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## Section: New Results

### Solving Systems in Finite Fields, Applications in Cryptology and Algebraic Number Theory.

#### Algebraic Cryptanalysis of a Quantum Money Scheme – The Noisy Case.

At STOC 2012, Aaronson and Christiano proposed a noisy and a noiseless version of the first public-key quantum money scheme endowed with a security proof. [5] addresses the so-called noisy hidden subspaces problem, on which the noisy version of their scheme is based. The first contribution of this work is a non-quantum cryptanalysis of the above-mentioned noisy quantum money scheme extended to prime fields $𝔽$, with $|𝔽|\ne 2$, that runs in randomised polynomial time. This finding is supported with experimental results showing that, in practice, the algorithm presented is efficient and succeeds with overwhelming probability. The second contribution is a non-quantum randomised polynomial-time cryptanalysis of the noisy quantum money scheme over ${𝔽}_{2}$ succeeding with a certain probability for values of the noise lying within a certain range. This result disproves a conjecture made by Aaronson and Christiano about the non-existence of an algorithm that solves the noisy hidden subspaces problem over ${𝔽}_{2}$ and succeeds with such probability.

#### On the Complexity of MQ in the Quantum Setting.

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 [21] we consider the quantum security of the problem of solving a system of $m$ 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, $O\left({2}^{0.462n}\right)$ quantum gates. To our knowledge this is the fastest algorithm for solving MQb.

#### MQsoft .

In 2017, NIST shook the cryptographic world by starting a process for standardizing post-quantum cryptography. Sixty-four submissions have been considered for the first round of the on-going NIST Post-Quantum Cryptography (PQC) process. Multivariate cryptography is a classical post-quantum candidate that turns to be the most represented in the signature category. At this stage of the process, it is of primary importance to investigate efficient implementations of the candidates. [17] presents MQsoft , an efficient library which permits to implement HFE -based multivariate schemes submitted to the NIST PQC process such as GeMSS, Gui and DualModeMS. The library is implemented in C targeting Intel 64-bit processors and using avx2 set instructions. We present performance results for our library and its application to GeMSS, Gui and DualModeMS. In particular, we optimize several crucial parts for these schemes. These include root finding for HFE polynomials and evaluation of multivariate quadratic systems in ${𝔽}_{2}$. We propose a new method which accelerates root finding for specific HFE polynomials by a factor of two. For GeMSS and Gui , we obtain a speed-up of a factor between 2 and 19 for the keypair generation, between $1.2$ and $2.5$ for the signature generation, and between $1.6$ and 2 for the verifying process. We have also improved the arithmetic in ${F}_{{2}^{n}}$ by a factor of 4 compared to the NTL library. Moreover, a large part of our implementation is protected against timing attacks.