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

TL;DR

The paper develops a Chebyshev-centric framework for local prime phenomena, recasting classical multiplicative-number-theory ideas in terms of $T_n(x)$ and the unit $rac{x}{+}$. Central to this is the Chebyshev-Euler primality criterion based on two quadratic characters $oldsymbol{ md}(a)$ and $oldsymbol{ me}(a)$ and the four-way residue partition into $A_{oldsymbol{ md}oldsymbol{ me}}$, enabling Chebyshev versions of primality testing, Wieferich-type primes, and primitive-root structure. It extends pseudoprimes, cyclotomic expansions, Diffie-Hellman-type protocols, and AKS-style primality testing to the Chebyshev setting, and demonstrates square-root cancellation in exponential sums over refined residue classes, revealing a richer local residue structure than the classical quadratic framework. The work highlights potential cryptographic analogues and primes of interest within the Chebyshev regime, while noting limitations such as the half-reach of Chebyshev-generated sets in DH and the need for further validation of density heuristics.

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 $ε_p(a)=\left ( \frac{a^2-1}{p} \right)$ and $δ_p(a)=\left( \frac{2(a+1)}{p} \right)$. This gives rise to a local partition of the integers mod an odd prime $p$ in two distinct ways into 4 disjoint sets $A_{εδ}$ of smaller size about $p/4$ over which the exponential sum still have squar root cancellations. 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 $ω_x=x+\sqrt{x^2-1}$, namely $ω_x^n=T_n(x)+U_{n-1}(x)\sqrt{x^2-1}$. There are obvious analogue of Chebyshev version of pseudoprimes, Wieferich primes, Lucas-Lehmer, AKS, Diffie-Hellman, cyclotomic expansions and probably many others which hopefully some may give improved resuts on the original version.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Conjecture 3.1
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5