By Carlos Cid

ISBN-10: 0387243631

ISBN-13: 9780387243634

ISBN-10: 0387368426

ISBN-13: 9780387368429

The Belgian block cipher Rijndael used to be selected in 2000 by way of the U.S. governments nationwide Institute of criteria and know-how (NIST) to be the successor to the knowledge Encryption common. Rijndael was once to that end standardized because the complex Encryption average (AES), that's almost certainly the worlds most vital block cipher. In 2002, a few new analytical ideas have been instructed which can have a dramatic impact at the defense of the AES. current analytical ideas for block ciphers rely seriously on a statistical procedure, while those new innovations are algebraic in nature.

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**Extra resources for Algebraic Aspects of the Advanced Encryption Standard**

**Example text**

This is termed the substitution layer. 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.

The AES array of bytes. either as a quotient ring or as an extension field. We refer to the field GF(2*) defined by the Rijndael polynomial as the Rijndael field and denote it by F throughout this monograph. Thus F = GF(2)[a:]/(m(a;)) or F = GF{2){8), where 6 denotes a root of the Rijndael polynomial, termed the Rijndael root. The representation of a byte b7b6b6b4b3b2bibo in F can then be given in cither of the following two equivalent ways. • Quotient Ring: h-jx^ + hex^ + bsx^ + h^x^^ + h^x^ + b2a;^ + bia; + bo.

This field is called the Galois field of order p and is denoted by GF(p). The Galois field GF(p) plays a fundamental role in the theory of finite fields. 57 Suppose that F and K are two fields. If F c K, then F is said to be a subfield of K, or equivalently K is said to be an extension field of F. 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.

### Algebraic Aspects of the Advanced Encryption Standard by Carlos Cid

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