Transform Domain Analysis of Sequences

Abstract: In cryptanalysis, security of ciphers vis-a-vis attacks is gauged against three criteria of complexities, i.e., computations, memory and time. Some features may not be so apparent in a particular domain, and their analysis in a transformed domain often reveals interesting patterns. Moreover, the complexity criteria in different domains are different and performance improvements are often achieved by transforming the problem in an alternate domain. Owing to the results of coding theory and signal processing, Discrete Fourier Transform (DFT) based attacks have proven to be efficient than algebraic attacks in terms of their computational complexity. Motivated by DFT based attacks, we present a transform domain analysis of Linear Feedback Shift Register(LFSR) based sequence generators. The time and frequency domain behavior of non-linear filter and combiner generators is discussed along with some novel observations based on the Chinese Remainder Theorem (CRT). CRT is exploited to establish patterns in LFSR sequences and underlying cyclic structures of finite fields. Application of DFT spectra attacks on combiner generators is also demonstrated. Our proposed method saves on the last stage computations of selective DFT attacks for combiner generators. The proposed approach is demonstrated on some examples of combiner generators and is scalable to general configuration of combiner generators.