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Diffusion Analysis of Message Expansion in STITCH-256

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Diffusion Analysis of Message Expansion in STITCH-256 Journal of Information Security, 2013, 4, 129-137 http://dx.doi.org/10.4236/jis.2013.43015 Published Online July 2013 (http://www.scirp.org/journal/jis) Diffusion Analysis of Message Expansion in STITCH-256 Norziana Jamil1,2, Ramlan Mahmod2, Muhammad Reza Z’aba3, Nur Izura Udzir2, 2 Zuriati Ahmad Zukarnain 1College of Information Technology, Universiti Tenaga Nasional, Kajang, Malaysia 2Faculty of Computer Science and Information Technology, Universiti Putra Malaysia, Seri Kembangan, Malaysia 3Cryptography Lab, MIMOS Berhad, Technology Park Malaysia, Bukit Jalil, Malaysia Email: [email protected], ramlan, Izura, [email protected], [email protected] Received May 7, 2013; revised June 9, 2013; accepted June 17, 2013 Copyright © 2013 Norziana Jamil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Cryptographic hash functions are built up from individual components, namely pre-processing, step transformation, and final processing. Some of the hash functions, such as SHA-256 and STITCH-256, employ non-linear message expan- sion in their pre-processing stage. However, STITCH-256 was claimed to produce high diffusion in its message expan- sion. In a cryptographic algorithm, high diffusion is desirable as it helps prevent an attacker finding collision-producing differences, which would allow one to find collisions of the whole function without resorting to a brute force search. In this paper, we analyzed the diffusion property of message expansion of STITCH-256 by observing the effect of a single bit difference over the output bits, and compare the result with that of SHA-256. We repeated the same procedure in 3 experiments of different round. The results from the experiments showed that the minimal weight in the message ex- pansion of STITCH-256 is very much lower than that in the message expansion of SHA-256, i.e. message expansion of STITCH-256 produce high diffusion. Significantly, we showed that the probability to construct differential characteris- tic in the message expansion of STITCH-256 is reduced. Keywords: STITCH-256; Message Expansion; Diffusion; Hash Function 1. Introduction ssage expansion. Briefly, message expansion in the SHA-family is perf- Recent advances in the cryptanalysis of hash functions ormed by recursive expansion. In SHA-1, for example, [1-12], to name a few, have led to the unexpected failure the message expansion accepts a 512-bit input that is of some popular algorithms, such as MD4, MD5, SHA-0, divided into sixteen 32-bit words WW015,, . Sixty-four SHA-1, HAVAL-128, and RIPEMD. From this crypt- additional expanded message words are generated as analysis, we understand that these broken hash functions follows: can have two distinct messages yielding the same hash WW W W W ROTL1 value, known as a collision. The new cryptanalysis tech- ii3 i 8 i 14 i 16 niques introduced by Wang et al. [13-16] provided this fori 16, ,79. breakthrough in cryptography, finding collisions for Message expansion in SHA-1 differs from that of MD4, MD5, SHA-0, SHA-1, HAVAL-128, and RIPEMD. SHA-0 only in the rotation of one bit to the left. The 80 It was found that the most successful attack on these hash words can be considered to constitute a linear code over functions is a differential attack, whereby a difference in F2. Due to the quasi-cyclic nature of message expansion the messages leads to zero difference in the output of the in SHA-0 and SHA-1, the full collision path can easily be hash function. In other words, the collision is obtained by constructed, as in the attack by Wang et al. This can be constructing a collision path, or characteristic, that ful- seen, for example, in SHA-0 message expansion, where fills certain conditions with respect to the message dif- differential characteristics with a probability of 1 can ferences. In SHA-0, SHA-1, and the MD-family of hash easily be constructed in the first 16 steps, as a single bit functions, a message with a small difference in the ex- difference affects fewer than 28 bits in the output. In panded keys is first obtained. This is then used to con- SHA-1, the code gives a minimum weight of no more struct a collision path in the step transformation. This than 44 for the full rounds. These traits were exploited by paper focuses only on the first part, which is the me- Wang et al. in order to find a collision path for the whole Copyright © 2013 SciRes. JIS 130 N. JAMIL ET AL. hash function with a complexity of 269 hash operations Table 1. Notation. [14]. Notation Description As a consequence of the work by Wang et al., the MD-family and SHA-0/1 hash functions are no longer A B XOR operation of A and B. suitable for secure communications. SHA-256 has now A + B Addition of A and B modulo 232. become the recommended hash function for many appli- Mi The i-th block of the 32-bit input message M. cations that require secure communication. The message Wi The i-th block of the 32-bit input message word W. expansion in SHA-256 is slightly different from its Bit rotation of A by n position/s to the right/left ROTR/Ln (A) predecessors. It is the first hash function to use nonlinear respectively. Bit shift of A by n position/s to the right/left modular addition in its message expansion, and success- SHFR/Ln (A) fully increases the minimum Hamming weight of the respectively. output bits from 44 (in a full round of SHA-1 message N Number of rounds in the message expansion. expansion) to 507 for a single-bit difference over a full round. To the best of our knowledge, no optimal lower 7183 0 x ROT x ROT x ROT x bound on the minimum weight of the output bits has yet where and x SHF17 x SHF19 x SHF10 x been found by the cryptographic community. However, it 1 . is important to have a high minimum weight for a sin- In total, there are 144 addition (modulo 32) operations gle-bit difference over a full round of message expansion and 192 XOR, rotation, and shift operations used in the to prevent an attacker from constructing a collision path. message expansion of SHA-256. This is because of a useful heuristic, often used in the analysis of SHA-0 and SHA-1, suggests that each weight 3. Message Expansion of STITCH-256 of the output bits lowers the probability of successful collision characteristics by, on average, a factor of 2−2.5 In this section, we describe the message expansion proce- [6]. dure of STITCH-256. We use the notation in Table 1. In STITCH-256 [17] is a dedicated cryptographic hash STITCH-256, the pre-processing again involves padding, function that also employs message expansion as a whereby the arbitrary length message is extended to an source of diffusion. In this paper, we describe the mes- exact multiple of 512-bits. This is followed by the mes- sage expansion process in STITCH-256 and compare it sage expansion, which works as follows: with that in SHA-256. This paper is organized as follows: WMii for 0 i 15 In Sections 2 and 3, we briefly describe the message ex- 11 pansion methods of SHA-256 and STITCH-256, respec- ROT Wiiiii s016151413 W,,, W W W tively. We then analyze the diffusion property of the two sW,,, W W W SV message expansion procedures in Section 4, and show 11211109iiii 0 13 that the minimum weight of the STITCH-256 output bits ROT s08765 Wiiii,,, W W W is higher than the minimum weight of the SHA-256 out- sW,,, W W W SV put bits. Finally, we offer some concluding remarks in 14321iiii 1 Section 5. for 16t 63 where 2. Message Expansion of SHA-256 wxyz,,, w x y z 0 In this section, we briefly describe the message expa- wxyz,,, w x y z and 1 nsion process of SHA-256. We use the notation shown in We use two salt values to support the message expan- Table 1 throughout this paper. In SHA-256, the pre-processing involves padding foll- sion of STITCH-256, where SV0 = 67452301 and SV1 = owed by message expansion. A message of arbitrary 41083726. In the message expansion of STITCH-256, length is first padded to form multiple 512-bit message every sixteen message words are taken into account to blocks. Each of the message blocks is denoted by a row form the (i-16)-th message word. This is to maximize the vector m represented by sixteen 32-bit words, bit propagation in the message expansion of STIT- CH-256. The bit rotations in the message expansion of M 015,, M . The input message is then expanded to sixty four 32-bit words by the message expansion process, and STITCH-256 are carefully selected to increase the diffu- this can be considered as a 2048-bit expanded message sion to the whole message expansion. The message ex- row vector w. The message words Wt are defined as foll- pansion of STITCH-256 is illustrated as in Figure 1. ows: In STITCH-256, the 512-bit message input is expan- ded to thirty-two 32-bit message words. This gives the Mi for 0i 15 W output of message expansion as 1024 bits. All the mes- i WW W W for 16 i 63 1tt 2 7 0 t 15 t 16 sage words WW1, , 31 are then reordered to cater to Copyright © 2013 SciRes.
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