Chebyshev polynomials and a refinement of the local residue/non-residue structure at a prime

TL;DR

This paper explores Chebyshev polynomials' properties and introduces a refined local residue/non-residue structure at a prime, extending classical number theory concepts and primality criteria using Chebyshev functions.

Contribution

It presents a Chebyshev-based refinement of residue classification at primes and develops analogues of primality tests and cryptographic protocols based on these polynomials.

Findings

Refined local partition of residue classes using quadratic characters

Chebyshev version of Euler's primality criterion involving quadratic characters

Potential extensions to pseudoprimes, Wieferich primes, and cryptographic protocols

Abstract

The basic power function $t_n(x)=x^n$ is in some sense a classical limit for large $x$, of the monictised Chebyshev polynomial of the first kind $T_n(x)/2^{n-1}$. A theorem of Ritt says they are the only two families of polynomials $p_n(x)$ over $\mathbb{C}$ which satisfies the commutativity relation $p_n(p_m(x))=p_m(p_n(x))$. The commutativity $t_n(t_m(x))=t_m(t_n(x))$ is the reason why the RSA scheme allow also digital signature but the Diffie-Hellman key exchange protocol depends only on the commutativity. The DH scheme and many results in elementary local (at a fixed prime) multiplicative number theory is about properties of the power function $t_n(x)$ and they have natural analogue extension to $T_n(x)$. Recently we discovered a Chebyshev version of Euler's primality criterion , which however depends on two quadratic characters $\epsilon_p(a)=\left ( \frac{a^2-1}{p} \right)$ and $\delta_p(a)=\left( \frac{2(a+1)}{p} \right)$. This gives rise to a local partition of $(\mathbb{Z}/p\mathbb{Z}) \setminus \{ \pm 1 \}$ into 4 disjoint sets $A_{\epsilon \delta}$. This can be thought of as a real refinement of the residue/non-residue as it arise from viewing $T_n(x)$ is the "real" part of the $n$th power of the unit $\omega_x=x+\sqrt{x^2-1}$, namely $\omega_x^n=T_n(x)+U_{n-1}(x)\sqrt{x^2-1}$. There are obvious analogue of Chebyshev version of pseudoprimes, Wieferich primes, Lucas-Lehmer, AKS, Diffie-Hellman, cyclotomic expansions and probably others.