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Security of Reduced Version of the Block Cipher Camellia Against Truncated and Impossible Differential Cryptanalysis

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Security of Reduced Version of the Block Cipher Camellia Against Truncated and Impossible Differential Cryptanalysis Security of Reduced Version of the Block Cipher Camellia against Truncated and Impossible Di®erential Cryptanalysis Makoto Sugita1, Kazukuni Kobara2, and Hideki Imai2 1 NTT Network Innovation Laboratories, NTT Corporation 1-1 Hikari-no-oka, Yokosuka-shi, Kanagawa, 239-0847 Japan [email protected] 2 Institute of Industrial Sciences, The University of Tokyo 4-6-1, Komaba, Meguro-ku, Tokyo, 153-8505, Japan fkobara,[email protected] Abstract. This paper describes truncated and impossible di®erential cryptanalysis of the 128-bit block cipher Camellia, which was proposed by NTT and Mitsubishi Electric Corporation. Our work improves on the best known truncated and impossible di®erential cryptanalysis. As a result, we show a nontrivial 9-round byte characteristic, which may lead to a possible attack of reduced-round version of Camellia without input/output whitening, F L or F L¡1 in a chosen plain text scenario. Previously, only 6-round di®erentials were known, which may suggest a possible attack of Camellia reduced to 8-rounds. Moreover, we show a nontrivial 7-round impossible di®erential, whereas only a 5-round im- possible di®erential was previously known. This cryptanalysis is e®ective against general Feistel structures with round functions composed of S-D (Substitution and Di®usion) transformation. Keywords: Block Cipher Camellia, Truncated Di®erential Cryptanaly- sis, Impossible Di®erential Cryptanalysis 1 Introduction Camellia is a 128-bit block cipher proposed by NTT and Mitsubishi Electric Corporation [1]. It was designed to withstand all known cryptanalytic attacks and to provide a su±cient headroom to allow its use over the next 10 ¡ 20 years. Camellia supports 128-bit block size and 128-, 192-, and 256-bit key lengths, i.e. the same interface speci¯cations as the Advanced Encryption Standard (AES). Camellia was proposed in response to the call for contributions from ISO/IEC JTC 1/SC27 with the aim of it being adopted as an international standard. Camellia was also submitted to NESSIE (New European Schemes for Signature, Integrity, and Encryption). Furthermore, Camellia was submitted to CRYP- TREC (CRYPTography Research & Evaluation Committee) in Japan and it is now being evaluated. Security of Reduced Version of the Block Cipher Camellia 195 Like E2 [5], which was submitted to AES, Camellia uses a combination of a Feistel structure and the SPN-(Substitution and Permutation Network)- structure, but it also includes new features such as the use of improved linear transformation in SPN-structures, the change of SPN-structures from three lay- ers into two, and the use of input/output whitening, F L and F L¡1. The result is improved immunity against truncated di®erential cryptanalysis, which was applied successfully against reduced-round version of E2 by Matsui and Tokita [12]. Truncated di®erential cryptanalysis was introduced by Knudsen [4], as a generalization of di®erential crypanalysis [3]. He de¯ned them as di®erentials where only a part of the di®erential can be predicted. The notion of truncated di®erentials as introduced by him is wide, but with a byte-oriented cipher such as E2 or Camellia, it is natural to study byte-wise di®erentials as truncated di®erentials. The initial analysis of the security of Camellia and its resistance to the trun- cated and impossible di®erential cryptanalysis is given in [1], [6]. They state that Camellia with more than 11 rounds is secure against truncated di®erential cryptanalysis, though they did not indicate the e®ective truncated di®erentials. Up to now, the e®ective cryptanalysis applicable to Camellia has been the higher order di®erential cryptanalysis proposed by Kawabata, et al.[7], which utilizes non-trivial 6-round higher order di®erentials, and the di®erential crypatanalysis which utilizes a 7-round di®erential [2]. Our analysis improves on the best known truncated and impossible crypt- analysis against Camellia. Our cryptanalysis ¯nds a nontrivial 9-round truncated di®erential, which may lead to a possible attack of Camellia reduced to 11-rounds without input/output whitening, F L, or F L¡1 by a chosen plain text scenario. Moreover, we show a nontrivial 7-round impossible di®erential, whereas only a 5-round impossible di®erentials were previously known. The contents of this paper are as follows. In Section 2, we describes the struc- tures of block ciphers, truncated di®erential probabilities, impossible di®erential cryptanalysis and the block cipher Camellia. In Section 3, we describe the previ- ous work on the security of block cipher Camellia. In Section 4, we cryptanalyze Camellia by truncated di®erential cryptanalysis. In Section 5, we cryptanalyze Camellia by impossible di®erential cryptanalysis. Section 6 concludes this paper. 2 Preliminaries In this section, we describe the general structures of block ciphers, truncated dif- ferential probabilities, impossible di®erential cryptanalysis and the block cipher Camellia. 2.1 Feistel structures n n Associate with a function f : GF(2) ! GF(2) , a function D2n;f (L; R) = n (R © f(L); L) for all L; R 2 GF(2) . D2n;f is called the Feistel transformation 196 Makoto Sugita, Kazukuni Kobara, and Hideki Imai n n associated with f. Furthermore, for functions f1; f2; ¢ ¢ ¢ ; fs : GF(2) ! GF(2) , de¯ne Ãn(f1; f2; ¢ ¢ ¢ ; fs) = D2n;fs ±¢ ¢ ¢±D2n;f2 ±D2n;f1 . We call F (f1; f2; ¢ ¢ ¢ ; fs) = Ãn(f1; f2; ¢ ¢ ¢ ; fs) the s-round Feistel structure. At this time, we call the functions f1; f2; ¢ ¢ ¢ ; fs the round functions of the Feistel structure F (f1; f2; ¢ ¢ ¢ ; fs). 2.2 SPN-structures [9] This structure consists of two kinds of layers: nonlinear layer and linear layer. Each layer has di®erent features as follows. Nonlinear (Substitution) layer: This layer is composed of m parallel n-bit bijective nonlinear transformations. Linear (Di®usion) layer: This layer is composed of linear transformations over the ¯eld GF(2n) (especially in the case of E2 and Camellia, GF(2)), where inputs are transformed linearly to outputs per word (n-bits). Next for positive integer s, we de¯ne the s-layer SPN-structure that consists of s layers. First is a nonlinear layer, second is a linear layer, third is a nonlinear layer, ¢ ¢ ¢ . 2.3 Word characteristics We de¯ne a word characteristic function  : GF(2n)m ! GF(2)m; (a1; ¢ ¢ ¢ ; am) 7¡! (b1; ¢ ¢ ¢ ; bm) by 0 if a = 0 b = i i ½ 1 otherwise; Hereafter, we call Â(a) the word characteristic of a 2 GF(2n)m. Especially in the case of n = 8, we call Â(a) the byte characteristic. 2.4 Truncated di®erential probability De¯nition 1. Let ¢x; ¢y 2 GF(2n)m denote the input and output di®erences of the function f, respectively. ¢x = (¢x1; ¢x2; ¢ ¢ ¢ ; ¢xm) ¢y = (¢y1; ¢y2; ¢ ¢ ¢ ; ¢ym) We de¯ne the input and output truncated di®erential (±x; ±y) 2 (GF(2)m)2 of the function f, where ±x = (±x1; ±x2; ¢ ¢ ¢ ; ±xm) ±y = (±y1; ±y2; ¢ ¢ ¢ ; ±ym) by ±x = Â(¢x); ±y = Â(¢y). Security of Reduced Version of the Block Cipher Camellia 197 Let pf (±x; ±y) denote the transition probability of the truncated di®erential induced by function f. pf (±x; ±y) is de¯ned on the truncated di®erential (±x; ±y). Truncated di®erential probability pf (±x; ±y) is de¯ned by n m pf (±x; ±y) = 1=c Pr(x 2 GF(2 ) jf(x) © f(x © ¢x) = ¢y); Â(¢x)=±Xx;Â(¢y)=±y where c is the number of ¢x that satisfy Â(¢x) = ±x. 2.5 Block cipher Camellia Fig. 1 shows the entire structure of Camellia. Fig. 2 shows its round functions, and Fig. 3. shows F L-function and F L¡1-function. M 128 ¡ kw1(64) kw2(64) L0(64) R0(64) L0(64) k1(64), k2(64), k3(64), kl (64) R0(64) k4(64), k5(64), k6(64) 6-round F L1(64) k 2(64) R1(64) kl kl F 1(64) FL FL-1 2(64) L2(64) k 3(64) R2(64) k7(64), k8(64), k9(64), k10(64), k11(64), k12(64) F 6-round L3(64) k 4(64) R3(64) F kl kl k L4(64) 3(64) FL FL-1 4(64) 5(64) R4(64) F k13(64), k14(64), k15(64), L5(64) k 6(64) R5(64) k16(64), k17(64), k18(64) 6-round F kw3(64) kw4(64) C(128) Fig. 1. Block cipher Camellia 198 Makoto Sugita, Kazukuni Kobara, and Hideki Imai ki(64) y8 x8(8) z8 z’ S1 8(8) y7 z x7(8) 7 z’ S4 7(8) y6 z x6(8) 6 z’ S3 6(8) x y5 z5 5(8) S z’5(8) y 2 z x4(8) 4 4 z’ S4 4(8) y3 z3 x3(8) S z’3(8) y 3 z x2(8) 2 2 z’ S2 2(8) y z x1(8) 1 1 S1 z’1(8) S-Function P-Function Fig. 2. F function of Camellia 3 Previous security evaluation of Camellia 3.1 Security evaluation against truncated di®erential cryptanalysis [10] In [10], an algorithm to search for the e®ective truncated di®erentials of Feistel ciphers was proposed. This search algorithm consists of recursive procedures. Feistel ciphers are assumed to have S rounds and input and output block size is 2m bits. Algorithm 1 [11] Let ¢X(r); ¢Y (r) 2 GF(2)m be the input and output truncated di®erence of the r-th round functions fr. (¢L; ¢R) is the truncated di®erence of the plaintext. Let Pr((¢X(0); ¢X(1))j(¢L; ¢R))) be the r-round truncated di®erential proba- bilities.
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