Paper 2025/1590 Original: eprint.iacr.org/2025/1590 ↗

The AIIP Problem: Toward a Post-Quantum Hardness Assumption from Affine Iterated Inversion over Finite Fields

Abstract

We introduce the Affine Iterated Inversion Problem (AIIP), a new candidate hard problem for post-quantum cryptography, based on inverting iterated polynomial maps over finite fields. Given a polynomial f ∈ Fq[x] of degree d ≥ 2, an iteration parameter n, and a target y ∈ Fq, AIIP requires finding an input x such that f(n)(x) = y, where f(n) denotes the n-fold composi tion of f. We establish the computational hardness of AIIP through two independent analytical frameworks: first, by establishing a formal connection to the Discrete Logarithm Problem in the Jacobian of hyperelliptic curves of exponentially large genus; second, via a polynomial time reduction to solving structured systems of multivariate quadratic (MQ) equations. The f irst construction provides number-theoretic evidence for hardness by embedding an AIIP in stance into the arithmetic of a high-genus curve, while the second reduction proves worst-case hardness relative to the NP-hard MQ problem. For the quadratic case f(x) = x2 + α, we show that the induced MQ system is heuristically indistinguishable from a random system, and we formalize a sufficient condition for its pseudorandomness under a standard cryptographic assumption. We provide a detailed security analysis against classical and quantum attacks, derive concrete parameters for standard security levels, and discuss the potential of AIIP as a foundation for digital signatures and public-key encryption. This dual hardness foundation, rooted in both algebraic geometry and multivariate algebra, positions AIIP as a versatile and promising primitive for post-quantum cryptography.