A Differential Geometry and Algebraic Topology Based Public-Key Cryptographic Algorithm in Presence of Quantum Adversaries

TL;DR

Z-Sigil addresses the threat of quantum adversaries to public-key cryptography by leveraging differential geometry and algebraic topology in a blockwise, serial encryption scheme. The core idea places cryptographic keys in the tangent fibers of a Calabi–Yau manifold and uses a fiberwise, potentially noncommutative operation paired with analytic inputs such as determinants and zeta-regularization to support encryption and decryption. The authors prove invertibility through a fiberwise identity and show that recovering plaintext without the private key reduces to an exponentially large unstructured search, providing quantum resistance under Grover-type speedups. By enforcing blockwise dependencies, the scheme suppresses naive quantum parallelism and differentiates itself from RSA-based approaches through a multilevel geometric–analytic security architecture with no direct discrete-algebraic analogue.

Abstract

In antiquity, the seal embodied trust, secrecy, and integrity in safeguarding the exchange of letters and messages. The purpose of this work is to continue this tradition in the contemporary era, characterized by the presence of quantum computers, classical supercomputers, and increasingly sophisticated artificial intelligence. We introduce Z-Sigil, an asymmetric public-key cryptographic algorithm grounded in functional analysis, differential geometry, and algebraic topology, with the explicit goal of achieving resistance against both classical and quantum attacks. The construction operates over the tangent fiber bundle of a compact Calabi-Yau manifold [13], where cryptographic keys are elements of vector tangent fibers, with a binary operation defined on tangent spaces of the base manifold giving rise to a groupoid structure. Encryption and decryption are performed iteratively on message blocks, enforcing a serial architecture designed to limit quantum parallelism [9,10]. Each block depends on secret geometric and analytic data, including a randomly chosen base point on the manifold, a selected section of the tangent fiber bundle, and auxiliary analytic data derived from operator determinants and Zeta function regularization [11]. The correctness and invertibility of the proposed algorithm are proven analytically. Furthermore, any adversarial attempt to recover the plaintext without the private key leads to an exponential growth of the adversarial search space,even under quantum speedups. The use of continuous geometric structures,non-linear operator compositions,and enforced blockwise serialization distinguishes this approach from existing quantum-safe cryptographic proposals based on primary discrete algebraic assumptions.