Internet DraftIndependent Submission S. KiyomotoIntended status: InformationalRequest for Comments: 7008 W. ShinExpires: December 2013Category: Informational KDDI R&D Laboratories, Inc.June 18,ISSN: 2070-1721 August 2013 A Description of the KCipher-2 Encryption Algorithmdraft-kiyomoto-kcipher2-09.txtAbstract 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 asAs of thetimepublication of thisdocument was written.document, no security vulnerabilities have been found. 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 ofthis MemoThisInternet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.Memo This documentmay not be modified, and derivative works of it mayis notbe created, andan Internet Standards Track specification; itmay not beis publishedexcept as an Internet-Draft. Internet-Drafts are working documents offor informational purposes. This is a contribution to theInternet Engineering Task Force (IETF). Note thatRFC Series, independently of any othergroups may also distribute working documents as Internet-Drafts.RFC stream. Thelist of current Internet- Drafts isRFC Editor has chosen to publish this document athttp://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents validits discretion and makes no statement about its value for implementation or deployment. 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Please review these documents carefully, as they describe your rights and restrictions with respect to this document.Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License.Table of Contents 1.Introduction...................................................3Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. AlgorithmDescription..........................................4Description . . . . . . . . . . . . . . . . . . . . 3 2.1.Notations.................................................4Notations . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. InternalState............................................4State . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1. Feedback ShiftRegisters.............................5Registers . . . . . . . . . . . . . . . 4 2.2.2. Internalregisters...................................5Registers . . . . . . . . . . . . . . . . . . 5 2.3.Operations................................................5Operations . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.1.next()...............................................5next() . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.2.init()...............................................7init() . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.3.stream().............................................8stream() . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.Subroutines...............................................9Subroutines . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1.NLF()................................................9NLF() . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2.sub_K2().............................................9sub_K2() . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.3.S_box().............................................10S_box() . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.4. Multiplications inGF(2#32).........................11GF(2#32) . . . . . . . . . . . . . 11 2.5.Encryption/Decryption scheme.............................13Encryption and Decryption Scheme . . . . . . . . . . . . . 13 2.5.1. Keystream generation...............................13Stream Generation . . . . . . . . . . . . . . . . 13 2.5.2.Encryption/DecryptionEncryption and Decryption of amessage..................14Message . . . . . . . . 14 3. SecurityConsiderations.......................................14Considerations . . . . . . . . . . . . . . . . . . . 14 4.References....................................................14References . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.1. NormativeReferences.....................................14References . . . . . . . . . . . . . . . . . . . 14 4.2. InformativeReferences...................................15References . . . . . . . . . . . . . . . . . . 14 Appendix A. Tables formultiplicationMultiplication inGF(2#32)................16GF(2#32) . . . . . . . . 16 A.1. The tableamul0..........................................16amul0 . . . . . . . . . . . . . . . . . . . . . 16 A.2. The tableamul1..........................................17amul1 . . . . . . . . . . . . . . . . . . . . . 17 A.3. The tableamul2..........................................19amul2 . . . . . . . . . . . . . . . . . . . . . 19 A.4. The tableamul3..........................................20amul3 . . . . . . . . . . . . . . . . . . . . . 20 Appendix B. Asimple implementation exampleSimple Implementation Example ofKCipher-2.........22KCipher-2 . . . . 22 B.1. CodecomponentsComponents I - Definitions anddeclarations.........22Declarations . . . . . 22 B.2. CodecomponentsComponents II -Functions...........................23Functions . . . . . . . . . . . . . . 23 B.3. Usecase.................................................28Case . . . . . . . . . . . . . . . . . . . . . . . . . 28 Appendix C. TestVectors.........................................29Vectors . . . . . . . . . . . . . . . . . . . . 28 C.1. Keystream generation examples...........................29Stream Generation Examples . . . . . . . . . . . . . . 28 C.2. Anotherkey stream generationKey Stream Generation with thestate values......30State Values . . . 29 C.2.1. S afterinit(1).....................................30init(1) . . . . . . . . . . . . . . . . . . . 30 C.2.2. S afterinit(2).....................................30init(2) . . . . . . . . . . . . . . . . . . . 30 C.2.3. S afterinit(3).....................................31init(3) . . . . . . . . . . . . . . . . . . . 30 C.2.4. S afterinit(4).....................................31init(4) . . . . . . . . . . . . . . . . . . . 31 C.2.5. S afterinit(5).....................................31init(5) . . . . . . . . . . . . . . . . . . . 31 C.2.6. S afterinit(6).....................................31init(6) . . . . . . . . . . . . . . . . . . . 31 C.2.7. S afterinit(7).....................................32init(7) . . . . . . . . . . . . . . . . . . . 31 C.2.8. S afterinit(8).....................................32init(8) . . . . . . . . . . . . . . . . . . . 32 C.2.9. S afterinit(9).....................................32init(9) . . . . . . . . . . . . . . . . . . . 32 C.2.10. S afterinit(10)...................................32init(10) . . . . . . . . . . . . . . . . . . . 32 C.2.11. S afterinit(11)...................................33init(11) . . . . . . . . . . . . . . . . . . . 32 C.2.12. S afterinit(12)...................................33init(12) . . . . . . . . . . . . . . . . . . . 33 C.2.13. S afterinit(13)...................................33init(13) . . . . . . . . . . . . . . . . . . . 33 C.2.14. S afterinit(14)...................................33init(14) . . . . . . . . . . . . . . . . . . . 33 C.2.15. S afterinit(15)...................................34init(15) . . . . . . . . . . . . . . . . . . . 33 C.2.16. S afterinit(16)...................................34init(16) . . . . . . . . . . . . . . . . . . . 34 C.2.17. S afterinit(17)...................................34init(17) . . . . . . . . . . . . . . . . . . . 34 C.2.18. S afterinit(18)...................................34init(18) . . . . . . . . . . . . . . . . . . . 34 C.2.19. S afterinit(19)...................................35init(19) . . . . . . . . . . . . . . . . . . . 34 C.2.20. S afterinit(20)...................................35init(20) . . . . . . . . . . . . . . . . . . . 35 C.2.21. S afterinit(21)...................................35init(21) . . . . . . . . . . . . . . . . . . . 35 C.2.22. S afterinit(22)...................................35init(22) . . . . . . . . . . . . . . . . . . . 35 C.2.23. S afterinit(23)...................................36init(23) . . . . . . . . . . . . . . . . . . . 35 C.2.24. S(0) afterinit(24)................................36init(24) . . . . . . . . . . . . . . . . . 36 C.2.25. S(1) and thekey streamKey Stream atS(1)....................36S(1) . . . . . . . . . . . 36 C.2.26. S(2) and thekey streamKey Stream atS(2)....................37S(2) . . . . . . . . . . . 36 1. Introduction KCipher-2 is a stream cipher that uses a 128-bit secret key and a128- bit128-bit initialization vector. Since the algorithm for KCipher-2 was published in 2007 [SASC07], it hasreceived attention from academiabeen evaluated in academic andindustries.industrial studies. The security and performance of KCipher-2 have been rigorously evaluated bytheits developers and other institutions[SECRYPT07], [ICETE07], [CRYPTEC],[SECRYPT07] [ICETE07] [CRYPTEC] [SIIS11].NoAs of the publication of this document, no attackhas been foundon KCipher-2as of this date.has been successful. KCipher-2 can be efficiently implemented in software to provide fast encryption and decryption, owing totheits 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 parallelaiming forto yield greater efficiency. Moreover, since its internal state representation only amounts to severalhundreds ofhundred bits, KCipher-2 is suitable for resource-limited environments. KCipher-2 has been actively used in several industrial applications inJapan andJapan, has been publishedtoby an international standardization body (ISO/IEC18033-4) [ISO18033]18033-4 [ISO18033]), andevaluated to behas been designated a Japanesee- Governmente-Government recommended cipher [CRYPTECLIST]. 2. Algorithm Description In this section, we describe the internal components of KCipher-2 and define the operations for deriving key streams from an input key and an initialization vector. We illustrate thedetail operationsdetailed operations, mostly inpseudopseudocode format, but also provide code snippets written in the C language syntaxifwhen necessary. 2.1. Notations All values in this document are stored in big-endian order(a. k. a.,(aka network byte order). We use the following notations in the description of KCipher-2. ^ Bitwise exclusive-OR n#m mth power of n +n Integer addition modulo 2#n <<_r n n-bit left circular shift in an r-bit register 0x Hexadecimal representation E[i] The (i + 1)th element of E when E is composed of consecutive multiple elements GF Galois field. GF(n#m) means the finite field of exactly n#m elements ** Multiplication of elements on the finite field GF(2#32)*NOTE: Many texts denote "the mth power of n" by "n^m", but we write it using '#', instead of '^', to avoidreaders'reader confusionoverwith the power operator and the XOR operator of the C language syntax. 2.2. Internal State The internal state of KCipher-2 can be denoted by S. The internal state consists ofitssix 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 Brespectivelyrefer to FSR-A andFSR-B.FSR-B, respectively. 2.2.1. Feedback Shift Registers The two feedback shift registers(FSR)(FSRs) 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. Eachof the unitsunit 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. InternalregistersRegisters 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. OperationsThere are threeThree major operationsthatconstitute 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 to the internal state. The stream() operation derives a key stream from the current state S. 2.3.1. next() The next() operation takes the current state 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 (NLFs) for every cycle of KCipher-2.Besides,Additionally, it isalsoused to initialize the system. The behaviors are distinguished by the input parameter that indicates the operation modes. Inside the next() operation, the internal registers are updated by the result of the substitution function described in Section 2.4.2. The feedback shift registersalsoare also updated by feedback functions. The feedback functions include the multiplication of register units and the fixed elements a0, a1, a2, and a3 in a finite field. The fixed elements a0, ..., a3 are carefully chosen to provide themaximum-maximum length of the feedback shift registers. The theory behind the selection of fixed elements and the wayof simplifyingto simplify the necessary multiplications are briefly described in Section 2.4.4. The operation takes the following inputs: o S = (A, B, L1, R1, L2, R2) o mode = {INIT, NORMAL}, where INIT means the operation is used forinitializationinitialization, and NORMAL means it is used for generating secure key streams.ItThe operation outputs a new state, o S' = (A', B', L1', R1', L2', R2') by performing thebelowfollowing steps: 1. Set registers in the nonlinear functionssetL1' = sub_K2(R2 +32 B[4]); R1' = sub_K2(L2 +32 B[9]); L2' = sub_K2(L1); R2' = sub_K2(R1); for m from 0 to 3 A'[m] = A[m + 1]; for m from 0 to 9 B'[m] = B[m + 1];*NOTE: sub_K2 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 NORMAL, A'[4] and B'[10] are computed as follows: A'[4] = (a0 ** A[0]) ^ A[3]; if A[2][30] is 1: if A[2][31] is 1: B'[10] = (a1 ** B[0]) ^ B[1] ^ B[6] ^ (a3 ** B[8]); else if A[2][31] is 0: B'[10] = (a1 ** B[0]) ^ B[1] ^ B[6] ^ B[8]; else if A[2][30] is 0: if A[2][31] is 1: B'[10] = (a2 ** B[0]) ^ B[1] ^ B[6] ^ (a3 ** B[8]); else if A[2][31] is 0: B'[10] = (a2 ** B[0]) ^ B[1] ^ B[6] ^ B[8]; b. When the mode is INIT, A'[4] and B'[10] are XOR-ed with the non-linear function output described in Section 2.4.1. A'[4] = (a0 ** A[0]) ^ A[3] ^ NLF(B[0], R2, R1, A[4]); if A[2][30] is 1: if A[2][31] is 1: B'[10] = (a1 ** B[0]) ^ B[1] ^ B[6] ^ (a3 ** B[8]) ^ NLF(B[10], L2, L1, A[0]); else if A[2][31] is 0: B'[10] = (a1 ** B[0]) ^ B[1] ^ B[6] ^ B[8] ^ NLF(B[10], L2, L1, A[0]); else if A[2][30] is 0: if A[2][31] is 1: B'[10] = (a2 ** B[0]) ^ B[1] ^ B[6] ^ (a3 ** B[8]) ^ NLF(B[10], L2, L1, A[0]); else if A[2][31] is 0: B'[10] = (a2 ** B[0]) ^ B[1] ^ B[6] ^ B[8] ^ NLF(B[10], L2, L1, A[0]); 3. Output S' = (A', B', L1', R1', L2', R2').*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.*The corresponding code snippets written in the C language syntax can be found in Section 2.4.4 andinAppendix B. 2.3.2. init() The init() operation takes a 128-bit key (K) and a 128-bit initialization vector(IV),(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. Expand Kis expandedto 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] ^ IK[m - 1]; else if m is 4: IK[4] = IK[0] ^ sub_K2(IK[3] <<_32 8) ^ (0x01, 0x00, 0x00, 0x00); else if m is 8: IK[8] = IK[4] ^ sub_K2(IK[7] <<_32 8) ^ (0x02, 0x00, 0x00, 0x00);*NOTE: sub_K2 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() operationtwenty- four24 timesrepeatedlyas 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 theseveralfunctions used above: sub_K2(), NLF(), and S_box(). 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 calculatedbyas follows. Q = (A +32 B) ^ C ^ D; 2.4.2. sub_K2() sub_K2() is a substitutionfunction, whichfunction that is a permutation of GF(2#32), based on components from the Advanced EncryptionStandard(AES)Standard (AES) [FIPS-AES]. Its input is a 32-bit value divided into four 8-bit strings. Inside sub_K2(), an8-to-8 bit8-to-8-bit substitution function, S_box(), is applied to each 8-bit string separately, and then a32-to-32 bit32- 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 sub_K2 as a 32-bit value W = (w[3], w[2], w[1], w[0]), where eachsub-elementsubelement of w is an 8-bit unit. Prepare two32- bit32-bit temporarystoragesstorages, 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() is 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 isan 1 by 4a 1x4 matrix, (q[0], q[1], q[2], q[3)) o M is a4 by 44x4 matrix, (02, 03, 01, 01, 01, 02, 03, 01, 01, 01, 02, 03, 03, 01, 01, 02) o T isan 1 by 4a 1x4 matrix, (t[0], t[1], t[2], t[3]). Namely, the procedure that calculates (q[3], q[2], q[1], q[0]) can be written in the C language syntax as: q[0] = GF_mult_by_2(t[0]) ^ GF_mult_by_3(t[1]) ^ t[2] ^ t[3]; q[1] = t[0] ^ GF_mult_by_2(t[1]) ^ GF_mult_by_3(t[2]) ^ t[3]; q[2] = t[0] ^ t[1] ^ GF_mult_by_2(t[2]) ^ GF_mult_by_3(t[3]); q[3] = GF_mult_by_3(t[0]) ^ t[1] ^ t[2] ^ GF_mult_by_2(t[3]);,where GF_mult_by_2 and GF_mult_by_3 are multiplication functions in GF(2#8), defined as follows: o Thefunction, GF_mult_by_2(t),function GF_mult_by_2(t) multiplies 2toby the given 8-bit value t inGF(2#8),GF(2#8) and returns an 8-bit value q as follows (lq is a temporary 32-bit variable): lq = t << 1; if ((lq & 0x100) != 0) lq ^= 0x011B; q = lq ^ 0xFF; o Thefunction, GF_mult_by_3(t),function GF_mult_by_3(t) multiplies 3toby the given 8-bit value t inGF(2#8),GF(2#8) and returns an 8-bit value q as follows (lq is a temporary 32-bit variable): lq = (t << 1) ^ t; if ((lq & 0x100) != 0) lq ^= 0x011B; q = lq ^ 0xFF; 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 }; 2.4.4. Multiplications in GF(2#32) FSR-A and FSR-B are word-oriented linear feedback shift registers(LFSR).(LFSRs). In the next() operation of Section 2.3.1, the feedback functions to the two LFSRs are shown, which includethemultiplication of fixed elementsofa0, a1, a2, or a3 in GF(2#32). The fixed elements are carefully chosen to maximize the period of the key stream generated by the two registers.HereHere, we briefly explain how weobtainedobtain the fixed elements. Further details and theories can be found in [SECRYPT07].In case of a0, how we obtainedWe obtain a0isasfollows:follows. First, to guaranteethe maximizethat the period is maximized for an 8-bit unit, we consider pisas therootsroot of the primitive polynomial: x#8 + x#7 + x#6, + x + 1 in GF(2). Therefore, an 8-bit string y = (y7, ..., y0), where y7 is the most significant bit, can be written as: y = y7(p#7) + y6(p#6) + ... + y1(p) + y0 Next, a0 is the root of irreducible polynomial of degree four: x#4 + p#24(x#3) + p#3(x#2) + p#12(x) + p#71 in GF(2#8). Then, hierarchically, a 32-bit unit Y = (Y3, Y2, Y1, Y0), where Y3 is the most significant byte, can be written as: Y3(a0#3) + Y2(a0#2) + Y1(a0) + Y0 The feedback polynomial to FSR-A, f(x) = a0(x#5) + x#2 + 1 produces themaximum lengthmaximum-length period of the key stream with a0. Similarly, a1, a2, and a3 are the roots of irreducible polynomials of degree four of x#4 + q#230(x#3) + q#156(x#2) + q#93(x) + q#29 in GF(2#8) x#4 + r#34(x#3) + r#16(x#2) + r#199(x) + r#248 in GF(2#8) x#4 + s#157(x#3) + s#253(x#2) + s#56(x) + s#16 in GF(2#8),respectively. The feedback polynomial to FSR-B that uses a1, a2, and a3 can produce the maximum-length period. The feedback polynomials to FSR-A and FSR-B are as written intheStep 2 of the next() operation, and the mathematical notations of these polynomialsalsocan also be found in [SECRYPT07]. Calculation of the original feedback polynomials mighttake long sincebe time- consuming because it includes multiplications in finite fields. However, these multiplications can be done faster if the multiples of a0, ..., a3 were already calculated for all possible inputs. The tables of amul0, ..., amul3 in Appendix A provide suchpre-calculationpre- calculation results. As shown intheStep 2 of next(), we can utilize these tables to finish the necessary calculations efficiently. For example, consider the input as a 32-bit value w, which represents an element ofGF(2#32), the outputGF(2#32). The 32-bit output string w' = a0 ** w can be obtained using the amul0 table in Appendix A.1 as follows: w' = (w << 8) ^ amul0[w >> 24]; Likewise, multiplications of (a1 ** w), (a2 ** w), and (a3 ** w) can be obtained in the samewayway, simply by using the amul1, amul2, and amul3 tables that we provide inAppendixAppendixes A.2, A.3, and A.4. Eventually,theStep 2 of the next() operation, which updates A'[4] and B'[10], can be written in the C language syntax asfollows (nA[4]follows. Note that nA[4] and nB[10] correspond to A'[4] and B'[10],respectively.respectively, and temp1 and temp2 are 32-bitvariables):variables. nA[4] = ((A[0] << 8) ^ amul0[(A[0] >> 24)]) ^ A[3]; if (mode == INIT) nA[4] ^= NLF(B[0], R2, R1, A[4]); if (A[2] & 0x40000000) { temp1 = (B[0] << 8) ^ amul1[(B[0] >> 24)]; } else { temp1 = (B[0] << 8) ^ amul2[(B[0] >> 24)]; } if (A[2] & 0x80000000) { temp2 = (B[8] << 8) ^ amul3[(B[8] >> 24)]; } else { temp2 = B[8]; } nB[10] = temp1 ^ B[1] ^ B[6] ^ temp2; if (mode == INIT) nB[10] ^= NLF(B[10], L2, L1, A[0]); 2.5.Encryption/Decryption schemeEncryption and Decryption Scheme In this section, we use the notation S(i) to specifically reference the values of the internal state at i (where i >= 0), which is an arbitrary, discrete temporal moment(a.k.a., a(aka cycle)i (i >= 0)after the initialization. 2.5.1. Keystream generationStream Generation Given a 128-bit key K, a 128-bit initialization vectorIV,(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)); 2.5.2.Encryption/DecryptionEncryption and Decryption of amessageMessage Given a 64-bit message block M and a key stream X, an encrypted message E is obtained by E = M ^ X; Conversely, the decrypted message D is obtained by D = E ^ X; The original message M and the decrypted message D are identical when the same key stream is used. 3. Security Considerations We recommendthat re-initializingreinitializing andre-keyingrekeying after 2#58 cycles of KCipher-2, which means after generating 2#64 key stream bits. It is important to make sure that no IV is ever reused under the same key. 4. References 4.1. Normative References [ISO18033] "Information technology--- Security techniques--- Encryption algorithms--- Part 4: Stream ciphers",ISO/IECISO/ IEC 18033-4:2012 Ed. 2, December 2012. [FIPS-AES]"Specification for the AdvancedNational Institute of Standards and Technology, "Advanced Encryption Standard (AES)",Federal Information Processing Standard (FIPS) PublicationFIPS PUB 197, November2001.2001, <http://csrc.nist.gov/publications/ fips/fips197/fips-197.pdf>. 4.2. Informative References [SECRYPT07]S.Kiyomoto,T.S., Tanaka, T., and K. Sakurai, "K2: A Stream Cipher Algorithm Using Dynamic Feedback Control", Proc. SECRYPT20072007, pp. 204-213. [ICETE07]S.Kiyomoto,T.S., Tanaka, T., and K. Sakurai, "K2 Stream Cipher", Proc. ICETE20072007, pp. 214-226. [CRYPTEC]A.Bogdanov,B.A., Preneel, B., and V. Rijmen, "Security Evaluation of the K2 Stream Cipher",2010. http://www.cryptrec.go.jp/english/estimation.html2010, <http://www.cryptrec.go.jp/english/estimation.html>. [CRYPTECLIST]Cryptography"Cryptography Research and EvaluationCommittees. http://www.cryptrec.go.jp/english/estimation.htmlCommittees", <http://www.cryptrec.go.jp/english/estimation.html>. [SIIS11]D.Priemuth-Schmid, D., "Attacks on Simplified Versions of K2", Proc.SIIS2011,SIIS 2011, LNCS 7053,pp.117-127.pp. 117-127. [SASC07]S.Kiyomoto,T.S., Tanaka, T., and K. Sakurai, "AWord-OrientedWord- Oriented Stream Cipher Using Clock Control", Proc. SASC20072007, pp. 260-274. Appendix A. Tables formultiplicationMultiplication 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, 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, 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 }; 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, 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, 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 }; Appendix B. Asimple implementation exampleSimple Implementation Example of KCipher-2 We provide an exampleembodimentimplementation of KCipher-2 written in C. The implementation issimple, which meanssimple; we do notconcernconsider storage or timecomplexity in the example. Neithercomplexity, nor do we consider software engineering-related issues, such as encapsulation, modularity, and so on. B.1. CodecomponentsComponents I - Definitions anddeclarationsDeclarations #include <stdio.h> #include <stdint.h> #define INIT 0 #define NORMAL 1 void init (unsigned int *, unsigned int *); void next(int); void stream (unsigned int *, unsigned int *); static const uint8_t S_box[256] = { ... // as defined in Section 2.4.3 }; static const uint32_t amul0[256] = { ... // as defined in Appendix A.1 }; static const uint32_t amul1[256] = { ... // as defined in Appendix A.2 }; static const uint32_t amul2[256] = { ... // as defined in Appendix A.3 }; static const uint32_t amul3[256] = { ... // as defined in Appendix A.4 }; /* Global variables */ // State S uint32_t A[5]; // five 32-bit units uint32_t B[11]; // eleven 32-bit units uint32_t L1, R1, L2, R2; // one 32-bit unit for each // The internal key (IK) and the initialization vector (IV) uint32_t IK[12]; //(12 * 32)-bit(12*32) bits uint32_t IV[4]; //(4 * 32)-bit(4*32) bits B.2. CodecomponentsComponents II - Functions /** * Do multiplication in GF(2#8) of the irreducible polynomial, * f(x) = x#8 + x#4 + x#3 + x + 1. The given parameter is multiplied * by 2. * @param t : (INPUT).8-bit.8 bits. The number will be multiplied by 2 * @return : (OUTPUT).8-bit.8 bits. The multiplication result */ uint8_t GF_mult_by_2 (uint8_t t) { uint8_t q; uint32_t lq; lq = t << 1; if ((lq & 0x100) != 0) lq ^= 0x011B; q = lq ^ 0xFF; return q; } /** * Do multiplication in GF(2#8) of the irreducible polynomial, * f(x) = x#8 + x#4 + x#3 + x + 1. The given parameter is multiplied * by 3. * @param t : (INPUT).8-bit.8 bits. The number will be multiplied by 3 * @return : (OUTPUT).8-bit.8 bits. The multiplication result */ uint8_t GF_mult_by_3 (uint8_t t) { uint8_t q; uint32_t lq; lq = (t << 1) ^ t; if ((lq & 0x100) != 0) lq ^= 0x011B; q = lq ^ 0xFF; return q; } /** * Do substitution on a given input. See Section 2.4.2. * @param t : (INPUT),(1*32)-bit(1*32) bits * @return : (OUTPUT),(1*32)-bit(1*32) bits */ uint32_t sub_k2 (uint32_t in) { uint32_t out; uint8_t w0 = in & 0x000000ff; uint8_t w1 = (in >> 8) & 0x000000ff; uint8_t w2 = (in >> 16) & 0x000000ff; uint8_t w3 = (in >> 24) & 0x000000ff; uint8_t t3, t2, t1, t0; uint8_t q3, q2, q1, q0; t0 = S_box[w0]; t1 = S_box[w1]; t2 = S_box[w2]; t3 = S_box[w3]; q0 = GF_mult_by_2(t0) ^ GF_mult_by_3(t1) ^ t2 ^ t3; q1 = t0 ^ GF_mult_by_2(t1) ^ GF_mult_by_3(t2) ^ t3; q2 = t0 ^ t1 ^ GF_mult_by_2(t2) ^ GF_mult_by_3(t3); q3 = GF_mult_by_3(t0) ^ t1 ^ t2 ^ GF_mult_by_2(t3); out = (q3 << 24) | (q2 << 16) | (q1 << 8) | q0; return out; } /** * Expand a given 128-bit key (K) to a 384-bit internal key * information (IK). * See Step 1 of init() in Section 2.3.2. * @param key[4] : (INPUT),(4*32)-bit(4*32) bits * @param iv[4] : (INPUT),(4*32)-bit(4*32) bits * @modify IK[12] : (OUTPUT),(12*32)-bit(12*32) bits * @modify IV[12] : (OUTPUT),(4*32)-bit(4*32) bits */ void key_expansion (uint32_t *key, uint32_t *iv) { // copy iv to IV IV[0] = iv[0]; IV[1] = iv[1]; IV[2] = iv[2]; IV[3] = iv[3]; // m = 0 ... 3 IK[0] = key[0]; IK[1] = key[1]; IK[2] = key[2]; IK[3] = key[3]; // m = 4 IK[4] = IK[0] ^ sub_k2((IK[3] << 8) ^ (IK[3] >> 24)) ^ 0x01000000; // m = 4 ... 11, but not 4 nor 8 IK[5] = IK[1] ^ IK[4]; IK[6] = IK[2] ^ IK[5]; IK[7] = IK[3] ^ IK[6]; // m = 8 IK[8] = IK[4] ^ sub_k2((IK[7] << 8) ^ (IK[7] >> 24)) ^ 0x02000000; // m = 4 ... 11, but not 4 nor 8 IK[9] = IK[5] ^ IK[8]; IK[10] = IK[6] ^ IK[9]; IK[11] = IK[7] ^ IK[10]; } /** * Set up the initial state value using IK and IV. See Step 2 of * init() in Section 2.3.2. * @param key[4] : (INPUT),(4*32)-bit(4*32) bits * @param iv[4] : (INPUT),(4*32)-bit(4*32) bits * @modify S : (OUTPUT), (A, B, L1, R1, L2, R2) */ void setup_state_values (uint32_t *key, uint32_t *iv) { // setting up IK and IV by calling key_expansion(key, iv) key_expansion(key, iv); // setting up the internal state values A[0] = IK[4]; A[1] = IK[3]; A[2] = IK[2]; A[3] = IK[1]; A[4] = IK[0]; 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; } /** * Initialize the system with a 128-bit key (K) and a 128-bit * initialization vector (IV). It sets up the internal state value * andinvokeinvokes next(INIT) iterativelyfor24 times. After this, * the system is ready to produce key streams. See Section 2.3.2. * @param key[12] : (INPUT),(4*32)-bit(4*32) bits * @param iv[4] : (INPUT),(4*32)-bit(4*32) bits * @modify IK :(12*32)-bit,(12*32) bits, by calling setup_state_values() * @modify IV :(4*32)-bit,(4*32) bits, by calling setup_state_values() * @modify S : (OUTPUT), (A, B, L1, R1, L2, R2) */ void init (uint32_t *k, uint32_t *iv) { int i; setup_state_values(k, iv); for(i=0; i < 24; i++) { next(INIT); } } /** * Non-linear function. See Section 2.4.1. * @param A : (INPUT),8-bit8 bits * @param B : (INPUT),8-bit8 bits * @param C : (INPUT),8-bit8 bits * @param D : (INPUT),8-bit8 bits * @return : (OUTPUT),8-bit8 bits */ uint32_t NLF (uint32_t A, uint32_t B, uint32_t C, uint32_t D ) { uint32_t Q; Q = (A + B) ^ C ^ D; return Q; } /** * Derive a new state from the current state values. * See Section 2.3.1. * @param mode : (INPUT) INIT (= 0) or NORMAL (= 1) * @modify S : (OUTPUT) */ void next (int mode) { uint32_t nA[5]; uint32_t nB[11]; uint32_t nL1, nR1, nL2, nR2; uint32_t temp1, temp2; nL1 = sub_k2(R2 + B[4]); nR1 = sub_k2(L2 + B[9]); nL2 = sub_k2(L1); nR2 = sub_k2(R1); // m = 0 ... 3 nA[0] = A[1]; nA[1] = A[2]; nA[2] = A[3]; nA[3] = A[4]; // m = 0 ... 9 nB[0] = B[1]; nB[1] = B[2]; nB[2] = B[3]; nB[3] = B[4]; nB[4] = B[5]; nB[5] = B[6]; nB[6] = B[7]; nB[7] = B[8]; nB[8] = B[9]; nB[9] = B[10]; // update nA[4] temp1 = (A[0] << 8) ^ amul0[(A[0] >> 24)]; nA[4] = temp1 ^ A[3]; if (mode == INIT) nA[4] ^= NLF(B[0], R2, R1, A[4]); // update nB[10] if (A[2] & 0x40000000) /* if A[2][30] == 1 */ { temp1 = (B[0] << 8) ^ amul1[(B[0] >> 24)]; } else /*if A[2][30] == 0*/ { temp1 = (B[0] << 8) ^ amul2[(B[0] >> 24)]; } if (A[2] & 0x80000000) /* if A[2][31] == 1 */ { temp2 = (B[8] << 8) ^ amul3[(B[8] >> 24)]; } else /* if A[2][31] == 0 */ { temp2 = B[8]; } nB[10] = temp1 ^ B[1] ^ B[6] ^ temp2; if (mode == INIT) nB[10] ^= NLF(B[10], L2, L1, A[0]); /* copy S' to S */ A[0] = nA[0]; A[1] = nA[1]; A[2] = nA[2]; A[3] = nA[3]; A[4] = nA[4]; B[0] = nB[0]; B[1] = nB[1]; B[2] = nB[2]; B[3] = nB[3]; B[4] = nB[4]; B[5] = nB[5]; B[6] = nB[6]; B[7] = nB[7]; B[8] = nB[8]; B[9] = nB[9]; B[10] = nB[10]; L1 = nL1; R1 = nR1; L2 = nL2; R2 = nR2; } /** * Obtain a key stream = (ZH, ZL) from the current state values. * See Section 2.3.3. * @param ZH : (OUTPUT) (1 * 32)-bit * @modify ZL : (OUTPUT) (1 * 32)-bit */ void stream (uint32_t *ZH, uint32_t *ZL) { *ZH = NLF(B[10], L2, L1, A[0]); *ZL = NLF(B[0], R2, R1, A[4]); } B.3. UsecaseCase void main (void) { // Set the key and the iv uint32_t key[4] = ...; uint32_t iv[4] = ...; init(key, iv); // produce a key stream stream(&zh, &zl); next(NORMAL); // produce another key stream stream(&zh, &zl); next(NORMAL); ... } Appendix C. Test Vectors This appendix provides running examples of KCipher-2 obtained from the naive implementation. All values are written in hexadecimal form. C.1. Keystream generation examplesStream Generation Examples Thefollowings demonstratefollowing is a series of the 64-bit key streams generated from the given128-bit8-bit keys (K) and 128-bit initialization vectors(IV).(IVs). - K : 00000000 00000000 00000000 00000000 - IV: 00000000 00000000 00000000 00000000 - Generated key streams at S(i) are asfollows;follows S(0): F871EBEF 945B7272 S(1): E40C0494 1DFF0537 S(2): 0B981A59 FBC8AC57 S(3): 566D3B02 C179DBB4 S(4): 3B46F1F0 33554C72 S(5): 5DE68BCC 9872858F S(6): 57549602 4062F0E9 S(7): F932C998 226DB6BA ... - K : A37B7D01 2F897076 FE08C22D 142BB2CF - IV: 33A6EE60 E57927E0 8B45CC4C A30EDE4A - Generated key streams at S(i) are asfollows;follows S(0): 60E9A6B6 7B4C2524 S(1): FE726D44 AD5B402E S(2): 31D0D1BA 5CA233A4 S(3): AFC74BE7 D6069D36 S(4): 4A75BB6C D8D5B7F0 S(5): 38AAAA28 4AE4CD2F S(6): E2E5313D FC6CCD8F S(7): 9D2484F2 0F86C50D ... - K : 3D62E9B1 8E5B042F 42DF43CC 7175C96E - IV: 777CEFE4 541300C8 ADCACA8A 0B48CD55 - Generated key streams at S(i) are asfollows;follows S(0): 690F108D 84F44AC7 S(1): BF257BD7 E394F6C9 S(2): AA1192C3 8E200C6E S(3): 073C8078 AC18AAD1 S(4): D4B8DADE 68802368 S(5): 2FA42076 83DEA5A4 S(6): 4C1D95EA E959F5B4 S(7): 2611F41E A40F0A58 ... C.2. Anotherkey stream generationKey Stream Generation with thestate valuesState Values In this section, the initialization procedure and the key stream generation are illustrated in detail. The given 128-bit key (K) and the 128-bit initialization vector (IV) are as follows: - K : 0F1E2D3C 4B5A6978 8796A5B4 C3D2E1F0 - IV: F0E0D0C0 B0A09080 70605040 30201000. Based on K and IV, the init()operation, in Section 2.3.2,operation (Section 2.3.2) sets up the internal state values, S = (A, B, L1, R1, L2, R2), as follows: A[0]: 7993A6A2 A[1]: C3D2E1F0 A[2]: 8796A5B4 A[3]: 4B5A6978 A[4]: 0F1E2D3C B[0]: 38AB371B B[1] : 4E26BC85 B[2]: F0E0D0C0 B[3]: B0A09080 B[4] : BF3D92AF B[5]: 8DF45D75 B[6]: 70605040 B[7] : 30201000 B[8]: 768D8B9E B[9]: 32C9CFDA B[10]: B55F6A6E L1: 00000000 R1: 00000000 L2: 00000000 R2: 00000000 To complete the initialization, the next() operation isrepeatedlyapplied to the state valuesfor24 times (in Section 2.3.2, Step 3). Let us denote eachof therepeated application of the next() operation by init(i), where 1 <= i <= 24. The internal state values resulting from each init(i) are shown inSection B.2.1Appendixes C.2.1 -B.2.24.C.2.24. C.2.1. S after init(1) A[0]: C3D2E1F0 A[1]: 8796A5B4 A[2]: 4B5A6978 A[3]: 0F1E2D3C A[4]: 37070F7F B[0]: 4E26BC85 B[1] : F0E0D0C0 B[2]: B0A09080 B[3]: BF3D92AF B[4] : 8DF45D75 B[5]: 70605040 B[6]: 30201000 B[7] : 768D8B9E B[8]: 32C9CFDA B[9]: B55F6A6E B[10]: 64DEFF24 L1: F360860C R1: E81907D5 L2: 63636363 R2: 63636363 C.2.2. S after init(2) A[0]: 8796A5B4 A[1]: 4B5A6978 A[2]: 0F1E2D3C A[3]: 37070F7F A[4]: 25BCF981 B[0]: F0E0D0C0 B[1] : B0A09080 B[2]: BF3D92AF B[3]: 8DF45D75 B[4] : 70605040 B[5]: 30201000 B[6]: 768D8B9E B[7] : 32C9CFDA B[8]: B55F6A6E B[9]: 64DEFF24 B[10]: 7E65CB6A L1: 1B9542ED R1: 9B259D28 L2: 971610F6 R2: 39C36E1D C.2.3. S after init(3) A[0]: 4B5A6978 A[1]: 0F1E2D3C A[2]: 37070F7F A[3]: 25BCF981 A[4]: FA2DD9D3 B[0]: B0A09080 B[1] : BF3D92AF B[2]: 8DF45D75 B[3]: 70605040 B[4] : 30201000 B[5]: 768D8B9E B[6]: 32C9CFDA B[7] : B55F6A6E B[8]: 64DEFF24 B[9]: 7E65CB6A B[10]: 08573732 L1: 1F41CDFB R1: CFAE13F3 L2: BCC7DC5B R2: 1528DDA1 C.2.4. S after init(4) A[0]: 0F1E2D3C A[1]: 37070F7F A[2]: 25BCF981 A[3]: FA2DD9D3 A[4]: AB820031 B[0]: BF3D92AF B[1] : 8DF45D75 B[2]: 70605040 B[3]: 30201000 B[4] : 768D8B9E B[5]: 32C9CFDA B[6]: B55F6A6E B[7] : 64DEFF24 B[8]: 7E65CB6A B[9]: 08573732 B[10]: 40941D82 L1: 8D7100A7 R1: AA6C8F89 L2: B4F43081 R2: 81264AF3 C.2.5. S after init(5) A[0]: 37070F7F A[1]: 25BCF981 A[2]: FA2DD9D3 A[3]: AB820031 A[4]: D8F5995F B[0]: 8DF45D75 B[1] : 70605040 B[2]: 30201000 B[3]: 768D8B9E B[4] : 32C9CFDA B[5]: B55F6A6E B[6]: 64DEFF24 B[7] : 7E65CB6A B[8]: 08573732 B[9]: 40941D82 B[10]: 1A8DA7FB L1: D315A91D R1: 751BC887 L2: 9E8539E3 R2: 929B1D3C C.2.6. S after init(6) A[0]: 25BCF981 A[1]: FA2DD9D3 A[2]: AB820031 A[3]: D8F5995F A[4]: F697B5BB B[0]: 70605040 B[1] : 30201000 B[2]: 768D8B9E B[3]: 32C9CFDA B[4] : B55F6A6E B[5]: 64DEFF24 B[6]: 7E65CB6A B[7] : 08573732 B[8]: 40941D82 B[9]: 1A8DA7FB B[10]: 13B5E7F3 L1: 88658E94 R1: 7F1C023D L2: B16F9402 R2: 5F06AB3F C.2.7. S after init(7) A[0]: FA2DD9D3 A[1]: AB820031 A[2]: D8F5995F A[3]: F697B5BB A[4]: 6B0A7012 B[0]: 30201000 B[1] : 768D8B9E B[2]: 32C9CFDA B[3]: B55F6A6E B[4] : 64DEFF24 B[5]: 7E65CB6A B[6]: 08573732 B[7] : 40941D82 B[8]: 1A8DA7FB B[9]: 13B5E7F3 B[10]: D76ABD2C L1: 21BF8813 R1: 743F68DE L2: A1F603E6 R2: 3D1EA499 C.2.8. S after init(8) A[0]: AB820031 A[1]: D8F5995F A[2]: F697B5BB A[3]: 6B0A7012 A[4]: 23995B7E B[0]: 768D8B9E B[1] : 32C9CFDA B[2]: B55F6A6E B[3]: 64DEFF24 B[4] : 7E65CB6A B[5]: 08573732 B[6]: 40941D82 B[7] : 1A8DA7FB B[8]: 13B5E7F3 B[9]: D76ABD2C B[10]: 997C3F70 L1: B48EA08C R1: 657C8FFD L2: AAB50B58 R2: 281F9A12 C.2.9. S after init(9) A[0]: D8F5995F A[1]: F697B5BB A[2]: 6B0A7012 A[3]: 23995B7E A[4]: F8532F87 B[0]: 32C9CFDA B[1] : B55F6A6E B[2]: 64DEFF24 B[3]: 7E65CB6A B[4] : 08573732 B[5]: 40941D82 B[6]: 1A8DA7FB B[7] : 13B5E7F3 B[8]: D76ABD2C B[9]: 997C3F70 B[10]: 95FFF657 L1: A2040C44 R1: EF19DC4E L2: 543A1967 R2: 05D0CF60 C.2.10. S after init(10) A[0]: F697B5BB A[1]: 6B0A7012 A[2]: 23995B7E A[3]: F8532F87 A[4]: BEDF1DEF B[0]: B55F6A6E B[1] : 64DEFF24 B[2]: 7E65CB6A B[3]: 08573732 B[4] : 40941D82 B[5]: 1A8DA7FB B[6]: 13B5E7F3 B[7] : D76ABD2C B[8]: 997C3F70 B[9]: 95FFF657 B[10]: 6D2C2FA3 L1: C7AE66B0 R1: 9C075DB9 L2: 5554CBE7 R2: 866080C4 C.2.11. S after init(11) A[0]: 6B0A7012 A[1]: 23995B7E A[2]: F8532F87 A[3]: BEDF1DEF A[4]: 983D37. B[0]: 64DEFF24 B[1] : 7E65CB6A B[2]: 08573732 B[3]: 40941D82 B[4] : 1A8DA7FB B[5]: 13B5E7F3 B[6]: D76ABD2C B[7] : 997C3F70 B[8]: 95FFF657 B[9]: 6D2C2FA3 B[10]: A02127BE L1: 29F322A2 R1: 01F771D9 L2: 725670A2 R2: D4F24463 C.2.12. S after init(12) A[0]: 23995B7E A[1]: F8532F87 A[2]: BEDF1DEF A[3]: 983D37CB A[4]: 526A110D B[0]: 7E65CB6A B[1] : 08573732 B[2]: 40941D82 B[3]: 1A8DA7FB B[4] : 13B5E7F3 B[5]: D76ABD2C B[6]: 997C3F70 B[7] : 95FFF657 B[8]: 6D2C2FA3 B[9]: A02127BE B[10]: 49F99042 L1: 51536DF4 R1: 66111E6A L2: 8147B572 R2: 6CC2AC80 C.2.13. S after init(13) A[0]: F8532F87 A[1]: BEDF1DEF A[2]: 983D37CB A[3]: 526A110D A[4]: A5EEB8AE B[0]: 08573732 B[1] : 40941D82 B[2]: 1A8DA7FB B[3]: 13B5E7F3 B[4] : D76ABD2C B[5]: 997C3F70 B[6]: 95FFF657 B[7] : 6D2C2FA3 B[8]: A02127BE B[9]: 49F99042 B[10]: 406CE62C L1: 9582D912 R1: 6953AFE8 L2: B22A3A1D R2: 903A4823 C.2.14. S after init(14) A[0]: BEDF1DEF A[1]: 983D37CB A[2]: 526A110D A[3]: A5EEB8AE A[4]: 70A5B5BA B[0]: 40941D82 B[1] : 1A8DA7FB B[2]: 13B5E7F3 B[3]: D76ABD2C B[4] : 997C3F70 B[5]: 95FFF657 B[6]: 6D2C2FA3 B[7] : A02127BE B[8]: 49F99042 B[9]: 406CE62C B[10]: C57BED5B L1: EB77DD2D R1: 633CFD8F L2: 32A4BCEF R2: CB33BCB2 C.2.15. S after init(15) A[0]: 983D37CB A[1]: 526A110D A[2]: A5EEB8AE A[3]: 70A5B5BA A[4]: B1145F18 B[0]: 1A8DA7FB B[1] : 13B5E7F3 B[2]: D76ABD2C B[3]: 997C3F70 B[4] : 95FFF657 B[5]: 6D2C2FA3 B[6]: A02127BE B[7] : 49F99042 B[8]: 406CE62C B[9]: C57BED5B B[10]: 7BE2C520 L1: E11420CC R1: 6730A956 L2: 8EC8ACEF R2: C7FC060A C.2.16. S after init(16) A[0]: 526A110D A[1]: A5EEB8AE A[2]: 70A5B5BA A[3]: B1145F18 A[4]: FA752FDC B[0]: 13B5E7F3 B[1] : D76ABD2C B[2]: 997C3F70 B[3]: 95FFF657 B[4] : 6D2C2FA3 B[5]: A02127BE B[6]: 49F99042 B[7] : 406CE62C B[8]: C57BED5B B[9]: 7BE2C520 B[10]: 1F48829C L1: 0D95C94D R1: 8238B05F L2: 7B00D356 R2: 0EFE8596 C.2.17. S after init(17) A[0]: A5EEB8AE A[1]: 70A5B5BA A[2]: B1145F18 A[3]: FA752FDC A[4]: DB29190A B[0]: D76ABD2C B[1] : 997C3F70 B[2]: 95FFF657 B[3]: 6D2C2FA3 B[4] : A02127BE B[5]: 49F99042 B[6]: 406CE62C B[7] : C57BED5B B[8]: 7BE2C520 B[9]: 1F48829C B[10]: F95DD14F L1: 262687B5 R1: 9B9AC5E9 L2: 7C08EB5C R2: 8C1300A3 C.2.18. S after init(18) A[0]: 70A5B5BA A[1]: B1145F18 A[2]: FA752FDC A[3]: DB29190A A[4]: 35623CDA B[0]: 997C3F70 B[1] : 95FFF657 B[2]: 6D2C2FA3 B[3]: A02127BE B[4] : 49F99042 B[5]: 406CE62C B[6]: C57BED5B B[7] : 7BE2C520 B[8]: 1F48829C B[9]: F95DD14F B[10]: D939E13E L1: E478DEF0 R1: 06F84503 L2: 71350E88 R2: 14EF8E61 C.2.19. S after init(19) A[0]: B1145F18 A[1]: FA752FDC A[2]: DB29190A A[3]: 35623CDA A[4]: 746B4AE8 B[0]: 95FFF657 B[1] : 6D2C2FA3 B[2]: A02127BE B[3]: 49F99042 B[4] : 406CE62C B[5]: C57BED5B B[6]: 7BE2C520 B[7] : 1F48829C B[8]: F95DD14F B[9]: D939E13E B[10]: 9970C980 L1: C2AC94C4 R1: C708FAE8 L2: FC4900F1 R2: 7C260B6A C.2.20. S after init(20) A[0]: FA752FDC A[1]: DB29190A A[2]: 35623CDA A[3]: 746B4AE8 A[4]: 2EB9213A B[0]: 6D2C2FA3 B[1] : A02127BE B[2]: 49F99042 B[3]: 406CE62C B[4] : C57BED5B B[5]: 7BE2C520 B[6]: 1F48829C B[7] : F95DD14F B[8]: D939E13E B[9]: 9970C980 B[10]: 3C517031 L1: 8F007DE9 R1: B2AE0889 L2: DD68D5EA R2: 3C8757AC C.2.21. S after init(21) A[0]: DB29190A A[1]: 35623CDA A[2]: 746B4AE8 A[3]: 2EB9213A A[4]: BE3CA984 B[0]: A02127BE B[1] : 49F99042 B[2]: 406CE62C B[3]: C57BED5B B[4] : 7BE2C520 B[5]: 1F48829C B[6]: F95DD14F B[7] : D939E13E B[8]: 9970C980 B[9]: 3C517031 B[10]: D1439B63 L1: AFC4E32F R1: 98FBC87F L2: 58B22D36 R2: 481DC7D6 C.2.22. S after init(22) A[0]: 35623CDA A[1]: 746B4AE8 A[2]: 2EB9213A A[3]: BE3CA984 A[4]: 974E6719 B[0]: 49F99042 B[1] : 406CE62C B[2]: C57BED5B B[3]: 7BE2C520 B[4] : 1F48829C B[5]: F95DD14F B[6]: D939E13E B[7] : 9970C980 B[8]: 3C517031 B[9]: D1439B63 B[10]: 9334E221 L1: F9C43357 R1: E5539EA2 L2: C0B76A7C R2: 06EE4ED5 C.2.23. S after init(23) A[0]: 746B4AE8 A[1]: 2EB9213A A[2]: BE3CA984 A[3]: 974E6719 A[4]: 86916EFF B[0]: 406CE62C B[1] : C57BED5B B[2]: 7BE2C520 B[3]: 1F48829C B[4] : F95DD14F B[5]: D939E13E B[6]: 9970C980 B[7] : 3C517031 B[8]: D1439B63 B[9]: 9334E221 B[10]: 50EF13E7 L1: 309527ED R1: C473D814 L2: 1B107B6D R2: 0180D95D C.2.24. S(0) after init(24) A[0]: 2EB9213A A[1]: BE3CA984 A[2]: 974E6719 A[3]: 86916EFF A[4]: F52DACF9 B[0]: C57BED5B B[1] : 7BE2C520 B[2]: 1F48829C B[3]: F95DD14F B[4] : D939E13E B[5]: 9970C980 B[6]: 3C517031 B[7] : D1439B63 B[8]: 9334E221 B[9]: 50EF13E7 B[10]: E0BD9F91 L1: 4370D8E6 R1: DABED76C L2: 11C1ACCB R2: C3BAAEDF Note that the result of init(24) is also referred to as S(0) (in Section 2.3.2). Since the state is S(0), the stream() operation (in Section 2.3.3) can be applied and generate key streams. Key stream at S(0) : 9FB6B580A6A5E7AF Henceforth, a new key stream can be producedby;by 1)obtainobtaining a new state by applying the next() operation to the current state, and 2)generategenerating a new key stream by applying the stream() operation to the new state. C.2.25. S(1) and thekey streamKey Stream at S(1) A[0]: BE3CA984 A[1]: 974E6719 A[2]: 86916EFF A[3]: F52DACF9 A[4]: 960329B5 B[0]: 7BE2C520 B[1] : 1F48829C B[2]: F95DD14F B[3]: D939E13E B[4] : 9970C980 B[5]: 3C517031 B[6]: D1439B63 B[7] : 9334E221 B[8]: 50EF13E7 B[9]: E0BD9F91 B[10]: 5318AEE1 L1: 8FD86092 R1: 4BBDC0F6 L2: 8D63A5EF R2: FEE0F24B Key stream at S(1) : D1989DC6A77D5E28 C.2.26. S(2) and thekey streamKey Stream at S(2) A[0]: 974E6719 A[1]: 86916EFF A[2]: F52DACF9 A[3]: 960329B5 A[4]: 1A3DB24E B[0]: 1F48829C B[1] : F95DD14F B[2]: D939E13E B[3]: 9970C980 B[4] : 3C517031 B[5]: D1439B63 B[6]: 9334E221 B[7] : 50EF13E7 B[8]: E0BD9F91 B[9]: 5318AEE1 B[10]: C86C2C77 L1: 9686FE8C R1: FAF89251 L2: 86C824E7 R2: 7BC21098 Key stream at S(2) : 4EFCC8CB7BCFB32B Authors' Addresses Shinsaku Kiyomoto KDDI R&D Laboratories, Inc. 2-1-15Ohara,Ohara Fujimino-shi, Saitama356-8502, Japan.356-8502 Japan Phone: +81-49-278-7885 Fax: +81-49-278-7510Email:EMail: kiyomoto@kddilabs.jp Wook Shin KDDI R&D Laboratories, Inc. 2-1-15Ohara,Ohara Fujimino-shi, Saitama356-8502, Japan. Email:356-8502 Japan EMail: ohpato@hanmail.net