Quantum Money from Quaternion Algebras

Abstract

We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We perform some basic analysis of this black box system and show that it is resistant to black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. We explain why we believe this instantiation is likely to be secure.

Failing to hash into supersingular isogeny graphs

“…Leaving these concerns aside for now, we present below a novel mathematical approach to producing random supersingular curves. We use the idea of continuous-time quantum walks on isogeny graphs of supersingular elliptic curves in characteristic p. The idea was first proposed by Kane, Sharif and Silverberg [36,37] for constructing public-key quantum money. In their scheme, quantum walks are carried out over the ideal class group of a quaternion algebra; we adapt these walks to isogeny graphs.…”

“…If we repeat this procedure but now with the operator U 2 and the input state |ψ 1 , we get a new state |ψ 2 that is the projection of |ψ 1 onto a smaller subspace X 2 ⊂ X 1 . If r is large enough, repeating this procedure for all the remaining T i we end up with some eigenstate |φ j with probability | E 0 |φ j | 2 ; see [36,37] for a detailed analysis of this claim. Now, if we measure |φ j in the basis {|E } E∈S , we obtain a curve E with probability | E|φ j | 2 .…”

Quantum Versus Classical Noise Cryptography

“…A particularly well-known example is a scheme from [FGH + 12] that relies on knot theory; while the scheme remains unbroken, it is difficult to analyze and lacks a security proof. Also in this category are the schemes presented in [Kan18,KSS21], though some analysis on [KSS21] has been done in [BDG23].…”