[Paper Review] The 4-Adic Complexity of A Class of Quaternary Cyclotomic Sequences with Period 2p

Shiyuan Qiang, Yan Li+2 more|arXiv (Cornell University)|Nov 24, 2020

Coding theory and cryptography6 references5 citations

TL;DR

This paper determines the 4-adic complexity of a class of quaternary cyclotomic sequences with period $2p$, using a novel quadratic Gauss sum $G_p$ in $\mathbb{Z}_{4^N - 1}$. It shows that the 4-adic complexity reaches the maximum $\log_4(4^N - 1)$ when $5 \nmid p-2$, and $\log_4\left(\frac{4^N - 1}{5}\right)$ otherwise, demonstrating near-optimal complexity for cryptographic applications.

ABSTRACT

In cryptography, we hope a sequence over $\mathbb{Z}_m$ with period $N$ having larger $m$-adic complexity. Compared with the binary case, the computation of 4-adic complexity of knowing quaternary sequences has not been well developed. In this paper, we determine the 4-adic complexity of the quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main method we utilized is a quadratic Gauss sum $G_{p}$ valued in $\mathbb{Z}_{4^N-1}$ which can be seen as a version of classical quadratic Gauss sum. Our results show that the 4-adic complexity of this class of quaternary cyclotomic sequences reaches the maximum if $5 mid p-2$ and close to the maximum otherwise.

Motivation & Objective

• To determine the 4-adic complexity of quaternary cyclotomic sequences with period $2p$.
• To address the lack of developed methods for computing 4-adic complexity in quaternary sequences compared to binary sequences.
• To establish that the proposed sequences achieve near-maximal 4-adic complexity, crucial for cryptographic security.
• To introduce and utilize a modified quadratic Gauss sum $G_p$ in $\mathbb{Z}_{4^N - 1}$ as a key analytical tool.

Proposed method

• Define a quaternary sequence $A$ over $\mathbb{Z}_4$ with period $2p$ based on cyclotomic classes modulo $2p$ and Legendre symbol partitions.
• Introduce a 4-adic Gauss sum $G_p = \sum_{a=1}^{p-1} \left(\frac{a}{p}\right) 4^{2a} \mod (4^N - 1)$ as a version of the classical quadratic Gauss sum.
• Express the 4-adic complexity generator $S_A(4) = \sum_{i=0}^{N-1} a(i) 4^i$ in terms of $G_p$ modulo $4^N - 1$.
• Analyze $\gcd(S_A(4), 4^N - 1)$ by splitting it into components $d_+ = \gcd(S_A(4), 4^p + 1)$ and $d_- = \gcd(S_A(4), 4^p - 1)$.
• Use properties of Legendre symbols and modular arithmetic to evaluate $d_+$ and $d_-$, particularly focusing on divisibility by 5 and prime divisors $\ell \geq 5$.
• Establish that $d_+ = 5$ if $5 \mid p-2$, and $d_+ = 1$ otherwise, while $d_- = 1$ always, leading to the final complexity expression.

Experimental results

Research questions

• RQ1 What is the 4-adic complexity of quaternary cyclotomic sequences with period $2p$?
• RQ2 How does the 4-adic complexity behave when $p \equiv 2 \pmod{5}$ compared to other primes?
• RQ3 Can a modified quadratic Gauss sum in $\mathbb{Z}_{4^N - 1}$ be used to compute the 4-adic complexity of such sequences?
• RQ4 Is the 4-adic complexity of this sequence close to the theoretical maximum?
• RQ5 What role does the prime 5 play in determining the complexity value?

Key findings

• The 4-adic complexity of the quaternary cyclotomic sequence reaches the maximum possible value $\log_4(4^N - 1)$ when $5 \nmid p - 2$.
• When $5 \mid p - 2$, the 4-adic complexity is $\log_4\left(\frac{4^N - 1}{5}\right)$, which is close to the maximum.
• The value $d = \gcd(S_A(4), 4^N - 1)$ is 1 if $5 \nmid p - 2$, and 5 if $5 \mid p - 2$, determining the complexity level.
• The analysis confirms that $d_- = \gcd(S_A(4), 4^p - 1) = 1$ for all odd primes $p$, ensuring no additional reduction in complexity from this component.
• The Gauss sum $G_p$ in $\mathbb{Z}_{4^N - 1}$ satisfies a property analogous to the classical quadratic Gauss sum, enabling the derivation of $S_A(4)$.
• The result shows that the sequence has strong cryptographic properties due to its high 4-adic complexity, especially when $p \not\equiv 2 \pmod{5}$.