Complexity computations in code cracking problems

Abstract

The linear complexity of a sequence is the size of the shortest feedback drift register, which generates the sequence. A method of estimating this complexity consists in successive processing of the sequence and obtaining a monotone increasing sequence of estimators (linear complexity profile), of which the limit is the linear complexity. The results presented here have applications in cryptography; more exactly, they can be used to find the linear equivalent complexity of a cipher algorithm and this to estimate the cryptographic resistance. In this paper, we present some efficient methods of estimating and evaluating the linear equivalent complexity of a cipher algorithm. Some interesting and new results are presented with regard to the complexity of the combinations of linear feedback shift registers. These combinations can be described in terms of Boolean function theory using logical operators like sum and product. The notion of splitting (a generalization of the classical term of decimation) and the inverse operator called interleave are also introduced. The proofs of the theorems are also given. This work can be easily generalized to quadratic complexity of high order complexity.