Paper 2025/1391

Inverse Discrete Logarithm - Post-Quantum take on a classical problem.

Abstract

We introduce the \(Inverse\ Discrete\ Logarithm\ Problem\) (iDLP) framework, which inverts traditional discrete logarithm assumptions by making the exponent public but deliberately non-invertible modulo the group order, while hiding the base. This creates a many-to-one algebraic mapping that is computationally infeasible under both classical and quantum attack models. Within this framework, we define three post-quantum cryptographic primitives: Inverse Discrete Diffie–Hellman (IDDH), Inverse Discrete Key Encapsulation (IDKE), and Inverse Discrete Data Encapsulation (IDDE). Using a 512-bit modulus (prime or semiprime), a random generator \( g \), and a public exponent \( y \) with \(\gcd(y, \varphi(m)) = d\), the masking function \[ \mathsf{Mask}_{g,y}(x) := g^{x y} \bmod m \] induces a many-to-one mapping that renders discrete logarithm inversion infeasible. Our security analysis shows that known quantum algorithms yield only multiple candidates, requiring exhaustive search among equivalence classes, which remains intractable at 512-bit parameters. We demonstrate efficient prototype implementations with sub-millisecond key operations and AES-GCM-level data throughput. Full source code and parameters are publicly available at \url{https://github.com/AdamaSoftware/InverseDiscrete/}.