Algebraic Aspects of the Advanced Encryption Standard by Carlos Cid

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 expertise (NIST) to be the successor to the knowledge Encryption average. Rijndael was once as a consequence standardized because the complex Encryption ordinary (AES), that is very likely the worlds most crucial block cipher. In 2002, a few new analytical suggestions have been urged which may have a dramatic impression at the protection of the AES. latest analytical recommendations for block ciphers count seriously on a statistical technique, while those new strategies are algebraic in nature.

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The quotient ring K = jjj^ IS a field of order q"^ = p"'', which is an extension field of degree d of F. 25, its elements can be represented as Ud-ix'''^^ + • • • + a2X^ + a\x + ao, where ai G F. 25. 59 states that any finite field of order p"'^ is isomorphic to K. We can also construct GF(p"'^) directly as an extension field of F. Let 0 denote a root of the irreducible polynomial f{x) of degree d. The set F(6) of all quotients (with nonzero denominator) of polynomials in 9 with coefficients in F is the smallest field containing both 9 and F.

The roots of f{x) arc the conjugates of a. We now consider some functions of interest on finite fields. 68 Let F be a finite field of order q and K be an extension field of F of degree d. The trace function on K with respect to F is the function Tr: K —> F defined by Ti{a) = a + a" + a"'' + ... + a'i''"\ Thus the trace of an element a £ K is the sum of all conjugates of a. 43). In fact, any linear functional on K is of the form a >—> Tr(/3a) for some /3 £ K. 69 Let F be a finite field of order q and K be an extension field of F of degree d.

Extension Field: b / ^ ^ + be^*^ + bg^^ + b46''* + hsO^ + 'o-^B'^ + bi6' + bo. In the AES, bytes are represented as elements of the Rijndael field F and are combined using addition (which is equivalent to bitwise XOR) and multiplication in the field. We use the common practice of representing a byte using hexadecimal notation, and we interpret such hexadecimal notation as a vector or field clement depending on the context. For example, 24 represents the bit string 00100100, the column vector ( 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 ) ^ , or the element (fi + 0^ in the extension field, according to the context.

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