## Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

Research output: Contribution to journalArticle

## Author(s)

• Jean-Charles Faug`ere
• Kelsey Horan
• Delaram Kahrobaei
• Marc Kaplan
• Elham Kashefi
• Ludovic Perret

## Department/unit(s)

### Publication details

Journal QUANTUM INFORMATION COMPUTATION Published - 19 Dec 2017 English

### Abstract

In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$ variables} (\MQb); a central problem in post-quantum cryptography. When $n=m$, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, $O(2^{0.462n})$ quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}.

### Research areas

• cs.CR, quant-ph

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