Paper 2026/2108

$\Delta$-KLPT$^+$: Improved Norm Bounds for the Quaternion Isogeny Path Problem and an Application to Ring Signatures

Abstract

Computing smooth-degree isogenies between supersingular elliptic curves with known endomorphism rings is a fundamental task in isogeny-based cryptography. Via the Deuring correspondence, this task is equivalent to finding a connecting ideal of smooth norm between the corresponding quaternion maximal orders. KLPT-type algorithms address this problem by transforming a known connecting ideal into an equivalent ideal of smooth norm. Although connecting ideals of norm approximately $p$ are expected to exist, the best known such algorithms for general input only achieve norm $\widetilde{O}(p^{4.5})$. In this paper, we propose $\Delta$-KLPT$^+$, a new KLPT-type algorithm that lowers this bound to $\widetilde{O}(p^4)$. Our construction builds on $\Delta$-KLPT, which attains a smaller norm bound but requires two input ideals of the same prime norm. To extend $\Delta$-KLPT to arbitrary inputs, we introduce a new subroutine IdealNormReduce, which reduces the common norm of an ideal pair. By iterating IdealNormReduce, we obtain two ideals with a common norm below $p$. Then we apply $\Delta$-KLPT to the resulting ideal pair to obtain a desired connecting ideal of smooth norm. As an application, we construct Delfar, a fully anonymous isogeny-based ring signature scheme. Compared to Erebor-full, Delfar achieves shorter signatures and lower signing and verification costs for large rings.