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Unit# Laboratoire de communications mobiles

Laboratory

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It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu in STOC'21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.

Post-quantum cryptography is a branch of cryptography which deals with cryptographic algorithms whose hardness assumptions are not based on problems known to be solvable by a quantum computer, such as the RSA problem, factoring or discrete logarithms.This thesis treats two such algorithms and provides theoretical and practical attacks against them.The first protocol is the generalised Legendre pseudorandom function - a random bit generator computed as the Legendre symbol of the evaluation of a secret polynomial at an element of a finite field. We introduce a new point of view on the protocol by analysing the action of the group of Möbius transformations on the set of secret keys (secret polynomials).We provide a key extraction attack by creating a table which is cubic in the number of the function queries, an improvement over the previous algorithms which only provided a quadratic yield. Furthermore we provide an ever stronger attack for a new set of particularly weak keys.The second protocol that we cover is SIKE - supersingular isogeny key encapsulation.In 2017 the American National Institute of Standards and Technology (NIST) opened a call for standardisation of post-quantum cryptographic algorithms. One of the candidates, currently listed as an alternative key encapsulation candidate in the third round of the standardisation process, is SIKE.We provide three practical side-channel attacks on the 32-bit ARM Cortex-M4 implementation of SIKE.The first attack targets the elliptic curve scalar multiplication, implemented as a three-point ladder in SIKE. The lack of coordinate randomisation is observed, and used to attack the ladder by means of a differential power analysis algorithm.This allows us to extract the full secret key of the target party with only one power trace.The second attack assumes coordinate randomisation is implemented and provides a zero-value attack - the target party is forced to compute the field element zero, which cannot be protected by randomisation. In particular we target both the three-point ladder and isogeny computation in two separate attacks by providing maliciously generated public keys made of elliptic curve points of irregular order.We show that an order-checking countermeasure is effective, but comes at a price of 10% computational overhead. Furthermore we show how to modify the implementation so that it can be protected from all zero-value attacks, i.e., a zero-value is never computed during the execution of the algorithm.Finally, the last attack targets a point swapping procedure which is a subroutine of the three-point ladder. The attack successfully extracts the full secret key with only one power trace even if the implementation is protected with coordinate randomisation or order-checking. We provide an effective countermeasure --- an improved point swapping algorithm which protects the implementation from our attack.

Eric Bezzam, Paul Hurley, Sepand Kashani, Matthieu Martin Jean-André Simeoni, Martin Vetterli

Fourier transforms are an often necessary component in many computational tasks, and can be computed efficiently through the fast Fourier transform (FFT) algorithm. However, many applications involve an underlying continuous signal, and a more natural choice would be to work with e.g. the Fourier series (FS) coefficients in order to avoid the additional overhead of translating between the analog and discrete domains. Unfortunately, there exists very little literature and tools for the manipulation of FS coefficients from discrete samples. This paper introduces a Python library called pyFFS for efficient FS coefficient computation, convolution, and interpolation. While the libraries SciPy and NumPy provide efficient functionality for discrete Fourier transform coefficients via the FFT algorithm, pyFFS addresses the computation of FS coefficients through what we call the fast Fourier series (FFS). Moreover, pyFFS includes an FS interpolation method based on the chirp Z-transform that can make it more than an order of magnitude faster than the SciPy equivalent when one wishes to perform interpolation. GPU support through the CuPy library allows for further acceleration, e.g. an order of magnitude faster for computing the 2-D FS coefficients of 1000 x 1000 samples and nearly two orders of magnitude faster for 2-D interpolation. As an application, we discuss the use of pyFFS in Fourier optics. pyFFS is available as an open source package at https://github.com/imagingofthings/pyFFS, with documentation at https://pyffs.readthedocs.io.

2022