### Algebraic attacks and annihilators

*Frederik Armknecht*

#### Abstract

Algebraic attacks on block ciphers and stream ciphers have gained more and more attention in cryptography. Their idea is to express a cipher by a system of equations whose solution reveals the secret key. The complexity of an algebraic attack generally increases with the degree of the equations. Hence, low-degree equations are crucial for the efficiency of algebraic attacks. In the case of simple combiners over $GF(2)$, it was proved in [9] that the existence of low-degree equations is equivalent to the existence of low-degree annihilators, and the term ā€¯algebraic immunityā€¯ was introduced. This result was extended to general finite fields GF (q) in [4]. In this paper, which improves parts of the unpublished eprint paper [2], we present a generalized framework which additionally covers combiners with memory and S- Boxes over GF (q). In all three cases, the existence of low-degree equations can be reduced to the existence of certain annihilators. This might serve as a starting point for further research.

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