[Resource Topic] 2004/081: Rewriting Variables: the Complexity of Fast Algebraic Attacks on Stream Ciphers

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Rewriting Variables: the Complexity of Fast Algebraic Attacks on Stream Ciphers

Authors: Philip Hawkes, Gregory G. Rose


Recently proposed algebraic attacks [AK03,CM03] and fast algebraic attacks [A04,C03] have provided the best analyses against some deployed LFSR-based ciphers. The process complexity is exponential in the degree of the equations. Fast algebraic attacks were introduced [C03] as a way of reducing run-time complexity by reducing the degree of the system of equations. Previous reports on fast algebraic attacks [A04,C03] have underestimated the complexity of substituting the keystream into the system of equations, which in some cases dominates the attack. We also show how the Fast Fourier Transform (FFT) [CT65] can be applied to decrease the complexity of the substitution step, although in one case this step still dominates the attack complexity. Finally, we show that all functions of degree d satisfy a common, function-independent linear combination that may be used in the pre-computation step of the fast algebraic attack and provide an explicit factorization of the corresponding characteristic polynomial. This yields the fastest known method for performing the pre-computation step.

ePrint: https://eprint.iacr.org/2004/081

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