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

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arxiv 2306.08585 v1 pith:WSOA3HXU submitted 2023-06-14 quant-ph

classification quant-ph
keywords connectionscostnon-localalgorithmalgorithmiccurved-localelliptic
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exploring the landscape of compact magic-state distillation factories

    quant-ph 2026-06 unverdicted novelty 8.0 of 10

    Classical repetition-code framing plus SAT search yields no-go theorems (no d>3 T-to-T on <8 qubits) and the smallest known unitary factories for d=4,5 T-states (10–11 qubits) and d=3,4 CCZ-states (9–10 qubits).

  2. A distillation-teleportation protocol for fault-tolerant QRAM

    quant-ph 2025-05 accept novelty 8.0 of 10

    An adaptive distillation-teleportation protocol implements a fault-tolerant QRAM query with poly(n) quantum resources and 1/poly(n) device fidelity, at the cost of an exponential classical dataset update each round.

  3. CaliScalpel: In-Situ and Fine-Grained Qubit Calibration Integrated with Surface Code Quantum Error Correction

    quant-ph 2024-12 conditional novelty 6.0 of 10

    CaliScalpel uses surface-code deformation to isolate drifted qubits for calibration while computation continues, with a new instruction set for heavy-hexagon hardware and drift-aware scheduling.

  4. Strategic Plan for Neutral Atom Quantum Computation

    quant-ph 2026-07 conditional novelty 3.0 of 10

    If qubit-count growth (~1.8x/yr) and gate-error reduction (~0.62x/yr) continue, neutral-atom quantum computers could reach practical quantum advantage within a decade, this roadmap projects.

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