Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
The Grain Family of Stream Ciphers
New Stream Cipher Designs
New Stream Cipher Designs
On the decomposition of an NFSR into the cascade connection of an NFSR into an LFSR
Journal of Complexity
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Recently nonlinear feedback shift registers (NFSRs) have frequently been used as building blocks for designing stream ciphers. Let NFSR (g) be an m-stage NFSR with characteristic function $${g=x_{0}\oplus g_{1}(x_{1},\cdots ,x_{m-1})\oplus x_{m}}$$ . Up to now there has been no known method to determine whether the family of output sequences of the NFSR (g), denoted by S(g), contains a sub-family of sequences that are exactly the output sequences of an NFSR(f) of stage n m. This paper studies affine cases, that is, finding an affine function f such that S(f) is a subset of S(g). If S(g) contains an affine sub-family S(f) whose order n is close to m, then a large number of sequences generated by the NFSR (g) have low linear complexities. First, we give two methods to bound the maximal order of affine sub-families included in S(g). Experimental data indicate that if S(g) contains an affine sub-family of order not smaller than m/2, then the upper bound given in the paper is tight. Second, we propose two algorithms to solve affine sub-families of a given order n included in S(g), both of which aim at affine sub-families with the maximal order. Algorithm 1 is applicable when n is close to m, while the feasibility of Algorithm 2 relies on the distribution of nonlinear terms of g. In particular, if Algorithm 2 works, then its computation complexity is less than that of Algorithm 1 and it is quite efficient for a number of cases.