Collision Attacks on AES-192/256, Crypton-192/256, Mcrypton-96/128, and Anubis
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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 713673, 10 pages http://dx.doi.org/10.1155/2013/713673 Research Article Collision Attacks on AES-192/256, Crypton-192/256, mCrypton-96/128, and Anubis Jinkeon Kang,1 Kitae Jeong,1 Jaechul Sung,2 Seokhie Hong,1 and Kyungho Lee1 1 Center for Information Security Technologies (CIST), Korea University, Anam-dong, Seongbuk-gu, Seoul 136-713, Republic of Korea 2 Department of Mathematics, University of Seoul, Jeonnong-dong, Dongdaemun-gu, Seoul 130-743, Republic of Korea Correspondence should be addressed to Kyungho Lee; [email protected] Received 19 July 2013; Accepted 22 August 2013 Academic Editor: Jongsung Kim Copyright © 2013 Jinkeon Kang 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. At AES’00, a collision attack on 7-round reduced AES was proposed. In this paper, we apply this idea to seven SPN block ciphers, AES-192/256, Crypton-192/256, mCrypton-96/128, and Anubis. Applying our attacks on AES-192/256, we improve the attack result based on meet-in-the-middle attack (AES-192) and the attack result proposed in AES’00 (AES-256), respectively. Our attack result on Anubis is superior to known cryptanalytic result on it. In the cases of Crypton-192/256 and mCrypton-96/128, our attacks are applicable to 8-round reduced versions. The attack results on mCrypton-96/128 are more practical than known cryptanalytic results on them. 1. Introduction attack results on them. Rijndael was announced as the Advanced Encryption Standard (AES) in 2001. After DES, it Recently, meet-in-the-middle attack has received attention. isoneofthemostwidelyusedandanalyzedciphersinthe The attack procedure of meet-in-the-middle attack can be world. AES is a 128-bit block cipher and accepts key sizes of summarized as follows. Let M be the message space and K 128, 192,and256 bits. These versions of AES are called AES- thekeyspace.Supposethat and : M × K → M are 128/192/256 and the number of rounds for these versions two block ciphers and let = ∘. In this attack, an is 10, 12,and14, respectively. Our attacks on AES-192/256 69.2 attacker tries to deduce from a given plaintext/ciphertext require a computational complexity of 2 encryptions with −1 32 74 pair =() by trying to solve () = ().In 2 chosen plaintexts and 2 memory bytes, respectively. somecases,theequationisnottestedforallthebitsof Though our attacks on them are not applicable to the full the intermediate encryption value, but rather for only some AES-192/256, these are superior to the attack results of [1] of them. So far, many meet-in-the-middle attack results on (AES-192)and[3](AES-256), respectively. block cipher have been proposed [1, 2]. On the other hand, Crypton is a 128-bit block cipher submitted as a candidate Gilbert and Minier proposed a collision attack on 7-round for the AES proposal of NIST. This algorithm has a 12-round of AES in [3]. This attack is a type of meet-in-the middle SPN structure with 128/192/256-bit secret keys. According attack and exploits a 4-round distinguisher. In general, block to the length of the secret keys, we call them Crypton- cipher is completely deterministic, that is, if it is run again 128/192/256, respectively. Our attack is applicable to 8-round with the exact same input values, identical output values will reduced Crypton-192/256. To our knowledge, truncated dif- be produced. Similarly, the different input values result in ferential cryptanalysis on 8-round reduced Crypton-192/256 the different output values. However, it is possible that the has the best results on them, so far [8]. The attacks on an different input values result in the output values of which the 8-round reduced Crypton-192/256 need the computational 161.6 96 specific parts are the same. We call this case a collision. complexity of 2 encryptions with 2 chosen plaintexts 138 In this paper, we apply the main idea of [3]toseven and 2 memory bytes, respectively. Compared to the attack SPN block ciphers, such as AES-192/256 [4], Crypton [5], results of [8], our attacks decrease the data complexity but mCrypton-96/128 [6], and Anubis [7]. Table 1 shows our increase the computational complexity. 2 Journal of Applied Mathematics secret key, the number of rounds is 12 and 14,respectively. 4×4 S00 S01 S02 S03 Each internal state is treated as a byte matrix of size , 8 where each byte represents a value in GF(2 ) (see Figure 1). In round , input/output values and round key are S10 S11 S12 S13 denoted by , ,andRK,respectively(=1,...,12(AES- 0 192)or14 (AES-256)). A whitening key is denoted by RK . The round function of AES-192/256 applies the following S20 S21 S22 S23 four operations to the state matrix: (i) SubBytes (SB): applying the same 8×8invertible S- box 16 timesinparalleloneachbyteofthestate; S30 S31 S32 S33 (ii) ShiftRows (SR): cyclic shift of each row (the th row is shifted by bytes to the left); Figure 1: Internal state of AES-192/256 and Anubis. (iii) MixColumns (MC): multiplication of each column by 8 aconstant4×4matrix over GF(2 ); (iv) AddRoundKey (ARK): XORing the state with a 128- A 64-bit block cipher mCrypton was designed as a bit round key. reduced version of Crypton for a ubiquitous environment. It provides low-resource hardware implementation, which is TheSB,SR,MC,andARKtransformationsareapplied suitable for low-end devices such as RFID tags and wireless to each round except that MC is omitted in the last round. sensor network. Several security aspects related to RFID Besides, before the first round, an extra ARK is applied, tags and wireless sensor network have already been studied. which we call a whitening key step. For the more detailed mCrypton has 12 rounds and supports the 64/96/128-bit descriptions of AES-192/256, we refer to [4]. We omit the key secret keys. Our attack is applicable to mCrypton-96/128. schedules of them, as they do not affect our attacks. 81.6 We require the computational complexity of 2 encryptions 48 65 with 2 chosen plaintexts and 2 memory bytes. However, 2.2. Finding a 4-Round Distinguisher of AES. In our attack this is not the best result on them, since related-key impossi- on AES, we use a 4-rounddistinguisherofround2∼5. We 2 ble differential cryptanalysis on 9-round reduced mCrypton- consider an input value of round 2 in AES which is 96/128 was proposed in [9]. However, considering the attack constructed as follows: assumption of a related-key attack, our attacks are more 2 2 2 2 (i) ( , , , ):four8-bit variables; practical than it. 00 10 20 30 2 Anubis is a block cipher submitted to the NESSIE project (ii) :afixed8-bit constant (0≤≤3,1≤≤3). 128 and operates on data blocks of bits, accepting keys of 2 length 32 bits ( = 4,...,10).ItisaRijndaelvariant By using ,wecanconstructthefollowingequationson 3 ( = 0,...,3) that uses involutions for the various operations. Our attack .Here, means the entry of the coefficient (=0,...,3) is applicable to Anubis where the length of the secret key is matrix of MC : longer than 192 bits. We need the computational complexity 3 2 2 2 161.6 96 138 0 =0 ∘ SB [00]⊕1 ∘ SB [11]⊕2 ∘ SB [22] of 2 encryptions with 2 chosen plaintexts and 2 memory bytes. This result is superior to known cryptanalytic 2 2 ⊕3 ∘ SB [33]⊕RK0, results on this algorithm. The rest of this paper is organized as follows. In Section 2, 3 2 2 2 1 =0 ∘ SB [01]⊕1 ∘ SB [12]⊕2 ∘ SB [23] weproposecollisionattacksonAES-192/256.Collision attacks on Crypton-192/256 and mCrypton-96/128 are intro- 2 2 ⊕3 ∘ SB [30]⊕RK1, duced in Section 3.InSection 4,wepresentacollisionattack (1) on Anubis. In each section, we start with a brief description 3 2 2 2 =0 ∘ SB [ ]⊕1 ∘ SB [ ]⊕2 ∘ SB [ ] of the block cipher concerned. Finally, we give our conclusion 2 02 13 20 in Section 5. 2 2 ⊕3 ∘ SB [31]⊕RK2, 3 2 2 2 2. Collision Attacks on 7-Round Reduced 3 =0 ∘ SB [03]⊕1 ∘ SB [10]⊕2 ∘ SB [21] AES-192/256 2 2 ⊕3 ∘ SB [32]⊕RK3. First, we briefly present AES-192/256.Andthenweintroduce 3 3 3 3 collision attacks on AES-192/256.Notethattheattackproce- Note that each 0, 1, 2,and3 depends on only one 192 256 2 2 2 2 dure on AES- isthesameasthatonAES- .Thus,forthe 8-bit variable and one 8-bit round key: (00, RK0), (30, RK1), 192 256 2 2 2 2 simplicity of notations, we just call them AES- / AES. (20, RK2),and(10, RK3). Applying the above procedure repeatedly, as shown in (2), 5 2.1. AES-192/256. AES-192/256 are 128-bit block ciphers wecanobtainanequationonaninputvalue00 of round 5 by 192 256 2 2 2 2 with / -bit secret keys. According to the length of the using . Here, bold characters indicate (10,20,30) Journal of Applied Mathematics 3 5 2 2 2 2 2 00 =00 ∘ SB [00 ∘ SB (00 ∘ SB [00]⊕01 ∘ SB [11]⊕02 ∘ SB [22]⊕03 ∘ SB [33]⊕RK00) 2 2 2 2 2 ⊕01 ∘ SB (10 ∘ SB [01]⊕11 ∘ SB [12]⊕12 ∘ SB [23]⊕13 ∘ SB [I30]⊕RK11) 2 2 2 2 2 ⊕02 ∘ SB (20 ∘ SB [02]⊕21 ∘ SB [13]⊕22 ∘ SB [I20]⊕23 ∘ SB [31]⊕RK22) 2 2 2 2 2 3 ⊕03 ∘ SB (30 ∘ SB [03]⊕31 ∘ SB [I10]⊕32 ∘ SB [21]⊕33 ∘ SB [32]⊕RK33)⊕RK00] 2 2 2 2 2 ⊕01 ∘ SB [10 ∘ SB (00 ∘ SB [01]⊕01 ∘ SB [02]⊕2 ∘ SB [23]⊕03 ∘ SB [I30]⊕RK01) 2 2 2 2 2 ⊕11 ∘ SB (10 ∘ SB [02]⊕11 ∘ SB [13]⊕12 ∘ SB [I20]⊕13 ∘ SB [31]⊕RK12) 2 2 2 2 2 ⊕12 ∘ SB (20 ∘ SB [03]⊕21 ∘ SB [I10]⊕22 ∘ SB [21]⊕23 ∘ SB [32]⊕RK23) 2 2 2 2 2 3 ⊕13 ∘ SB (30 ∘ SB [ ]⊕31 ∘ SB [ ]⊕32 ∘ SB [ ]⊕33 ∘ SB [ ]⊕RK )⊕RK ] 00 11 22 33 30 11 (2) 2 2 2 2 2 ⊕02 ∘ SB [20 ∘ SB (00 ∘ SB [02]⊕01 ∘ SB [13]⊕02 ∘ SB [I20]⊕03 ∘ SB [31]⊕RK02) 2 2 2 2 2 ⊕21 ∘ SB (10 ∘ SB [03]⊕11 ∘ SB [I10]⊕12 ∘ SB [21]⊕13 ∘ SB [32]⊕RK13) 2 2 2 2 2 ⊕22 ∘ SB (20 ∘ SB [00]⊕21 ∘ SB [11]⊕22 ∘ SB [22]⊕23 ∘ SB [33]⊕RK20) 2 2 2 2 2 3 ⊕23 ∘ SB (30 ∘ SB [01]⊕31 ∘ SB [12]⊕32 ∘ SB [23]⊕33 ∘ SB [I30]⊕RK31)⊕RK22] 2 2 2 2 2 ⊕03 ∘ SB [30 ∘ SB (00 ∘ SB [03]⊕01 ∘ SB [I10]⊕02 ∘ SB [21]⊕03 ∘ SB [32]⊕RK03) 2 2 2 2 2 ⊕31 ∘ SB (10 ∘ SB [00]⊕11 ∘ SB [11]⊕12 ∘ SB [22]⊕13 ∘ SB [33]⊕RK10) 2 2 2 2 2 ⊕32 ∘ SB (20 ∘ SB [01]⊕21 ∘ SB [12]⊕22 ∘ SB [23]⊕23 ∘ SB [I30]⊕RK21) 2 2 2 2 2 3 4 ⊕33 ∘ SB (30 ∘ SB [02]⊕31 ∘ SB [13]⊕32 ∘ SB [I20]⊕33 ∘ SB [31]⊕RK32)⊕RK33]⊕RK00.