A Probabilistic Analysis on a Lattice Attack against DSA

Analyzing the security of cryptosystems under attacks based on the malicious modification of memory registers is a research topic of high importance. This type of attacks may affect the randomness of the secret parameters by forcing a limited number of bits to a certain value which can be unknown to the attacker. In this context, we revisit the attack on DSA presented by Faug\`ere, Goyet and Renault during the conference SAC 2012: we simplify their method and we provide a probabilistic approach in opposition to the heuristic proposed in the former to measure the limits of the attack. More precisely, the main problem is formulated as the search for a closest vector to a lattice, then we study the distribution of the vectors with bounded norms in a this family of lattices and we apply the result to predict the behavior of the attack. We validated this approach by computational experiments.