How to compute a 256-bit elliptic curve private key with only 50 million Toffoli gates

TL;DR

This paper estimates the quantum resources needed to compute a 256-bit elliptic curve private key using Shor's algorithm in a silicon-photonics-inspired architecture, showing significant reductions with non-local connections.

Contribution

It provides resource estimates for quantum elliptic curve key computation in a novel architecture, highlighting the impact of non-local connections and algorithmic modifications.

Findings

Non-local connections reduce key computation cost by 300-700 times.

A quantum device with non-local connections can generate a key every 10 minutes.

Algorithmic modifications can cut the Toffoli count by up to a factor of 5.

Abstract

We use Shor's algorithm for the computation of elliptic curve private keys as a case study for resource estimates in the silicon-photonics-inspired active-volume architecture. Here, a fault-tolerant surface-code quantum computer consists of modules with a logarithmic number of non-local inter-module connections, modifying the algorithmic cost function compared to 2D-local architectures. We find that the non-local connections reduce the cost per key by a factor of 300-700 depending on the operating regime. At 10% threshold, assuming a 10-$\mu$s code cycle and non-local connections, one key can be generated every 10 minutes using 6000 modules with 1152 physical qubits each. By contrast, a device with strict 2D-local connectivity requires more qubits and produces one key every 38 hours. We also find simple architecture-independent algorithmic modifications that reduce the Toffoli count per key by up to a factor of 5. These modifications involve reusing the stored state for multiple keys and spreading the cost of the modular division operation over multiple parallel instances of the algorithm.