By Carlos Cid
The Belgian block cipher Rijndael was once selected in 2000 by way of the U.S. governments nationwide Institute of criteria and know-how (NIST) to be the successor to the information Encryption regular. Rijndael was once in this case standardized because the complex Encryption commonplace (AES), that is very likely the worlds most vital block cipher. In 2002, a few new analytical thoughts have been steered which can have a dramatic impact at the safety of the AES. current analytical recommendations for block ciphers count seriously on a statistical technique, while those new recommendations are algebraic in nature.
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Extra resources for Algebraic Aspects of the Advanced Encryption Standard
2. The GF(2)-linear mapping on the entire state space is thus given by a 128x128 matrix L over GF(2), where L is a block diagonal matrix with blocks given by this circulant 8 x 8 matrix. Sliif tRows a n d MixColumns o p e r a t i o n s T h e Shif tRows operation is based on the rotation of rows of the state array. 2 V0 0 0 0 \ 0 0 R^ By re-ordering the rows and columns of this matrix, we can obtain a 16x16 matrix R over F that represents the Shif tRows operation with respect to the standard state array ordering by column.
1. The AES GF(2)-linear mapping within the S-box. 7. S-box constant. The output byte y of the GF(2)-linear mapping is regarded as an clement of the Rijndael field F and added to the field clement 63 to produce the output from the S-box. T h e rationale for using the inversion operation is that it provides good local resistance [98, 99] to the standard block cipher cryptanalytic techniques of differential [11, 10, 68] and linear  cryptanalysis. The rationale for the use of the GF(2)-linear mapping and the S-box constant is to increase the algebraic complexity of the S-box and to remove fixed points respectively [37, 39].
Multiplication by an element of F is a linear transformation of F considered as the vector space GF(2)®, and so multiplication is described by an 8 x 8 matrix over GF(2). 65). T h e linear diffusion required by the wide trail strategy is therefore given by the 128x128 matrix CR over GF(2). The algebraic properties of R and C, and hence CR, are directly given by those of R and C. A u g m e n t e d linear diffusion We have seen t h a t the final two parts of the SubBytes operation, namely the GF(2)-linear mapping and the addition of the S-box constant, form an affine operation over GF(2).