Symmetric measures of pseudorandomness for binary sequences

TL;DR

This paper compares symmetric and ordinary measures of pseudorandomness for binary sequences, revealing that symmetrization can significantly reduce complexity measures and affect their expected values, with implications for sequence analysis.

Contribution

It introduces and analyzes symmetric variants of classical pseudorandomness measures, demonstrating their differences and effects on binary sequences in both periodic and aperiodic cases.

Findings

Symmetric 2-adic complexity can be smaller than the ordinary version for certain sequences.

Linear complexity remains invariant under reversal, matching its symmetric version.

Symmetric complexities generally have lower expected values, especially on an exponential scale.

Abstract

We compare ordinary and symmetric variants of two classical measures of pseudorandomness for binary sequences, the $2$-adic complexity and the linear complexity. In the periodic setting, we show that for binary periodic sequences constructed from the binary expansions of non-palindromic primes, the symmetric $2$-adic complexity can be strictly smaller than the ordinary $2$-adic complexity. We also give a direct proof (of the known result) that the linear complexity of a periodic binary sequence is invariant under reversal, and hence coincides with its symmetric version. In the aperiodic setting, we provide explicit families of finite binary sequences for which both the $N$th symmetric 2-adic complexity and the $N$th symmetric linear complexity are substantially smaller than their ordinary counterparts. Furthermore, we show that the expected values of the $N$th rational complexity and of the $N$th exponential linear complexity exceed those of their symmetric analogues by at least a term of order of magnitude $N$. Thus, the effect of symmetrization is clearly visible on an exponential scale. We also establish lower bounds for the expected values of the symmetric rational complexity, symmetric $2$-adic complexity, symmetric linear complexity, and symmetric exponential linear complexity.