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

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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 El