By Carlos Cid

The Belgian block cipher Rijndael used to be selected in 2000 through the U.S. governments nationwide Institute of criteria and expertise (NIST) to be the successor to the information Encryption regular. Rijndael was once thus standardized because the complex Encryption general (AES), that is probably the worlds most crucial block cipher. In 2002, a few new analytical concepts have been instructed which may have a dramatic impression at the safeguard of the AES. latest analytical innovations for block ciphers count seriously on a statistical procedure, while those new ideas are algebraic in nature.

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58 A finite field of characteristic p (prime) has a unique minimal subfield isomorphic to GF(p). If K is a extension field of the field F, then K is also a vector space over F. T h e dimension of this vector space is the degree of the extension. 58 that every finite field has order p " for some prime p and some positive integer n. 59 For every prime number p and every positive integer n, there exists a finite field of order p " . Furthermore, any two finite fields of order p " are isomorphic. Thus finite fields of order p " arc unique up to isomorphism.

Thus r is a field automorphism of GF(p'^), known as the Frobenius automorphism. The set of all automorphisms of G¥{p'^) under the operation of composition is the cyclic group of order d generated by r . We note t h a t T fixes all elements of the subfield GF(p) of GF(p''). Thus the automorphisms of GF(p'^) are also hncar transformations over GF(p). 5. Varieties and Grobner Bases A large part of this monograph is concerned with expressing an AES encryption as a system of polynomial equations and considering methods of solution for such equations.

M are polynomials in F [ a ; i , . . , x„]. ,a„) = 0 This variety is denoted by V ( / i , . . , fm)- for i = 1 , . . , m } . 72 describes the set of solutions in F of the polynomial equation system / i ( a ; i , . . , a ; „ ) = 0 , . . , / „ i ( a ; i , . . ,a;„) = 0. 73 Consider the polynomial ring IR[x,j/] in two variables, and let / ( x , y) = x^ -\- xp — \ and g{x, y) = x — 1 he two polynomials in lR[a;,y]. The affinc variety V ( / ) consists of the points in the circle of radius 1 in R^ and is the solution set of the equation x"^ + y"^ = 1.