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Internet Draft                                              S. Kiyomoto
Intended status: Informational                                  W. Shin
Expires: September 2011                     KDDI R&D Laboratories, Inc.
                                                         March 28, 2011



              A Description of KCipher-2 Encryption Algorithm
                      draft-kiyomoto-kcipher2-00.txt


Abstract

   This document describes the KCipher-2 encryption algorithm. KCipher-2
   is a stream cipher with a 128-bit key and a 128-bit initialization
   vector. Since the algorithm for KCipher-2 was published in 2007,
   security and efficiency have been rigorously evaluated through
   academic and industrial studies. No security vulnerability has been
   found as of the time this document was written. KCipher-2 offers fast
   encryption and decryption by means of simple operations that enable
   efficient implementation. KCipher-2 has been used for industrial
   applications, especially for mobile health monitoring and diagnostic
   services in Japan.

Status of this Memo

   This Internet-Draft is submitted in full conformance with the
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Copyright Notice

   Copyright (c) 2011 IETF Trust and the persons identified as the
   document authors. All rights reserved.




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   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document. Please review these documents
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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. Algorithm Description..........................................3
      2.1. Notations.................................................3
      2.2. Internal State............................................4
         2.2.1. Feedback Shift Registers.............................4
         2.2.2. Internal registers...................................4
      2.3. Operations................................................4
         2.3.1. The Next operation...................................4
         2.3.2. Init.................................................6
         2.3.3. Stream...............................................7
      2.4. Subroutines...............................................7
         2.4.1. NLF..................................................8
         2.4.2. SubK2................................................8
         2.4.3. S-Box................................................9
         2.4.4. Multiplications in GF(2^32).........................10
      2.5. Encryption/Decryption scheme.............................10
         2.5.1. Key stream generation...............................10
         2.5.2. Encryption/Decryption of a message..................11
   3. Security Considerations.......................................11
   4. References....................................................11
      4.1. Normative References.....................................11
      4.2. Informative References...................................11
   Appendix A. Tables for multiplication in GF(2^32)................13
      A.1. The table amul0..........................................13
      A.2. The table amul1..........................................14
      A.3. The table amul2..........................................16
      A.4. The table amul3..........................................17

1. Introduction

   KCipher-2 is a stream cipher that uses 128-bit secret key and a 128-
   bit initialization vector. Since the algorithm for KCipher-2 was
   published in 2007 [SASC07], it has received attention from academia
   and industries. The security and performance of KCipher-2 have been


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   rigorously evaluated by the developers and other institutions
   [SECRYPT07], [ICETE07], [CRYPTEC], [KDDI]. No attack has been found
   on KCipher-2 as of this date. KCipher-2 can be efficiently
   implemented in software to provide fast encryption and decryption,
   owing to the uncomplicated design. Only four simple operations are
   used: exclusive-OR, addition, shift, and table lookup. When the
   algorithm is implemented in hardware, internal computations can be
   parallel aiming for greater efficiency. Moreover, since its internal
   state representation only amounts to several hundreds of bits,
   KCipher-2 is suitable for resource-limited environments. KCipher-2
   has been actively used in several industrial applications in Japan
   and has been submitted to an international standardization body
   (ISO/IEC 18033) [ISO18033] and evaluated to be a Japanese e-
   Government recommended cipher [CRYPTECLIST].

2. Algorithm Description

   In this section, we will describe the internal components of KCipher-
   2 and define the operations for deriving key streams from an input
   key and an initialization vector.

2.1. Notations

   All values in this document are stored in big-endian order (a. k. a.,
   network byte order). We use the following notations in the
   description of KCipher-2.

      XOR   Bitwise exclusive OR

      ^     Power operator

      +n    Integer addition modulo 2^n

      <<n   Left circular rotation by n bits

      >>n   Right circular rotation by n bits

      0x    Hexadecimal representation

     E[i] The (i + 1)th element of E when E is composed of consecutive
           multiple elements.

     GF   Galois field. GF(2^x) means the finite field of exactly 2^n
           elements.

     **   Multiplication of elements on the finite field GF(2^32)



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2.2. Internal State

   The internal state of KCipher-2 can be denoted by S. The internal
   state consists of its six sub-components: two feedback shift
   registers, FSR-A and FSR-B, and four internal registers, L1, R1, L2,
   and R2. We, therefore, often write S = (A, B, L1, R1, L2, R2), where
   A and B respectively refer FSR-A and FSR-B.

2.2.1. Feedback Shift Registers

   The two feedback shift registers (FSR) are separately called Feedback
   Shift Register A (FSR-A) and Feedback Shift Register B (FSR-B). FSR-A
   is composed of five 32-bit units that are consecutively arranged.
   Each of the units can be identified by A[0], A[1], A[2], A[3], and
   A[4]. Likewise, FSR-B is composed of eleven consecutive 32-bit units,
   B[0], ..., B[10]. All values stored in each 32-bit unit of FSR is in
   GF(2^32).

2.2.2. Internal registers

   Besides FSR, KCipher-2 has four internal registers to store
   intermediate computation results during operation. The four registers
   are named L1, R1, L2, and R2.

2.3. Operations

   There are three major operations that constitute the behavior of
   KCipher-2: Init, Next, and Stream. The Init operation initializes the
   internal values of the system. The Next operation derives new values
   of S' from the values of S, where S' and S refer the internal state.
   The Stream operation derives a key stream from the current state S.

2.3.1. The Next operation

   The Next operation takes the current state input S = (A, B, L1, R1,
   L2, R2) as input. The size of the input amounts to twenty of the 32-
   bit units in total (five units for A, eleven for B, and one for L1,
   R1, L2, and R2). It produces the next state S' = (A', B', L1', R1',
   L2', R2'). This operation is mainly used to generate secure key
   streams by applying non-linear functions for every cycle of KCipher-2.
   Besides, it is also used to initialize the system. The behaviors are
   distinguished by the input parameter that indicates the operation
   modes.

   The operation takes the following inputs:

   o  S = (A, B, L1, R1, L2, R2)


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   o  mode = {INIT, NORMAL}, where INIT means the operation is used for
      initialization and NORMAL means it is used for generating secure
      key streams.

   It outputs a new state,

   o  S' = (A', B', L1', R1', L2', R2')

   by performing the below steps:

   1.  Set registers in the nonlinear functions set

      L1' = SubK2(R2 +32 B[4]);
      R1' = SubK2(L2 +32 B[9]);
      R2' = SubK2(R1);
      L2' = SubK2(L1);

      for m from 0 to 3
         A'[m] = A[m + 1];

      for m from 0 to 9
         B'[m] = B[m + 1];

   * NOTE: SubK2 is a substitution function described in Section 2.4.2.

   2. Depending on the value of the operation mode, do the following:

       a. When the mode is INIT, do as follows:

          A'[4] = (a0 ** A[0]) XOR A[3] XOR NLF(B[0], R2, R1, A[4]);

          if A[2][30] is 1:
            B'[10] = (a1 ** B[0]) XOR B[1] XOR  B[6] XOR
                     ((a3 ^ A[2][31]) ** B[8]) XOR
                     NLF(B[10], L2, L1, A[0]);
          else if A[2][30] is 0:
            B'[10] = (a2 ** B[0]) XOR  B[1] XOR B[6] XOR
                     ((a3 ^ A[2][31]) ** B[8]) XOR
                     NLF(B[10], L2, L1, A[0]);

   * NOTE: The function NLF is described in Section 2.4.1.

       b. When the mode is NORMAL, NLF is not used, as follows:

          A'4 = (a0 ** A[0]) XOR A[3];




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          if A[2][30] is 1:
            B'[10] = (a1 ** B[0]) XOR B[1] XOR  B[6] XOR
                     ((a3 ^ A[2][31]) ** B[8]);
          else if A[2][30] is 0:
            B'[10] = (a2 ** B[0]) XOR  B[1] XOR B[6] XOR
                     ((a3 ^ A[2][31]) ** B[8]);

   * Note that A[2] is a 32-bit unit. Thus, A[2][j] is the value of the
   jth least significant bit of A[2], where 0 <= j <= 31.

   * Note that the operation ** to elements a0, a1, a2, and a3 are
   described in Section 2.4.4.

   3. Output S' = (A', B', L1', R1', L2', R2').

2.3.2. Init

   The Init operation takes a 128-bit key (K) and a 128-bit
   initialization vector (IV), and prepares the values of the state
   variables for generating key streams.

   o  K = (K[0], K[1], K[2], K[3]), where each K[i] is a 32-bit unit and
      0 <= i <= 3

   o  IV =(IV[0], IV[1], IV[2], IV[3]), where each IV[i] is a 32-bit
      unit and 0 <= i <= 3,

   and the output is an initialized state S, which will be referenced as
   S(0). The output is derived from the following steps:

   1. K is expanded to the 384-bit internal key IK = (IK[0], ...,
      IK[11]), where IK[i] is a 32-bit unit and 0 <= i <= 11. The
      expansion procedure is as follows:

      for m from 0 to 11
         if m is 0, 1, 2, or 3:
            IK[m] = K[m];
         else if m is 5, 6, 7, 9, 10, or 11:
            IK[m] = IK[m - 4] XOR IK[m - 1];
         else if m is 4:
            IK[4] = IK[0] XOR
            SubK2((IK[3] <<32 8) XOR (IK[3] >>32 24)) XOR
            (0x01, 0x00, 0x00, 0x00);
         else if m is 8:
            IK[8] = IK[4] XOR
            SubK2((IK[7] <<32 8) XOR (IK[7] >>32 24)) XOR
            (0x02, 0x00, 0x00, 0x00);


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   * NOTE: SubK2 is the substitution function described in Section 2.4.2.

   2. Initialize the feedback shift registers and the internal registers
      using the values of IK and IV as follows:

      for m from 0 to 4
         A[m] = IK[4 - m];

      B[0] = IK[10]; B[1] = IK[11]; B[2] = IV[0];  B[3] = IV[1];
      B[4] = IK[8];  B[5] = IK[9];  B[6] = IV[2];  B[7] = IV[3];
      B[8] = IK[7];  B[9] = IK[5];  B[10] = IK[6];

      L1 = R1 = L2 = R2 = 0x00000000;

      Set S as (A, B, L1, R1, L2, R2).

   3. Prepare the state values by applying the Next operation twenty-
      four times repeatedly as follows:

      for m from 1 to 24
         Set S' as Next(S, INIT);
         Set S as S';

   4. Output S.

2.3.3. Stream

   The Stream function derives a 64-bit key stream, Z, from the state
   values. Its input is an initialized state,

   o  S = (A, B, L1, R1, L2, R2)

   , and its output is Z = (ZH, ZL), where ZH and ZL are 32-bit units.
   Stream performs the following:

   1. Set register values

     ZH = NLF(B[10], L2, L1, A[0]);
     ZL = NLF(B[0], R2, R1, A[4]);

   2. Output Z = (ZH, ZL).

   * NOTE: The function NLF is described in Section 2.4.1.

2.4. Subroutines

   We explain the several functions used above: SubK2, NLF, and S-Box.


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2.4.1. NLF

   NLF is a non-linear function that takes the four 32-bit values, A, B,
   C, D, and outputs the 32-bit value, Q. The output Q is calculated by

      Q = (A +32 B) XOR C XOR D;

2.4.2. SubK2

   SubK2 is a substitution function, which is a permutation of GF(2^32),
   based on components from the Advanced Encryption Standard(AES) [FIPS-
   AES]. Its input is a 32-bit value divided into four 8-bit strings.
   Inside SubK2, an 8-to-8 bit substitution function, S-Box, is applied
   to each 8-bit separately, and then a 32-to-32 bit linear permutation
   is applied to the whole 32-bit string. Our S-Box function is
   identical to the S-Box operation of AES, and our linear permutation
   is identical to the AES Mix Column operation.

   Consider the input of SubK2 as a 32-bit value W = (w[3], w[2], w[1],
   w[0]), where each sub-element of w is an 8-bit unit. Prepare two 32-
   bit temporary storages T = (t[3], t[2], t[1], t[0]) and Q = (q[3],
   q[2], q[1], q[0]), where t[i] and q[i] are 8-bit units and 0 <= i <=
   3.

   The 32-bit output Q is obtained from the following procedures:

   1. Apply S-Box to each 8-bit input string. Note that S-Box will be
      defined in Section 2.4.3.

      for m from 0 to 3
         t[m] = S-Box(w[m]);

   2. Calculate q by the matrix multiplication, Q = M * T in GF(2^8) of
      the irreducible polynomial f(x) = x^8 + x^4 + x^3 + x + 1, where

       o  Q is an 1 by 4 matrix, (q[0], q[1], q[2], q[3))

       o  M is a 4 by 4 matrix,
               (02,  03,  01,  01,
                01,  02,  03,  01,
                01,  01,  02,  03,
                03,  01,  01,  02)

       o  T is an 1 by 4 matrix, (t[0], t[1], t[2], t[3]).

      Namely, Q can be calculated like as follows:



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      q[0] = 2 * t[0] + 3 * t[1] + t[2] + t[3];
      q[1] = t[0] + 2 * t[1] + 3 * t[2] + t[3];
      q[2] = t[0] + t[1] + 2 * t[2] + 3 * t[3];
      q[3] = 3 * t[0] + t[1] + t[2] + 2 * t[3];

   3. Output Q = (q[3], q[2], q[1], q[0]).

2.4.3. S-Box

   S-Box is a substitution that can be done by a simple table lookup
   operation. Thus, S-Box can be defined by the following value table:

   S-Box[256] = {
      0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5,
      0x30, 0x01, 0x67, 0x2b, 0xfe, 0xd7, 0xab, 0x76,
      0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, 0x47, 0xf0,
      0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 0xc0,
      0xb7, 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc,
      0x34, 0xa5, 0xe5, 0xf1, 0x71, 0xd8, 0x31, 0x15,
      0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, 0x9a,
      0x07, 0x12, 0x80, 0xe2, 0xeb, 0x27, 0xb2, 0x75,
      0x09, 0x83, 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0,
      0x52, 0x3b, 0xd6, 0xb3, 0x29, 0xe3, 0x2f, 0x84,
      0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b,
      0x6a, 0xcb, 0xbe, 0x39, 0x4a, 0x4c, 0x58, 0xcf,
      0xd0, 0xef, 0xaa, 0xfb, 0x43, 0x4d, 0x33, 0x85,
      0x45, 0xf9, 0x02, 0x7f, 0x50, 0x3c, 0x9f, 0xa8,
      0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5,
      0xbc, 0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2,
      0xcd, 0x0c, 0x13, 0xec, 0x5f, 0x97, 0x44, 0x17,
      0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, 0x73,
      0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 0x88,
      0x46, 0xee, 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb,
      0xe0, 0x32, 0x3a, 0x0a, 0x49, 0x06, 0x24, 0x5c,
      0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79,
      0xe7, 0xc8, 0x37, 0x6d, 0x8d, 0xd5, 0x4e, 0xa9,
      0x6c, 0x56, 0xf4, 0xea, 0x65, 0x7a, 0xae, 0x08,
      0xba, 0x78, 0x25, 0x2e, 0x1c, 0xa6, 0xb4, 0xc6,
      0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a,
      0x70, 0x3e, 0xb5, 0x66, 0x48, 0x03, 0xf6, 0x0e,
      0x61, 0x35, 0x57, 0xb9, 0x86, 0xc1, 0x1d, 0x9e,
      0xe1, 0xf8, 0x98, 0x11, 0x69, 0xd9, 0x8e, 0x94,
      0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf,
      0x8c, 0xa1, 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68,
      0x41, 0x99, 0x2d, 0x0f, 0xb0, 0x54, 0xbb, 0x16 };




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2.4.4. Multiplications in GF(2^32)

   In the Init operation, 32-bit values are multiplied to a0, a1, a2, or
   a3 in GF(2^32). The multiplication can be simply done with the table
   look-up operation when the tables described in Appendix A are used.

   For example, consider the input as a 32-bit value w, which represents
   an element of GF(2^32), the output 32-bit string w' = a0 ** w can be
   obtained by

      w' = (w <<32 8) XOR amul0[w >>32 24];

   The amul0 table is defined in Appendix A.1.

   Likewise, multiplications of (a1 ** w), (a2 ** w), and (a3 ** w) can
   be obtained using the same procedure. The only difference is that
   each of the multiplications uses different tables, amul1, amul2, and
   amul3, and these tables are shown in Appendix A.2, A.3, and A.4,
   respectively.

2.5. Encryption/Decryption scheme

   In this section, we use the notation S(i) to specifically reference
   the values of the internal state at an arbitrary, discrete temporal
   moment (a.k.a., a cycle) i (i >= 0) after the initialization.

2.5.1. Key stream generation

   Given a 128-bit key K, a 128-bit initialization vector IV, KCipher-2
   is initialized as follows:

      S(0) = Init(K, IV);

   , where S(0) is a state representation. With an initialized state
   S(i), where i >= 0, a 64-bit key stream X(i) can be obtained using
   the Stream operation, as follows:

      X(i) = Stream(S(i));

   To generate a new key stream X(i + 1), use the Next operation and the
   Stream operation as follows:

      S(i + 1) = Next(S(i), NORMAL);
      X(i + 1) = Stream(S(i + 1));





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2.5.2. Encryption/Decryption of a message

   Given a 64-bit message block M and a key stream X, an encrypted
   message E is obtained by

      E = M XOR X;

   Conversely, the decrypted message D is obtained by

      D = E XOR X;

   The original message M and the decrypted message D are identical when
   the same key stream is used.

3. Security Considerations

   We recommend that re-initializing and re-keying after 2^58 cycles of
   KCipher-2, which means after generating 2^64 key stream bits.

4. References

4.1. Normative References

   [ISO18033] "Information technology - Security techniques - Encryption
             algorithms - Part 4: Stream ciphers", ISO/IEC 18033-4, 2011.

   [FIPS-AES] "Specification for the Advanced Encryption Standard (AES)",
             Federal Information Processing Standard (FIPS) Publication
             197, November 2001.

4.2. Informative References

   [SASC07] S. Kiyomoto, T. Tanaka, and K. Sakurai, "A Word-Oriented
             Stream Cipher Using Clock Control", Proc. SASC 2007 pp.
             260-274.

   [SECRYPT07] S. Kiyomoto, T. Tanaka, and K. Sakurai, "K2: A Stream
             Cipher Algorithm Using Dynamic Feedback Control", Proc.
             SECRYPT 2007 pp. 204-213.

   [ICETE07] S. Kiyomoto, T. Tanaka, and K. Sakurai, "K2 Stream Cipher",
             Proc. ICETE 2007 pp. 214-226.

   [CRYPTEC] A. Bogdanov, B. Preneel, and V. Rijmen, "Security
             Evaluation of the K2 Stream Cipher", 2010.
             http://www.cryptrec.go.jp/english/estimation.html



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   [CRYPTECLIST] Cryptography Research and Evaluation Committees.
             http://www.cryptrec.go.jp/english/estimation.html

   [KDDI] B. Roy, "Evaluation of the Word-Oriented Stream Cipher: K2",
             2009.
             http://www.kddilabs.jp/kcipher2/kcipher2.htm











































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Appendix A.                 Tables for multiplication in GF(2^32)

A.1. The table amul0

   amul0[256] = {
      0x00000000,0xB6086D1A,0xAF10DA34,0x1918B72E,
      0x9D207768,0x2B281A72,0x3230AD5C,0x8438C046,
      0xF940EED0,0x4F4883CA,0x565034E4,0xE05859FE,
      0x646099B8,0xD268F4A2,0xCB70438C,0x7D782E96,
      0x31801F63,0x87887279,0x9E90C557,0x2898A84D,
      0xACA0680B,0x1AA80511,0x03B0B23F,0xB5B8DF25,
      0xC8C0F1B3,0x7EC89CA9,0x67D02B87,0xD1D8469D,
      0x55E086DB,0xE3E8EBC1,0xFAF05CEF,0x4CF831F5,
      0x62C33EC6,0xD4CB53DC,0xCDD3E4F2,0x7BDB89E8,
      0xFFE349AE,0x49EB24B4,0x50F3939A,0xE6FBFE80,
      0x9B83D016,0x2D8BBD0C,0x34930A22,0x829B6738,
      0x06A3A77E,0xB0ABCA64,0xA9B37D4A,0x1FBB1050,
      0x534321A5,0xE54B4CBF,0xFC53FB91,0x4A5B968B,
      0xCE6356CD,0x786B3BD7,0x61738CF9,0xD77BE1E3,
      0xAA03CF75,0x1C0BA26F,0x05131541,0xB31B785B,
      0x3723B81D,0x812BD507,0x98336229,0x2E3B0F33,
      0xC4457C4F,0x724D1155,0x6B55A67B,0xDD5DCB61,
      0x59650B27,0xEF6D663D,0xF675D113,0x407DBC09,
      0x3D05929F,0x8B0DFF85,0x921548AB,0x241D25B1,
      0xA025E5F7,0x162D88ED,0x0F353FC3,0xB93D52D9,
      0xF5C5632C,0x43CD0E36,0x5AD5B918,0xECDDD402,
      0x68E51444,0xDEED795E,0xC7F5CE70,0x71FDA36A,
      0x0C858DFC,0xBA8DE0E6,0xA39557C8,0x159D3AD2,
      0x91A5FA94,0x27AD978E,0x3EB520A0,0x88BD4DBA,
      0xA6864289,0x108E2F93,0x099698BD,0xBF9EF5A7,
      0x3BA635E1,0x8DAE58FB,0x94B6EFD5,0x22BE82CF,
      0x5FC6AC59,0xE9CEC143,0xF0D6766D,0x46DE1B77,
      0xC2E6DB31,0x74EEB62B,0x6DF60105,0xDBFE6C1F,
      0x97065DEA,0x210E30F0,0x381687DE,0x8E1EEAC4,
      0x0A262A82,0xBC2E4798,0xA536F0B6,0x133E9DAC,
      0x6E46B33A,0xD84EDE20,0xC156690E,0x775E0414,
      0xF366C452,0x456EA948,0x5C761E66,0xEA7E737C,
      0x4B8AF89E,0xFD829584,0xE49A22AA,0x52924FB0,
      0xD6AA8FF6,0x60A2E2EC,0x79BA55C2,0xCFB238D8,
      0xB2CA164E,0x04C27B54,0x1DDACC7A,0xABD2A160,
      0x2FEA6126,0x99E20C3C,0x80FABB12,0x36F2D608,
      0x7A0AE7FD,0xCC028AE7,0xD51A3DC9,0x631250D3,
      0xE72A9095,0x5122FD8F,0x483A4AA1,0xFE3227BB,
      0x834A092D,0x35426437,0x2C5AD319,0x9A52BE03,
      0x1E6A7E45,0xA862135F,0xB17AA471,0x0772C96B,
      0x2949C658,0x9F41AB42,0x86591C6C,0x30517176,
      0xB469B130,0x0261DC2A,0x1B796B04,0xAD71061E,


Kiyomoto & Shin       Expires September 28, 2011              [Page 13]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


      0xD0092888,0x66014592,0x7F19F2BC,0xC9119FA6,
      0x4D295FE0,0xFB2132FA,0xE23985D4,0x5431E8CE,
      0x18C9D93B,0xAEC1B421,0xB7D9030F,0x01D16E15,
      0x85E9AE53,0x33E1C349,0x2AF97467,0x9CF1197D,
      0xE18937EB,0x57815AF1,0x4E99EDDF,0xF89180C5,
      0x7CA94083,0xCAA12D99,0xD3B99AB7,0x65B1F7AD,
      0x8FCF84D1,0x39C7E9CB,0x20DF5EE5,0x96D733FF,
      0x12EFF3B9,0xA4E79EA3,0xBDFF298D,0x0BF74497,
      0x768F6A01,0xC087071B,0xD99FB035,0x6F97DD2F,
      0xEBAF1D69,0x5DA77073,0x44BFC75D,0xF2B7AA47,
      0xBE4F9BB2,0x0847F6A8,0x115F4186,0xA7572C9C,
      0x236FECDA,0x956781C0,0x8C7F36EE,0x3A775BF4,
      0x470F7562,0xF1071878,0xE81FAF56,0x5E17C24C,
      0xDA2F020A,0x6C276F10,0x753FD83E,0xC337B524,
      0xED0CBA17,0x5B04D70D,0x421C6023,0xF4140D39,
      0x702CCD7F,0xC624A065,0xDF3C174B,0x69347A51,
      0x144C54C7,0xA24439DD,0xBB5C8EF3,0x0D54E3E9,
      0x896C23AF,0x3F644EB5,0x267CF99B,0x90749481,
      0xDC8CA574,0x6A84C86E,0x739C7F40,0xC594125A,
      0x41ACD21C,0xF7A4BF06,0xEEBC0828,0x58B46532,
      0x25CC4BA4,0x93C426BE,0x8ADC9190,0x3CD4FC8A,
      0xB8EC3CCC,0x0EE451D6,0x17FCE6F8,0xA1F48BE2 };

A.2. The table amul1

   amul1[256] = {
      0x00000000,0xA0F5FC2E,0x6DC7D55C,0xCD322972,
      0xDAA387B8,0x7A567B96,0xB76452E4,0x1791AECA,
      0x996B235D,0x399EDF73,0xF4ACF601,0x54590A2F,
      0x43C8A4E5,0xE33D58CB,0x2E0F71B9,0x8EFA8D97,
      0x1FD646BA,0xBF23BA94,0x721193E6,0xD2E46FC8,
      0xC575C102,0x65803D2C,0xA8B2145E,0x0847E870,
      0x86BD65E7,0x264899C9,0xEB7AB0BB,0x4B8F4C95,
      0x5C1EE25F,0xFCEB1E71,0x31D93703,0x912CCB2D,
      0x3E818C59,0x9E747077,0x53465905,0xF3B3A52B,
      0xE4220BE1,0x44D7F7CF,0x89E5DEBD,0x29102293,
      0xA7EAAF04,0x071F532A,0xCA2D7A58,0x6AD88676,
      0x7D4928BC,0xDDBCD492,0x108EFDE0,0xB07B01CE,
      0x2157CAE3,0x81A236CD,0x4C901FBF,0xEC65E391,
      0xFBF44D5B,0x5B01B175,0x96339807,0x36C66429,
      0xB83CE9BE,0x18C91590,0xD5FB3CE2,0x750EC0CC,
      0x629F6E06,0xC26A9228,0x0F58BB5A,0xAFAD4774,
      0x7C2F35B2,0xDCDAC99C,0x11E8E0EE,0xB11D1CC0,
      0xA68CB20A,0x06794E24,0xCB4B6756,0x6BBE9B78,
      0xE54416EF,0x45B1EAC1,0x8883C3B3,0x28763F9D,
      0x3FE79157,0x9F126D79,0x5220440B,0xF2D5B825,
      0x63F97308,0xC30C8F26,0x0E3EA654,0xAECB5A7A,


Kiyomoto & Shin       Expires September 28, 2011              [Page 14]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


      0xB95AF4B0,0x19AF089E,0xD49D21EC,0x7468DDC2,
      0xFA925055,0x5A67AC7B,0x97558509,0x37A07927,
      0x2031D7ED,0x80C42BC3,0x4DF602B1,0xED03FE9F,
      0x42AEB9EB,0xE25B45C5,0x2F696CB7,0x8F9C9099,
      0x980D3E53,0x38F8C27D,0xF5CAEB0F,0x553F1721,
      0xDBC59AB6,0x7B306698,0xB6024FEA,0x16F7B3C4,
      0x01661D0E,0xA193E120,0x6CA1C852,0xCC54347C,
      0x5D78FF51,0xFD8D037F,0x30BF2A0D,0x904AD623,
      0x87DB78E9,0x272E84C7,0xEA1CADB5,0x4AE9519B,
      0xC413DC0C,0x64E62022,0xA9D40950,0x0921F57E,
      0x1EB05BB4,0xBE45A79A,0x73778EE8,0xD38272C6,
      0xF85E6A49,0x58AB9667,0x9599BF15,0x356C433B,
      0x22FDEDF1,0x820811DF,0x4F3A38AD,0xEFCFC483,
      0x61354914,0xC1C0B53A,0x0CF29C48,0xAC076066,
      0xBB96CEAC,0x1B633282,0xD6511BF0,0x76A4E7DE,
      0xE7882CF3,0x477DD0DD,0x8A4FF9AF,0x2ABA0581,
      0x3D2BAB4B,0x9DDE5765,0x50EC7E17,0xF0198239,
      0x7EE30FAE,0xDE16F380,0x1324DAF2,0xB3D126DC,
      0xA4408816,0x04B57438,0xC9875D4A,0x6972A164,
      0xC6DFE610,0x662A1A3E,0xAB18334C,0x0BEDCF62,
      0x1C7C61A8,0xBC899D86,0x71BBB4F4,0xD14E48DA,
      0x5FB4C54D,0xFF413963,0x32731011,0x9286EC3F,
      0x851742F5,0x25E2BEDB,0xE8D097A9,0x48256B87,
      0xD909A0AA,0x79FC5C84,0xB4CE75F6,0x143B89D8,
      0x03AA2712,0xA35FDB3C,0x6E6DF24E,0xCE980E60,
      0x406283F7,0xE0977FD9,0x2DA556AB,0x8D50AA85,
      0x9AC1044F,0x3A34F861,0xF706D113,0x57F32D3D,
      0x84715FFB,0x2484A3D5,0xE9B68AA7,0x49437689,
      0x5ED2D843,0xFE27246D,0x33150D1F,0x93E0F131,
      0x1D1A7CA6,0xBDEF8088,0x70DDA9FA,0xD02855D4,
      0xC7B9FB1E,0x674C0730,0xAA7E2E42,0x0A8BD26C,
      0x9BA71941,0x3B52E56F,0xF660CC1D,0x56953033,
      0x41049EF9,0xE1F162D7,0x2CC34BA5,0x8C36B78B,
      0x02CC3A1C,0xA239C632,0x6F0BEF40,0xCFFE136E,
      0xD86FBDA4,0x789A418A,0xB5A868F8,0x155D94D6,
      0xBAF0D3A2,0x1A052F8C,0xD73706FE,0x77C2FAD0,
      0x6053541A,0xC0A6A834,0x0D948146,0xAD617D68,
      0x239BF0FF,0x836E0CD1,0x4E5C25A3,0xEEA9D98D,
      0xF9387747,0x59CD8B69,0x94FFA21B,0x340A5E35,
      0xA5269518,0x05D36936,0xC8E14044,0x6814BC6A,
      0x7F8512A0,0xDF70EE8E,0x1242C7FC,0xB2B73BD2,
      0x3C4DB645,0x9CB84A6B,0x518A6319,0xF17F9F37,
      0xE6EE31FD,0x461BCDD3,0x8B29E4A1,0x2BDC188F };






Kiyomoto & Shin       Expires September 28, 2011              [Page 15]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


A.3. The table amul2

   amul2[256] = {
      0x00000000,0x5BF87F93,0xB6BDFE6B,0xED4581F8,
      0x2137B1D6,0x7ACFCE45,0x978A4FBD,0xCC72302E,
      0x426E2FE1,0x19965072,0xF4D3D18A,0xAF2BAE19,
      0x63599E37,0x38A1E1A4,0xD5E4605C,0x8E1C1FCF,
      0x84DC5E8F,0xDF24211C,0x3261A0E4,0x6999DF77,
      0xA5EBEF59,0xFE1390CA,0x13561132,0x48AE6EA1,
      0xC6B2716E,0x9D4A0EFD,0x700F8F05,0x2BF7F096,
      0xE785C0B8,0xBC7DBF2B,0x51383ED3,0x0AC04140,
      0x45F5BC53,0x1E0DC3C0,0xF3484238,0xA8B03DAB,
      0x64C20D85,0x3F3A7216,0xD27FF3EE,0x89878C7D,
      0x079B93B2,0x5C63EC21,0xB1266DD9,0xEADE124A,
      0x26AC2264,0x7D545DF7,0x9011DC0F,0xCBE9A39C,
      0xC129E2DC,0x9AD19D4F,0x77941CB7,0x2C6C6324,
      0xE01E530A,0xBBE62C99,0x56A3AD61,0x0D5BD2F2,
      0x8347CD3D,0xD8BFB2AE,0x35FA3356,0x6E024CC5,
      0xA2707CEB,0xF9880378,0x14CD8280,0x4F35FD13,
      0x8AA735A6,0xD15F4A35,0x3C1ACBCD,0x67E2B45E,
      0xAB908470,0xF068FBE3,0x1D2D7A1B,0x46D50588,
      0xC8C91A47,0x933165D4,0x7E74E42C,0x258C9BBF,
      0xE9FEAB91,0xB206D402,0x5F4355FA,0x04BB2A69,
      0x0E7B6B29,0x558314BA,0xB8C69542,0xE33EEAD1,
      0x2F4CDAFF,0x74B4A56C,0x99F12494,0xC2095B07,
      0x4C1544C8,0x17ED3B5B,0xFAA8BAA3,0xA150C530,
      0x6D22F51E,0x36DA8A8D,0xDB9F0B75,0x806774E6,
      0xCF5289F5,0x94AAF666,0x79EF779E,0x2217080D,
      0xEE653823,0xB59D47B0,0x58D8C648,0x0320B9DB,
      0x8D3CA614,0xD6C4D987,0x3B81587F,0x607927EC,
      0xAC0B17C2,0xF7F36851,0x1AB6E9A9,0x414E963A,
      0x4B8ED77A,0x1076A8E9,0xFD332911,0xA6CB5682,
      0x6AB966AC,0x3141193F,0xDC0498C7,0x87FCE754,
      0x09E0F89B,0x52188708,0xBF5D06F0,0xE4A57963,
      0x28D7494D,0x732F36DE,0x9E6AB726,0xC592C8B5,
      0x59036A01,0x02FB1592,0xEFBE946A,0xB446EBF9,
      0x7834DBD7,0x23CCA444,0xCE8925BC,0x95715A2F,
      0x1B6D45E0,0x40953A73,0xADD0BB8B,0xF628C418,
      0x3A5AF436,0x61A28BA5,0x8CE70A5D,0xD71F75CE,
      0xDDDF348E,0x86274B1D,0x6B62CAE5,0x309AB576,
      0xFCE88558,0xA710FACB,0x4A557B33,0x11AD04A0,
      0x9FB11B6F,0xC44964FC,0x290CE504,0x72F49A97,
      0xBE86AAB9,0xE57ED52A,0x083B54D2,0x53C32B41,
      0x1CF6D652,0x470EA9C1,0xAA4B2839,0xF1B357AA,
      0x3DC16784,0x66391817,0x8B7C99EF,0xD084E67C,
      0x5E98F9B3,0x05608620,0xE82507D8,0xB3DD784B,
      0x7FAF4865,0x245737F6,0xC912B60E,0x92EAC99D,


Kiyomoto & Shin       Expires September 28, 2011              [Page 16]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


      0x982A88DD,0xC3D2F74E,0x2E9776B6,0x756F0925,
      0xB91D390B,0xE2E54698,0x0FA0C760,0x5458B8F3,
      0xDA44A73C,0x81BCD8AF,0x6CF95957,0x370126C4,
      0xFB7316EA,0xA08B6979,0x4DCEE881,0x16369712,
      0xD3A45FA7,0x885C2034,0x6519A1CC,0x3EE1DE5F,
      0xF293EE71,0xA96B91E2,0x442E101A,0x1FD66F89,
      0x91CA7046,0xCA320FD5,0x27778E2D,0x7C8FF1BE,
      0xB0FDC190,0xEB05BE03,0x06403FFB,0x5DB84068,
      0x57780128,0x0C807EBB,0xE1C5FF43,0xBA3D80D0,
      0x764FB0FE,0x2DB7CF6D,0xC0F24E95,0x9B0A3106,
      0x15162EC9,0x4EEE515A,0xA3ABD0A2,0xF853AF31,
      0x34219F1F,0x6FD9E08C,0x829C6174,0xD9641EE7,
      0x9651E3F4,0xCDA99C67,0x20EC1D9F,0x7B14620C,
      0xB7665222,0xEC9E2DB1,0x01DBAC49,0x5A23D3DA,
      0xD43FCC15,0x8FC7B386,0x6282327E,0x397A4DED,
      0xF5087DC3,0xAEF00250,0x43B583A8,0x184DFC3B,
      0x128DBD7B,0x4975C2E8,0xA4304310,0xFFC83C83,
      0x33BA0CAD,0x6842733E,0x8507F2C6,0xDEFF8D55,
      0x50E3929A,0x0B1BED09,0xE65E6CF1,0xBDA61362,
      0x71D4234C,0x2A2C5CDF,0xC769DD27,0x9C91A2B4 };

A.4. The table amul3

   amul3[256] = {
      0x00000000,0x4559568B,0x8AB2AC73,0xCFEBFAF8,
      0x71013DE6,0x34586B6D,0xFBB39195,0xBEEAC71E,
      0xE2027AA9,0xA75B2C22,0x68B0D6DA,0x2DE98051,
      0x9303474F,0xD65A11C4,0x19B1EB3C,0x5CE8BDB7,
      0xA104F437,0xE45DA2BC,0x2BB65844,0x6EEF0ECF,
      0xD005C9D1,0x955C9F5A,0x5AB765A2,0x1FEE3329,
      0x43068E9E,0x065FD815,0xC9B422ED,0x8CED7466,
      0x3207B378,0x775EE5F3,0xB8B51F0B,0xFDEC4980,
      0x27088D6E,0x6251DBE5,0xADBA211D,0xE8E37796,
      0x5609B088,0x1350E603,0xDCBB1CFB,0x99E24A70,
      0xC50AF7C7,0x8053A14C,0x4FB85BB4,0x0AE10D3F,
      0xB40BCA21,0xF1529CAA,0x3EB96652,0x7BE030D9,
      0x860C7959,0xC3552FD2,0x0CBED52A,0x49E783A1,
      0xF70D44BF,0xB2541234,0x7DBFE8CC,0x38E6BE47,
      0x640E03F0,0x2157557B,0xEEBCAF83,0xABE5F908,
      0x150F3E16,0x5056689D,0x9FBD9265,0xDAE4C4EE,
      0x4E107FDC,0x0B492957,0xC4A2D3AF,0x81FB8524,
      0x3F11423A,0x7A4814B1,0xB5A3EE49,0xF0FAB8C2,
      0xAC120575,0xE94B53FE,0x26A0A906,0x63F9FF8D,
      0xDD133893,0x984A6E18,0x57A194E0,0x12F8C26B,
      0xEF148BEB,0xAA4DDD60,0x65A62798,0x20FF7113,
      0x9E15B60D,0xDB4CE086,0x14A71A7E,0x51FE4CF5,
      0x0D16F142,0x484FA7C9,0x87A45D31,0xC2FD0BBA,


Kiyomoto & Shin       Expires September 28, 2011              [Page 17]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


      0x7C17CCA4,0x394E9A2F,0xF6A560D7,0xB3FC365C,
      0x6918F2B2,0x2C41A439,0xE3AA5EC1,0xA6F3084A,
      0x1819CF54,0x5D4099DF,0x92AB6327,0xD7F235AC,
      0x8B1A881B,0xCE43DE90,0x01A82468,0x44F172E3,
      0xFA1BB5FD,0xBF42E376,0x70A9198E,0x35F04F05,
      0xC81C0685,0x8D45500E,0x42AEAAF6,0x07F7FC7D,
      0xB91D3B63,0xFC446DE8,0x33AF9710,0x76F6C19B,
      0x2A1E7C2C,0x6F472AA7,0xA0ACD05F,0xE5F586D4,
      0x5B1F41CA,0x1E461741,0xD1ADEDB9,0x94F4BB32,
      0x9C20FEDD,0xD979A856,0x169252AE,0x53CB0425,
      0xED21C33B,0xA87895B0,0x67936F48,0x22CA39C3,
      0x7E228474,0x3B7BD2FF,0xF4902807,0xB1C97E8C,
      0x0F23B992,0x4A7AEF19,0x859115E1,0xC0C8436A,
      0x3D240AEA,0x787D5C61,0xB796A699,0xF2CFF012,
      0x4C25370C,0x097C6187,0xC6979B7F,0x83CECDF4,
      0xDF267043,0x9A7F26C8,0x5594DC30,0x10CD8ABB,
      0xAE274DA5,0xEB7E1B2E,0x2495E1D6,0x61CCB75D,
      0xBB2873B3,0xFE712538,0x319ADFC0,0x74C3894B,
      0xCA294E55,0x8F7018DE,0x409BE226,0x05C2B4AD,
      0x592A091A,0x1C735F91,0xD398A569,0x96C1F3E2,
      0x282B34FC,0x6D726277,0xA299988F,0xE7C0CE04,
      0x1A2C8784,0x5F75D10F,0x909E2BF7,0xD5C77D7C,
      0x6B2DBA62,0x2E74ECE9,0xE19F1611,0xA4C6409A,
      0xF82EFD2D,0xBD77ABA6,0x729C515E,0x37C507D5,
      0x892FC0CB,0xCC769640,0x039D6CB8,0x46C43A33,
      0xD2308101,0x9769D78A,0x58822D72,0x1DDB7BF9,
      0xA331BCE7,0xE668EA6C,0x29831094,0x6CDA461F,
      0x3032FBA8,0x756BAD23,0xBA8057DB,0xFFD90150,
      0x4133C64E,0x046A90C5,0xCB816A3D,0x8ED83CB6,
      0x73347536,0x366D23BD,0xF986D945,0xBCDF8FCE,
      0x023548D0,0x476C1E5B,0x8887E4A3,0xCDDEB228,
      0x91360F9F,0xD46F5914,0x1B84A3EC,0x5EDDF567,
      0xE0373279,0xA56E64F2,0x6A859E0A,0x2FDCC881,
      0xF5380C6F,0xB0615AE4,0x7F8AA01C,0x3AD3F697,
      0x84393189,0xC1606702,0x0E8B9DFA,0x4BD2CB71,
      0x173A76C6,0x5263204D,0x9D88DAB5,0xD8D18C3E,
      0x663B4B20,0x23621DAB,0xEC89E753,0xA9D0B1D8,
      0x543CF858,0x1165AED3,0xDE8E542B,0x9BD702A0,
      0x253DC5BE,0x60649335,0xAF8F69CD,0xEAD63F46,
      0xB63E82F1,0xF367D47A,0x3C8C2E82,0x79D57809,
      0xC73FBF17,0x8266E99C,0x4D8D1364,0x08D445EF };








Kiyomoto & Shin       Expires September 28, 2011              [Page 18]

Internet-Draft         KCipher-2 Stream Cipher               March 2011


Authors' Addresses

   Shinsaku Kiyomoto
   KDDI R&D Laboratories, Inc.
   2-1-15 Ohara, Fujimino-shi,
   Saitama 356-8502, Japan.

   Phone: +81-49-278-7885
   Fax: +81-49-278-7510
   Email: kiyomoto@kddilabs.jp


   Wook Shin
   KDDI R&D Laboratories, Inc.
   2-1-15 Ohara, Fujimino-shi,
   Saitama 356-8502, Japan.

   Email: wookshin@kddilabs.jp































Kiyomoto & Shin       Expires September 28, 2011              [Page 19]


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