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Algebraic Aspects of the Advanced Encryption Standard by Carlos Cid

By Carlos Cid

The Belgian block cipher Rijndael used to be selected in 2000 by means of the U.S. government’s nationwide Institute of criteria and know-how (NIST) to be the successor to the knowledge Encryption typical. Rijndael used to be thus standardized because the complicated Encryption typical (AES), that is probably the world’s most vital block cipher. In 2002, a few new analytical thoughts have been instructed which may have a dramatic impression at the safety of the AES. latest analytical options for block ciphers rely seriously on a statistical process, while those new concepts are algebraic in nature. Algebraic facets of the complicated Encryption commonplace, showing 5 years after book of the AES, offers the state-of-the-art for using such algebraic options in studying the AES. the first viewers for this paintings contains educational and researchers in cryptology; the e-book can also be appropriate for advanced-level scholars.

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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)^. Such an analysis of the augmented diffusion matrix M extends to parity checks. In this case, a parity check is a row vector e^ of length 128 over GF(2), and the parity check value of a vector x is e ^ x e GF(2). Furthermore, there are also 2^^ row vectors e t h a t are fixed by the augmented diffusion matrix M , that is e^M = e^.

If we have such a system fi{xi,.. ,Xn) = 0 , . . , / m ( a ; i , . . , a : „ ) = 0, then we can find its solution set by computing the Grobner basis for the ideal / = ( / i , . . , fm) and computing the associated variety V ( / ) . The Grobner basis of / provides implicit solutions to the equation system over the algebraic closure of the field F. A particularly useful monomial ordering for finding solutions to this polynomial equation system in F is the lex ordering, which is an example of an elimination ordering.

The property of a cipher being self-dual under non-trivial affine mappings is essentially equivalent to the property of a cipher possessing linear factors or structures. 1 The complementation property of the DES gives a nontrivial self-dual cipher or, equivalently, a hnear structure. Let 1^ and Ifc denote the vectors ( 1 , . . , 1)^ of lengths 64 and 56 respectively. Then take (J and 7 to be the mapping x 1-^ x + 1^, and K to be the mapping k ^> k + lk- These are known as the complementation mappings and give a non-trivial self-dual cipher for the DES.

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