Probabilistic results on the $2$-adic complexity

TL;DR

This paper addresses the average and asymptotic behavior of the $2$-adic and rational complexities of binary sequences, resolving open questions about their distribution. It develops counting tools and asymptotic bounds for sequences with small or large $2$-adic complexity, and proves sharp upper and lower bounds on the expected values. For random sequences, it establishes almost-sure results showing the $2$-adic complexity profile concentrates near $N/2$ with deviations only of order $\log N$, implying $\Lambda_S(N)=2^{N/2+O(\log N)}$ for all $N$ with probability $1$. Together, these results advance understanding of complexity measures relevant to cryptography and FCSR-based pseudorandom generators, and they settle several conjectures and questions from prior work.

Abstract

This work is devoted to solving some closely related open problems on the average and asymptotic behavior of the $2$-adic complexity of binary sequences. First, for fixed $N$, we prove that the expected value $E^{\mathrm{2-adic}}_N$ of the $2$-adic complexity over all binary sequences of length $N$ is close to $\frac{N}{2}$ and the deviation from $\frac{N}{2}$ is at most of order of magnitude $\log(N)$. More precisely, we show that $$\frac{N}{2}-1 \le E^{\mathrm{2-adic}}_N= \frac{N}{2}+O(\log(N)).$$ We also prove bounds on the expected value of the $N$th rational complexity. Our second contribution is to prove for a random binary sequence $\mathcal{S}$ that the $N$th $2$-adic complexity satisfies with probability $1$ $$ λ_{\mathcal{S}}(N)=\frac{N}{2}+O(\log(N)) \quad \mbox{for all $N$}. $$

Theorems & Definitions (10)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 2
• Theorem 3
• Lemma 6: Borel-Cantelli Lemma
• Theorem 4