Quantum Money from Class Group Actions on Elliptic Curves

Hart Montgomery; Shahed Sharif

Paper 2024/1585

Quantum Money from Class Group Actions on Elliptic Curves

Hart Montgomery

, Linux Foundation

Shahed Sharif, CSU San Marcos

Abstract

We construct a quantum money/quantum lightning scheme from class group actions on elliptic curves over $F_{p}$ . Our scheme, which is based on the invariant money construction of Liu-Montgomery-Zhandry (Eurocrypt '23), is simple to describe. We believe it to be the most instantiable and well-defined quantum money construction known so far. The security of our quantum lightning construction is exactly equivalent to the (conjectured) hardness of constructing two uniform superpositions over elliptic curves in an isogeny class which is acted on simply transitively by an exponentially large ideal class group. However, we needed to advance the state of the art of isogenies in order to achieve our scheme. In partcular, we show: 1. An efficient (quantum) algorithm for sampling a uniform superposition over a cryptographically large isogeny class. 2. A method for specifying polynomially many generators for the class group so that polynomial-sized products yield an exponential-sized subset of class group, modulo a seemingly very modest assumption. Achieving these results also requires us to advance the state of the art of the (pure) mathematics of elliptic curves, and we are optimistic that the mathematical tools we developed in this paper can be used to advance isogeny-based cryptography in other ways.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: A major revision of an IACR publication in ASIACRYPT 2024
Keywords: Quantum Money Elliptic Curve Isogenies
Contact author(s): hart montgomery @ gmail com
ssharif @ csusm edu
History: 2024-10-08: approved; 2024-10-07: received; See all versions
Short URL: https://ia.cr/2024/1585
License: CC BY

BibTeX

@misc{cryptoeprint:2024/1585,
      author = {Hart Montgomery and Shahed Sharif},
      title = {Quantum Money from Class Group Actions on Elliptic Curves},
      howpublished = {Cryptology {ePrint} Archive, Paper 2024/1585},
      year = {2024},
      url = {https://eprint.iacr.org/2024/1585}
}