Paper 2024/1585
Quantum Money from Class Group Actions on Elliptic Curves
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)
- Category
- Public-key cryptography
- Publication info
- A major revision of an IACR publication in ASIACRYPT 2024
- Keywords
- Quantum MoneyElliptic 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} }