Performance Improvement of the Chebyshev Polynomial Method through Matrix-Based Computation in Key Exchange: Results and Analysis

Resumen

Cryptographic key-exchange protocols are fundamental tools for securing modern communication systems. The classical Diffie–Hellman protocol constitutes the reference model, relying on the hardness of the discrete logarithm problem in finite groups. Parallel to this line, Chebyshev polynomials have attracted attention due to their remarkable commutative property,

$T_m(T_n(x)) = T_n(T_m(x)) = T_{mn}(x)$,

suggesting interesting analogies with modular exponentiation.

However, most existing Chebyshev-based schemes depend on recursive evaluation, which is computationally expensive and difficult to adapt reliably to finite fields.

In this work, we present a significant development in Chebyshev-based cryptography. We introduce new algebraic formulas together with a matrix-based formulation that allows Chebyshev polynomials to be computed efficiently using fast exponentiation. This framework eliminates recursion, improves stability, and provides exact evaluation over finite fields. We then integrate this formulation into a key-exchange construction and compare it to the classical Diffie–Hellman scheme.

Our results show that the proposed matrix formulation evaluates $T_n(x)$ in $O(log n)$ operations while preserving the structural properties required for key exchange. Although Diffie–Hellman remains more suitable for real-world deployment, our contribution offers a powerful and scalable alternative framework. These findings extend the current state of research on Chebyshev polynomials and highlight their potential for developing diversified cryptographic mechanisms.