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We use Shor's algorithm for the computation of elliptic curve private keys asa case study for resource estimates in the silicon-photonics-inspiredactive-volume architecture. Here, a fault-tolerant surface-code quantumcomputer consists of modules with a logarithmic number of non-localinter-module connections, modifying the algorithmic cost function compared to2D-local architectures. We find that the non-local connections reduce the costper 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 keycan be generated every 10 minutes using 6000 modules with 1152 physical qubitseach. By contrast, a device with strict 2D-local connectivity requires morequbits and produces one key every 38 hours. We also find simplearchitecture-independent algorithmic modifications that reduce the Toffolicount per key by up to a factor of 5. These modifications involve reusing thestored state for multiple keys and spreading the cost of the modular divisionoperation over multiple parallel instances of the algorithm.

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