Algebraic Aspects of the Advanced Encryption Standard by Carlos Cid

By Carlos Cid

The Belgian block cipher Rijndael used to be selected in 2000 via the U.S. governments nationwide Institute of criteria and know-how (NIST) to be the successor to the knowledge Encryption common. Rijndael used to be thus standardized because the complex Encryption general (AES), that's in all likelihood the worlds most vital block cipher. In 2002, a few new analytical recommendations have been advised which can have a dramatic influence at the safeguard of the AES. current analytical thoughts for block ciphers rely seriously on a statistical procedure, while those new suggestions are algebraic in nature.

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While these observations do not apply to the AES, they demonstrate that high probability differential effects can be observed in AES-like ciphers satisfying the demands of the wide trail strategy [37, 39]. Thus further analysis of some of the issues raised in [87-89] may yet be of interest in the analysis of the AES. 2. Algebraic Representations There can be many equivalent ways to describe a cryptosystcm. Although standardisation requires the same convention to be used for d a t a representation, alternative representations of the cipher operations can be of much interest.

The second part is Shif tRows followed by MixColumns, which gives difi^usion across the state array. This is termed the diffusion layer. The final part of an AES round introduces key material by AddRoundKey. We now discuss the substitution and diffusion layers. S u b s t i t u t i o n layer T h e substitution layer is based on the AES S-box which is, in turn, defined by the composition of three operations. Inversion. The AES inversion operation is inversion in the Rijndael field F , but extended so that 0 — i > 0.

Thus there are 2-"^*^ vectors fixed by the augmented diffusion. Furthermore suppose t h a t x and x' are two vectors such that the difference x -f x' e Vi, then M x + M x ' = M ( x + x') = X + x', and so the augmented diffusion of the AES also fixes 2^^ differences. In particular, there exist vectors t h a t are fixed by M and are nonzero for only 12 of the 16 bytes of the state. Thus the use of such a difference in an analysis of the AES would involve only 12 active S-boxes in each round. One such vector over GF(2) given in hexadecimal notation is (55336600 33550066 55336600 33550066)^.

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