If |A| > 0,
       then A = A_1 || ... || A*_h, A_j in V_n, for j = 1, ... , h - 1,
       A*_h in V_t, 1 <= t <= n.  If |A| = 0, then by definition A*_h is
       empty, and the h and t parameters are set as follows: h = 0, t =
       n.  The associated data is authenticated but is not encrypted.

   The MGM encryption and authentication procedure outputs the following
   parameters:

   1.  Initial counter nonce ICN.

   2.  Associated authenticated data A.

   3.  Ciphertext C in V_{|P|}.

   4.  Authentication tag T in V_S.

   The MGM encryption and authentication procedure consists of the
   following steps:

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   +----------------------------------------------------------------+
   |  MGM-Encrypt(K, ICN, P, A)                                     |
   |----------------------------------------------------------------|
   |  1. Encryption step:                                           |
   |      - Y_1 = E_K(0 || ICN),                                    |
   |      - For i = 2, 3, ... , q do                                |
   |              Y_i = incr_r(Y_{i-1}),                            |
   |      - For i = 1, 2, ... , q - 1 do                            |
   |              C_i = P_i (xor) E_K(Y_i),                         |
   |      - C*_q = P*_q (xor) MSB_u(E_K(Y_q)),                      |
   |      - C = C_1 || ... || C*_q.                                 |
   |                                                                |
   |  2. Padding step:                                              |
   |      - A_h = A*_h || 0^{n-t},                                  |
   |      - C_q = C*_q || 0^{n-u}.                                  |
   |                                                                |
   |  3. Authentication tag T generation step:                      |
   |      - Z_1 = E_K(1 || ICN),                                    |
   |      - sum = 0,                                                |
   |      - For i = 1, 2, ..., h do                                 |
   |              H_i = E_K(Z_i),                                   |
   |              sum = sum (xor) ( H_i (x) A_i ),                  |
   |              Z_{i+1} = incr_l(Z_i),                            |
   |      - For j = 1, 2, ..., q do                                 |
   |              H_{h+j} = E_K(Z_{h+j}),                           |
   |              sum = sum (xor) ( H_{h+j} (x) C_j ),              |
   |              Z_{h+j+1} = incr_l(Z_{h+j}),                      |
   |      - H_{h+q+1} = E_K(Z_{h+q+1}),                             |
   |      - T = MSB_S(E_K(sum (xor) H_{h+q+1} (x)                   |
   |                       (len(A) || len(C)))).                    |
   |                                                                |
   |  4. Return (ICN, A, C, T).                                     |
   |----------------------------------------------------------------+

   The ICN value for each message that is encrypted under the given key
   K must be chosen in a unique manner.  Using the same ICN values for
   two different messages encrypted with the same key eliminates the
   security properties of this mode.

   Users who do not wish to encrypt plaintext can provide a string P of
   zero length.  Users who do not wish to authenticate associated data
   can provide a string A of zero length.  The length of the associated
   data A and of the plaintext P MUST be such that 0 < |A| + |P| <
   2^{n/2}.

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4.2.  MGM Decryption and Authentication Check Procedure

   The MGM decryption and authentication procedure takes the following
   parameters as inputs:

   1.  The encryption key K in V_k.

   2.  The initial counter nonce ICN in V_{n-1}.

   3.  The associated authenticated data A, 0 <= |A| < 2^{n/2}. A =
       A_1 || ... || A*_h, A_j in V_n, for j = 1, ... , h - 1, A*_h in
       V_t, 1 <= t <= n.

   4.  The ciphertext C, 0 <= |C| < 2^{n/2}. C = C_1 || ... || C*_q, C_i
       in V_n, for i = 1, ... , q - 1, C*_q in V_u, 1 <= u <= n.

   5.  The authenticated tag T in V_S.

   The MGM decryption and authentication procedure outputs FAIL or the
   following parameters:

   1.  Plaintext P in V_{|C|}.

   2.  Associated authenticated data A.

   The MGM decryption and authentication procedure consists of the
   following steps:

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   +----------------------------------------------------------------+
   |  MGM-Decrypt(K, ICN, A, C, T)                                  |
   |----------------------------------------------------------------|
   |  1. Padding step:                                              |
   |      - A_h = A*_h || 0^{n-t},                                  |
   |      - C_q = C*_q || 0^{n-u}.                                  |
   |                                                                |
   |  2. Authentication tag T verification step:                    |
   |      - Z_1 = E_K(1 || ICN),                                    |
   |      - sum = 0,                                                |
   |      - For i = 1, 2, ..., h do                                 |
   |              H_i = E_K(Z_i),                                   |
   |              sum = sum (xor) ( H_i (x) A_i ),                  |
   |              Z_{i+1} = incr_l(Z_i),                            |
   |      - For j = 1,  2, ..., q do                                |
   |              H_{h+j} = E_K(Z_{h+j}),                           |
   |              sum = sum (xor) ( H_{h+j} (x) C_j ),              |
   |              Z_{h+j+1} = incr_l(Z_{h+j}),                      |
   |      - H_{h+q+1} = E_K(Z_{h+q+1}),                             |
   |      - T' = MSB_S(E_K(sum (xor) H_{h+q+1} (x)                  |
   |                       (len(A) || len(C)))),                    |
   |      - If T' != T then return FAIL.                            |
   |                                                                |
   |  3. Decryption step:                                           |
   |      - Y_1 = E_K(0 || ICN),                                    |
   |      - For i = 2, 3, ... , q do                                |
   |              Y_i = incr_r(Y_{i-1}),                            |
   |      - For i = 1, 2, ... , q - 1 do                            |
   |              P_i = C_i (xor) E_K(Y_i),                         |
   |      - P*_q = C*_q (xor) MSB_u(E_K(Y_q)),                      |
   |      - P = P_1 || ... || P*_q.                                 |
   |                                                                |
   |  4. Return (P, A).                                             |
   |----------------------------------------------------------------+

5.  Rationale

   The MGM mode was originally proposed in [PDMODE].

   From the operational point of view the MGM mode is designed to be
   parallelizable, inverse free, online and to provide availability of
   precomputations.

   Parallelizability of the MGM mode is achieved due to its counter-type
   structure and the usage of the multilinear function for
   authentication.  Indeed, both encryption blocks E_K(Y_i) and
   authentication blocks H_i are produced in the counter mode manner,

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   and the multilinear function determined by H_i is parallelizable in
   itself.  Additionally, the counter-type structure of the mode
   provides the inverse free property.

   The online property means the possibility to process message even if
   it is not completely received (so its length is unknown).  To provide
   this property the MGM mode uses blocks E_K(Y_i) and H_i which are
   produced basing on two independent source blocks Y_i and Z_i.

   Availability of precomputations for the MGM mode means the
   possibility to calculate H_i and E_K(Y_i) even before data is
   retrieved.  It is holds due to again the usage of counters for
   calculating them.

   The MGM mode incorporates some mechanisms for advancing cryptographic
   properties.  Further we note the main ones:

   Different functions generating the counter values:  The functions
      incr_r and incr_l are chosen to minimize intersection (if it
      happens) between the sets of counter values Y_i and Z_i.

   Ciphering of the multilinear function output:  This procedure allows
      to resist attacks based on padding and linear properties (see
      [Ferg05] for details).

   Multilinear function for authentication:  It allows to resist the
      small subgroup attacks [Saar12].

   Ciphering of the nonces (0 || ICN)  and (1 || ICN):  The aim of this
      ciphering is to minimize the number of plaintext/ciphertext pairs
      of blocks known to an adversary.  Small number of these pairs
      allows to resist attacks that need substantial amount of such
      material (e.g., linear and differential cryptanalysis, side-
      channel attacks).

6.  References

6.1.  Normative References

   [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>.

   [RFC7801]  Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher
              "Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016,
              <https://www.rfc-editor.org/info/rfc7801>.

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6.2.  Informative References

   [Ferg05]   Ferguson, N., "Authentication weaknesses in GCM", 2005.

   [GOST3412-2015]
              Federal Agency on Technical Regulating and Metrology,
              "Information technology. Cryptographic data security.
              Block ciphers", GOST R 34.12-2015, 2015.

   [PDMODE]   Nozdrunov, V., "Parallel and double block cipher mode of
              operation (PD-mode) for authenticated encryption", CTCrypt
              2017 proceedings, pp. 36-45, 2017.

   [Saar12]   Saarinen, O., "Cycling Attacks on GCM, GHASH and Other
              Polynomial MACs and Hashes", FSE 2012 proceedings, pp.
              216-225, 2012.

Appendix A.  Test Vectors

   Test vectors for the Kuznyechik block cipher (n = 128, k = 256)
   defined in [GOST3412-2015] (the English version can be found in
   [RFC7801]).

   Encryption key K:
   00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
   00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

   Associated authenticated data A:
   00000:   02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01
   00010:   04 04 04 04 04 04 04 04 03 03 03 03 03 03 03 03
   00020:   EA 05 05 05 05 05 05 05 05

   Plaintext P:
   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
   00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
   00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
   00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
   00040:   AA BB CC

   1. Encryption step:

   0^1 || ICN:
   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Y_1:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CD
   E_K(Y_1):

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   00000:   B8 57 48 C5 12 F3 19 90 AA 56 7E F1 53 35 DB 74

   Y_2:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CE
   E_K(Y_2):
   00000:   80 64 F0 12 6F AC 9B 2C 5B 6E AC 21 61 2F 94 33

   Y_3:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CF
   E_K(Y_3):
   00000:   58 58 82 1D 40 C0 CD 0D 0A C1 E6 C2 47 09 8F 1C

   Y_4:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED D0
   E_K(Y_4):
   00000:   E4 3F 50 81 B5 8F 0B 49 01 2F 8E E8 6A CD 6D FA

   Y_5:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED D1
   E_K(Y_5):
   00000:   86 CE 9E 2A 0A 12 25 E3 33 56 91 B2 0D 5A 33 48

   C:
   00000:   A9 75 7B 81 47 95 6E 90 55 B8 A3 3D E8 9F 42 FC
   00010:   80 75 D2 21 2B F9 FD 5B D3 F7 06 9A AD C1 6B 39
   00020:   49 7A B1 59 15 A6 BA 85 93 6B 5D 0E A9 F6 85 1C
   00030:   C6 0C 14 D4 D3 F8 83 D0 AB 94 42 06 95 C7 6D EB
   00040:   2C 75 52

   2. Padding step:

   A_1 || ... || A_h:
   00000:   02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01
   00010:   04 04 04 04 04 04 04 04 03 03 03 03 03 03 03 03
   00020:   EA 05 05 05 05 05 05 05 05 00 00 00 00 00 00 00

   C_1 || ... || C_q:
   00000:   A9 75 7B 81 47 95 6E 90 55 B8 A3 3D E8 9F 42 FC
   00010:   80 75 D2 21 2B F9 FD 5B D3 F7 06 9A AD C1 6B 39
   00020:   49 7A B1 59 15 A6 BA 85 93 6B 5D 0E A9 F6 85 1C
   00030:   C6 0C 14 D4 D3 F8 83 D0 AB 94 42 06 95 C7 6D EB
   00040:   2C 75 52 00 00 00 00 00 00 00 00 00 00 00 00 00

   3. Authentication tag T generation step:

   1^1 || ICN:
   00000:   91 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

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   Z_1:
   00000:   7F C2 45 A8 58 6E 66 02 A7 BB DB 27 86 BD C6 6F
   H_1:
   00000:   8D B1 87 D6 53 83 0E A4 BC 44 64 76 95 2C 30 0B
   current sum:
   00000:   4C F4 27 F4 AD B7 5C F4 C0 DA 39 D5 AB 48 CF 38

   Z_2:
   00000:   7F C2 45 A8 58 6E 66 03 A7 BB DB 27 86 BD C6 6F
   H_2:
   00000:   7A 24 F7 26 30 E3 76 37 21 C8 F3 CD B1 DA 0E 31
   current sum:
   00000:   94 95 44 0E F6 24 A1 DD C6 F5 D9 77 28 50 C5 73

   Z_3:
   00000:   7F C2 45 A8 58 6E 66 04 A7 BB DB 27 86 BD C6 6F
   H_3:
   00000:   44 11 96 21 17 D2 06 35 C5 25 E0 A2 4D B4 B9 0A
   current sum:
   00000:   A4 9A 8C D8 A6 F2 74 23 DB 79 E4 4A B3 06 D9 42

   Z_4:
   00000:   7F C2 45 A8 58 6E 66 05 A7 BB DB 27 86 BD C6 6F
   H_4:
   00000:   D8 C9 62 3C 4D BF E8 14 CE 7C 1C 0C EA A9 59 DB
   current sum:
   00000:   09 FE 3F 6A 83 3C 21 B3 90 27 D0 20 6A 84 E1 5A

   Z_5:
   00000:   7F C2 45 A8 58 6E 66 06 A7 BB DB 27 86 BD C6 6F
   H_5:
   00000:   A5 E1 F1 95 33 3E 14 82 96 99 31 BF BE 6D FD 43
   current sum:
   00000:   B5 DA 26 BB 00 EB A8 04 35 D7 97 6B C6 B5 46 4D

   Z_6:
   00000:   7F C2 45 A8 58 6E 66 07 A7 BB DB 27 86 BD C6 6F
   H_6:
   00000:   B4 CA 80 8C AC CF B3 F9 17 24 E4 8A 2C 7E E9 D2
   current sum:
   00000:   DD 1C 0E EE F7 83 C8 EB 2A 33 F3 58 D7 23 0E E5

   Z_7:
   00000:   7F C2 45 A8 58 6E 66 08 A7 BB DB 27 86 BD C6 6F
   H_7:
   00000:   72 90 8F C0 74 E4 69 E8 90 1B D1 88 EA 91 C3 31
   current sum:
   00000:   89 6C E1 08 32 EB EA F9 06 9F 3F 73 76 59 4D 40

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   Z_8:
   00000:   7F C2 45 A8 58 6E 66 09 A7 BB DB 27 86 BD C6 6F
   H_8:
   00000:   23 CA 27 15 B0 2C 68 31 3B FD AC B3 9E 4D 0F B8
   current sum:
   00000:   99 1A F5 C9 D0 80 F7 63 87 FE 64 9E 7C 93 C6 42

   Z_9:
   00000:   7F C2 45 A8 58 6E 66 0A A7 BB DB 27 86 BD C6 6F
   H_9:
   00000:   BC BC E6 C4 1A A3 55 A4 14 88 62 BF 64 BD 83 0D
   len(A) || len(C):
   00000:   00 00 00 00 00 00 01 48 00 00 00 00 00 00 02 18
   sum (xor) H_9 (x) (len(A) || len(C)):
   00000:   C0 C7 22 DB 5E 0B D6 DB 25 76 73 83 3D 56 71 28

   Tag T:
   00000:   CF 5D 65 6F 40 C3 4F 5C 46 E8 BB 0E 29 FC DB 4C

Appendix B.  Contributors

   o  Evgeny Alekseev
      CryptoPro
      alekseev@cryptopro.ru

   o  Ekaterina Smyshlyaeva
      CryptoPro
      ess@cryptopro.ru

   o  Lilia Akhmetzyanova
      CryptoPro
      lah@cryptopro.ru

   o  Grigory Marshalko
      TC 26
      marshalko_gb@tc26.ru

   o  Vladimir Rudskoy
      TC 26
      rudskoy_vi@tc26.ru

   o  Alexey Nesterenko
      National Research University Higher School of Economics
      anesterenko@hse.ru

Smyshlyaev, et al.      Expires December 20, 2019              [Page 12]
Internet-Draft        Multilinear Galois Mode (MGM)            June 2019

Authors' Addresses

   Stanislav Smyshlyaev (editor)
   CryptoPro

   Phone: +7 (495) 995-48-20
   Email: svs@cryptopro.ru

   Vladislav Nozdrunov
   TC 26

   Email: nozdrunov_vi@tc26.ru

   Vasily Shishkin
   TC 26

   Email: shishkin_va@tc26.ru

   Ekaterina Smyshlyaeva
   CryptoPro

   Email: ess@cryptopro.ru

Smyshlyaev, et al.      Expires December 20, 2019              [Page 13]