Secure Multi-party Quantum Computation with a Dishonest Majority Yfke Dulek, Alex Grilo, Stacey Je ff ery, Christian Majenz, Christian Scha ff ner
Introduction
Multi-party computation (MPC) Input (player i): x i 83 false Output: f(x 1 , …, x k ) false 102 e.g., what is the maximum input? 102 100 true Output (player i): f i (x 1 , …, x k ) false e.g., was my input the highest? false false This is the ideal situation. 11 10 What if there is no ? 55 Icons by Pixel perfect.
Multi-party computation (MPC) We want: 83 • privacy of inputs • correctness of outputs 102 100 x i ↦ f i (x 1 , …, x k ) We cannot prevent: • lying about inputs • unfairness 11 10 9001 55
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MPQC: two approaches 1. Secret sharing [CGS02] R 1 • distribute inputs R 6 R 2 • up to <k/2 dishonest 2. Authentication [DNS12] • protect inputs • hope: up to k-1 dishonest R 5 R 3 R 4 [CGS02] Crépeau, Gottesman, and Smith. Secure multi-party quantum computation. (STOC 2002) [DNS12] Dupuis, Nielsen, and Salvail. Actively secure two-party evaluation of any quantum operation. (CRYPTO 2012)
Introduction Authentication Computation Magic-state generation Summary
Clifford code Subgroup of unitaries Key: C ∈ R Cli ff ord n +1 Generated by H, √ Z, CNOT Looks “ random ” � | ψ i ⌦ | 0 i ⊗ n � Encoding: | ψ i 7! C traps Decoding: apply , measure traps C † Theorem (informal): for any on qubits, the A n + 1 probability that changes , but is not detected at | ψ i A decoding is very small ( ). 2 − n Bonus : the Cli ff ord code also provides privacy.
Clifford code in MPQC R 1 • What if the encoding player is dishonest? R 6 R 2 • How to do computation? Data is unalterable! Answers: use classical multi- party computation! R 5 R 3 R 4
Public authentication test C 1 ( | ψ i ⌦ | 0 2 n i ) C 6 C 5 C 4 C 3 C 2 R 1 R 6 R 2 R 5 R 3 R 4
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