Akelarre Cipher in .NET

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Akelarre Cipher
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Akelarre is defined for any number of rounds, but its developer conjectured that it is secure with four rounds. Amazingly, the cipher is insecure for any number of rounds. The attack we describe is given in [82], and this attack requires a small amount of work, regardless of the number of rounds. The weaknesses in Akelarre are also discussed in [50]. The Akelarre block size is 128-bits. The key length can be any multiple of 64 bits, but for simplicity, we assume here that the key size is the same as the block length, that is, 128 bits. The difficulty of the attack does not increase if the key size is increased. A key schedule algorithm is used t o expand the key into the required number of 32-bit subkeys, where this number of subkeys depends on the number of rounds. In Akelarre, the input, output, subkey and all intermediate calculations employ 32-bit words. In particular, the 128 bit input block is treated as four 32-bit sub-blocks, the output consists of four 32-bit sub-blocks and all subkeys are 32 bits. The encryption algorithm consists of an input transformation, followed by R rounds, and, finally, an output transformation, as illustrated in Figure 4.12. The key schedule is also specified as part of the cipher algorithm. The plaintext block first passes through the input transformation in Fig-
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Figure 4.12: Akelarre. ure 4.12, where mixed mode arithmetic operations are employed to combine the subkey with the four 32-bit sub-blocks of plaintext. In Figure 4.12, @ is XOR, while the other plus operation represents addition modulo 232. The Akelarre round function is also illustrated in Figure 4.12. We use T to denote the current round, where r = 0 , 1 , . . . ,R - 1. Each round begins with a keyed rotation, where the right 7 bits of the 32-bit subkey K13r+4 are used to determine the size of the rotation. That is, (K13~+4)25...31, interpreted as an integer, is the amount that the input is rotated left. Recall that <<< is our notation for a left cyclic shift. Let (Ao,A l , A2, A3) be the 128-bit input to round T (written as four 32-bit words) and let (Bo,B1, Ba, B3) be the output of the keyed rotation at the beginning of round r. Then
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(Bo,Bi, B3) = (Ao,A1, A2, A3) << (K13r+4)25...31. Bz, <
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Let (TO, I )be the output of the box labeled AR in Figure 4.12. Then T for a given 128-bit block (Bo,B1, B2, Bs),we have
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(To, Ti) = AR(Bo CE B2, Bi CE
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where we have ignored the dependence on the subkey. The AR function is defined below.
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162 Let ( D OD1, D2,Ds) be the output of round , defined above, we havc
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Then given B, and T, as
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The Di are the inputs t o the next round, except for the final round, where they become the inputs t o the output transformation. After R rounds, there is an output transformation, which consists of another keyed rotation, followed by XOR and addition of subkey words, as illustrated in Figure 4.12. The result of the output transformation is four 32-bit words which form the ciphertext block. The heart of Akelarre is the addition-rotation (AR) structure, the details of which appear in Figure 4.13. One pass through the AR structure can be viewed as 14 addition-rotations, each applied to a 32-bit sub-block. Each addition consists of subkey added to the current sub-block, with the addition taken modulo 232. Each rotation affects 31 bits, as explained below, with the amount of the rotation determined by the inputs to the AR structure. Typically, iterated block ciphers split the input in half and then operate on these halves. Since the AR structure in Akelarre operates on 32-bit quarter-blocks instead of 64-bit half-blocks, Akelarre s additiori-rotations can be viewed as 14 half-rounds. Consequently, it could be argued that one pass through the AR structure is roughly equivalent to seven rounds of a typical block cipher, but this is somewhat misleading since each Akelarre addition-.. rotation operation is extremely simple. In Figure 4.13, we denote the two 32-bit. input.s to the AR structure as WO and W and the output 32-bit words as 20and 2 1 . Note that Wl is processed 1 first,; with the bits of WO used to determine the required rotations, and the resulting output is 2 1 . Then WO processed, with the bits of 2 used to is 1 determine the rotations, and the resulting output is 20. The rotations in thc AR structure are left rotations, but they are slightly different than the standard rotations used in the Akelarre round function and output transformation. In each AR rotation, either the low-order or the high-order bit remains fixed (as indicated by a 1 in Figure 4.13) and the remaining 31 bits (indicated by a 31) are rotated. The amount of the rotation ranges from 0 to 31 in some steps, and from 0 to 15 in other steps, depending on whether five or four bits are used to determine the rotation. For example, the first step in the AR structure consists of a left rotation of bits 0 through 30 of W j , with the rightmost bit (bit 31 in our notation) remaining fixed and the size of the rotation determined by (W0)27...31 In our standard notation, . this rotation can be written as