What does this research mean for the field?

The discrete logarithm problem can be solved in polynomial time using ternary manifold embeddings, mathematically proving that Elliptic Curve Cryptography is structurally unstable and its private keys are deterministically extractable. Novelty: ClaimNovelty.CONTRADICTORY. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

What question did this study set out to answer?

The study aims to explore structural properties of the discrete logarithm problem under specific constraints and demonstrate vulnerabilities in elliptic curve cryptography.

April 21, 2026Open Access

On the Structural Reduction of Discrete Logarithms and the Polynomial-Time Collapse of Elliptic Curve Cryptography via Ternary Manifold Embeddings

Key Points

The study aims to explore structural properties of the discrete logarithm problem under specific constraints and demonstrate vulnerabilities in elliptic curve cryptography.
Introduced the P1→3 mapping to project cryptographic instances into a ternary manifold.
Applied admissibility conditions based on the ABC conjecture to analyze the parameter space.
Mathematically analyzed the search complexity associated with the structural framework.
Demonstrated a dimension-folding effect that leads to polynomial compression of the search domain.
Proved that elliptic curve cryptography based on classical assumptions is structurally unstable.
Showed that the transition from exponential to polynomial search complexity allows for deterministic extraction of private keys.

Abstract

This paper investigates the structural properties of the discrete logarithm problem (DLP)over finite fields when subjected to strict arithmetic and topological constraints. By intro-ducing the P1→3 mapping, cryptographic instances are projected into a metrized arithmeticternary manifold equipped with an ∅-state buffer. Imposing admissibility conditions analo-gous to the strong form of the ABC conjecture demonstrates a dimension-folding effect onthe parameter space. The effective search domain undergoes a polynomial compression rela-tive to its radical weight. Consequently, this framework mathematically proves that EllipticCurve Cryptography (ECC) relying on classical binary O(√p) assumptions is structurallyunstable. The theoretical transition from exponential to polynomial search complexity isformalized, rendering the private key deterministically extractable.

On the Structural Reduction of Discrete Logarithms and the Polynomial-Time Collapse of Elliptic Curve Cryptography via Ternary Manifold Embeddings

Key Points

Abstract

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