By Carlos Cid

ISBN-10: 0387243631

ISBN-13: 9780387243634

ISBN-10: 0387368426

ISBN-13: 9780387368429

The Belgian block cipher Rijndael was once selected in 2000 via the U.S. government’s nationwide Institute of criteria and expertise (NIST) to be the successor to the knowledge Encryption ordinary. Rijndael used to be thus standardized because the complex Encryption average (AES), that's in all likelihood the world’s most crucial block cipher. In 2002, a few new analytical options have been instructed which may have a dramatic impression at the safeguard of the AES. present analytical thoughts for block ciphers rely seriously on a statistical procedure, while those new concepts are algebraic in nature. Algebraic features of the complicated Encryption general, showing 5 years after e-book of the AES, offers the cutting-edge for using such algebraic innovations in examining the AES. the first viewers for this paintings comprises educational and researchers in cryptology; the publication can also be appropriate for advanced-level scholars.

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**Additional info for Algebraic Aspects of the Advanced Encryption Standard (Advances in Information Security)**

**Sample text**

8. An equivalent S-box over GF(2*) for small scale variants of the AES. viewed as an n^ x ric array of words of e bits. Useful small scale variants exist when both n^ and ric are restricted to 1, 2, or 4. Examples of such arrays with words numbered in the AES style are given below. m 0 1 0 1 2 3 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 The word sizes e = 4 and e = 8 are the most relevant and are defined with respect to the fields GF(2'') and GF(2*^). The field G¥{2^) is defined by the primitive polynomial x'^ + x + 1 over GF(2) with root p.

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, the input byte to the S-box is regarded as an element w £ F and for w 7^ 0 the output x satisfies x = w and wx = 1. 6. GF{2)-linear mapping. The GF(2)-Hnear mapping is a hnear transformation ^ : GF(2)* —» GF(2)^ specified by an 8 x 8 circulant matrix over GF(2).

The number of primitive polynomials of degree n is -<^((J'" — 1), where (/? is Euler's totient function. 63 There are i (^(1)2** + / i ( 2 ) 2 ' ' + / i ( 4 ) 2 2 + / i ( 8 ) 2 i ) = 60 irreducible polynomials of degree 8 in GF(2)[a;], of which \ip{2^-1) = 16 arc primitive polynomials. 64 A field F is said to be algebraically closed if every polynomial in F[a;] has a root in F. The algebraic closure of a field F is the smallest extension field K of F such t h a t K is algebraically closed. R e p r e s e n t a t i o n s of a finite field Let F be a field and K = ¥{0) be an extension field of F of degree d.

### Algebraic Aspects of the Advanced Encryption Standard (Advances in Information Security) by Carlos Cid

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