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Paper 1

The Universal Composable Security of Quantum Message Authentication with Key Recyling

Patrick Hayden, Debbie W. Leung, Dominic Mayers

Year

2016

Journal

arXiv preprint

DOI

arXiv:1610.09434

arXiv

1610.09434

Barnum, Crepeau, Gottesman, Tapp, and Smith (quant-ph/0205128) proposed methods for authentication of quantum messages. The first method is an interactive protocol (TQA') based on teleportation. The second method is a noninteractive protocol (QA) in which the sender first encrypts the message using a protocol QEnc and then encodes the quantum ciphertext with an error correcting code chosen secretly from a set (a purity test code (PTC)). Encryption was shown to be necessary for authentication. We augment the protocol QA with an extra step which recycles the entire encryption key provided QA accepts the message. We analyze the resulting integrated protocol for quantum authentication and key generation, which we call QA+KG. Our main result is a proof that QA+KG is universal composably (UC) secure in the Ben-Or-Mayers model (quant-ph/0409062). More specifically, this implies the UC-security of (a) QA, (b) recycling of the encryption key in QA, and (c) key-recycling of the encryption scheme QEnc by appending PTC. For an m-qubit message, encryption requires 2m bits of key; but PTC can be performed using only O(log m) + O(log e) bits of key for probability of failure e. Thus, we reduce the key required for both QA and QEnc, from linear to logarithmic net consumption, at the expense of one bit of back communication which can happen any time after the conclusion of QA and before reusing the key. UC-security of QA also extends security to settings not obvious from quant-ph/0205128. Our security proof structure is inspired by and similar to that of quant-ph/0205128, reducing the security of QA to that of TQA'. In the process, we define UC-secure entanglement, and prove the UC-security of the entanglement generating protocol given in quant-ph/0205128, which could be of independent interest.

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Paper 2

Quantum resource estimates for computing elliptic curve discrete logarithms

Martin Roetteler, Michael Naehrig, Krysta M. Svore, Kristin Lauter

Year

2017

Journal

arXiv preprint

DOI

arXiv:1706.06752

arXiv

1706.06752

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.

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