Faster SCALLOP from Non-prime Conductor Suborders in Medium Sized Quadratic Fields

Allombert, Bill and Biasse, Jean-François and Eriksen, Jonathan Komada and Kutas, Péter and Leonardi, Chris and Page, Aurel and Scheidler, Renate and Tot Bagi, Márton (2025) Faster SCALLOP from Non-prime Conductor Suborders in Medium Sized Quadratic Fields. LECTURE NOTES IN COMPUTER SCIENCE, 15676. pp. 333-363. ISSN 0302-9743

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Abstract

A crucial ingredient for many cryptographic primitives such as key exchange protocols and advanced signature schemes is a commutative group action where the structure of the underlying group can be computed efficiently. SCALLOP provides such a group action, based on oriented supersingular elliptic curves. We present PEARL-SCALLOP, a variant of SCALLOP that changes several parameter and design choices, thereby improving on both efficiency and security and enabling feasible parameter generation for larger security levels. Within the SCALLOP framework, our parameters are essentially optimal; the orientation is provided by a 2e -isogeny, where 2e is roughly equal to the discriminant of the acting class group. As an important subroutine we present a practical algorithm for generating oriented supersingular elliptic curves. To demonstrate our improvements, we provide a proof-of-concept implementation which instantiates PEARL-SCALLOP at record-sized security levels. For the previous largest parameter set, equivalent to CSIDH-1024, our timings are more than an order of magnitude faster than any other SCALLOP version.

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