The Continuous Logarithm in the Complex Circle for Post-Quantum Cryptographic Algorithms

TL;DR

The paper addresses the vulnerability of traditional discrete logarithm-based cryptography to quantum attacks and proposes a novel framework based on the continuous logarithm on the complex circle and $n$-th roots of unity to reframe DLP-style protocols. It formalizes the continuous logarithm via $L(r^k) = i\frac{2\pi k}{n}$ and analyzes its multi-valuedness and angular ambiguity to derive quantum-resistant properties, then outlines reformulations of Diffie-Hellman, ECDSA, and ElGamal in this geometric-spectral setting. The work highlights the geometric and spectral advantages of this approach, including potential for homomorphic schemes and unitary operator models in a Hilbert space $\mathcal{H}$. It lays groundwork for practical post-quantum cryptography based on the complex circle, with directions for richer generators and efficient spectral computations.

Abstract

This paper introduces a novel cryptographic approach based on the continuous logarithm in the complex circle, designed to address the challenges posed by quantum computing. By leveraging its multi-valued and spectral properties, this framework enables the reintroduction of classical algorithms (DH, ECDSA, ElGamal, EC) and elliptic curve variants into the post-quantum landscape. Transitioning from classical or elliptic algebraic structures to the geometric and spectral properties of the complex circle, we propose a robust and adaptable foundation for post-quantum cryptography.