Paper 2026/590 Original: eprint.iacr.org/2026/590 ↗

On the Security of Constraint-Friendly Map-to-Curve Relations

Abstract

Standard hash-to-curve constructions first hash the message to a field element through a cryptographic hash-to-field step, then map this field element to an elliptic-curve point. Inside constraint systems, this inner cryptographic hash is often the dominant cost. Groth, Malvai, Miller and Zhang (Asiacrypt 2025) introduced \emph{constraint-friendly map-to-elliptic-curve-group relations} that bypass this step, achieving substantial reductions in circuit size. Their security proof works in the Elliptic Curve Generic Group Model (EC-GGM). We identify three gaps. First, the security bound in terms of the message space size $M$ and tweak bound $T$ is not explicitly analyzed, and the bounds stated for the concrete instantiations are loose. Second, the EC-GGM does not capture the algebraic structure of most deployed curves; we exhibit a concrete signature forgery using the parameters claimed secure. Third, the construction requires a congruence condition on the field that is not satisfied by all deployed curves; we extend it to any field, although witness generation remains practical only for fields of low 2-adicity. As a countermeasure we propose a $y$-increment variant that, on the $j=0$ curves used in practice, neutralizes the algebraic attack, removes the field restriction in practice, and preserves a comparable constraint count. We implement and benchmark both constructions in the open-source gnark (Go) library; the attack is additionally demonstrated via a self-contained SageMath simulation and confirmed at the circuit level against the authors' own Noir (Rust) implementation.