Cryptography

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By Douglas Stinson

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A state A is organized as an array of 5 × 5 lanes each of length w ∈ {1, 2, 4, 8, 16, 32, 64}. e. b ∈ {25, 50, 100, 200, 400, 800, 1600}. The number of rounds nr depends on the state size and is given by nr = 12+2 where = log2 w. A round consists of a sequence of invertible steps on the state A[x][y][z]: – θ step is a linear map and can be divided into three steps: C[x] = A[x, 0] ⊕ A[x, 1] ⊕ A[x, 2] ⊕ A[x, 3] ⊕ A[x, 4] for x = 0, . . , 4 D[x] = C[x − 1] ⊕ (C[x + 1] ≫ 1) for x = 0, . . , 4 A[x, y] = A[x, y] ⊕ D[x] for x, y = 0, .

Q − 1. Note that if we start from an irreducible non-primitive polynomial g of degree n and order d and construct the ﬁnite ﬁeld F2n as the algebraic extension by β, a root of g, then there are several primitive elements α ∈ F2n which are solutions for β = αq (with q = (2n − 1)/d). Moreover these α need not all have the same minimal polynomial. 22 D. Gardner, A. -W. Phan Our goal is, given g, to produce one elementary sequence from each of the q equivalence classes; moreover these sequences should be in the correct phase relative to each other (as described by Theorem 4) such that they may be interleaved to generate an m-sequence.

It follows that a plaintext can be viewed as a vector of elements in GF(2d ) and vice-versa. Moreover, arithmetic over plaintexts correspond to element-wise arithmetic over -vectors. Elements of this -vector are usually referred to as slots. In [13] it is shown how to move the content of a slot in another one, using automorphisms over R2 . Recall that for any i ∈ Z∗m the automorphism κi over R2 is deﬁned as κi : a(x) → a(xi ) (mod Φm (x)). Up to a reordering of factors Fi , there exists an integer h such that if a ∈ R encodes the -vector (a0 , .