## Section: New Results

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

#### Linear Repairing Codes and Side-Channel Attacks

To strengthen the resistance of countermeasures based on secret sharing, several works have suggested to use the scheme introduced by Shamir in 1978, which proposes to use the evaluation of a random d-degree polynomial into $n$ $d+1$ public points to share the sensitive data. Applying the same principles used against the classical Boolean sharing, all these works have assumed that the most efficient attack strategy was to exploit the minimum number of shares required to rebuild the sensitive value; which is $d+1$ if the reconstruction is made with Lagrange's interpolation. In [2], we highlight first an important difference between Boolean and Shamir's sharings which implies that, for some signal-to-noise ratio, it is more advantageous for the adversary to observe strictly more than d + 1 shares. We argue that this difference is related to the existence of so-called exact linear repairing codes, which themselves come with reconstruction formulae that need (much) less information (counted in bits) than Lagrange's interpolation. In particular, this result implies that, contrary to what was believed, the choice of the public points in Shamir's sharing has an impact on the countermeasure strength. As another contribution, we exhibit a positive impact of the existence of linear exact repairing schemes; we indeed propose to use them to improve the state-of-the-art multiplication algorithms dedicated to Shamir's sharing. We argue that the improvement can be effective when the multiplication operation in the base field is at least two times smaller than in its sub-fields.

#### On the Use of Independent Component Analysis to Denoise Side-Channel Measurements

Independent Component Analysis (ICA) is a powerful technique for blind source separation. It has been successfully applied to signal processing problems, such as feature extraction and noise reduction , in many different areas including medical signal processing and telecommunication. In [17], we propose a framework to apply ICA to denoise side-channel measurements and hence to reduce the complexity of key recovery attacks. Based on several case studies, we afterwards demonstrate the overwhelming advantages of ICA with respect to the commonly used preprocessing techniques such as the singular spectrum analysis. Mainly, we target a software masked implementation of an AES and a hardware unprotected one. Our results show a significant Signal-to-Noise Ratio (SNR) gain which translates into a gain in the number of traces needed for a successful side-channel attack. This states the ICA as an important new tool for the security assessment of cryptographic implementations.