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Discrete Logarithm Cryptanalyses : Number Field Sieve and Lattice Tools for Side-Channel Attacks

Gabrielle de Micheli 1 
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Public-key cryptosystems are constructed using one-way functions which ensure both the security and the efficiency of the schemes. One of the two main candidates originally considered to construct public-key cryptosystems is modular exponentiation with its hard inverse operation, computing discrete logarithms. In this thesis, we study the security of protocols that make use of modular exponentiation where the exponent is a secret of the protocol. To assess the security of such protocols, one can either estimate the hardness of directly solving the discrete logarithm problem (DLP) in the groups considered by the protocols or look at implementation vulnerabilities from fast exponentiation algorithms. One way of estimating the security of protocols based on the hardness of the discrete logarithm problem is to directly study the complexity of the algorithms that solve the latter. In this thesis, we first study the asymptotic complexity of algorithms that solve DLP over finite fields “\F_{p^n}” precisely of the form where pairings take their values. These algorithms come from the index-calculus family from which the Number Field Sieve (NFS) is an example. This study allows us to draw conclusions on the security of pairing-based protocols. We also propose a first implementation of the variant Tower Number Field Sieve (TNFS) of NFS, which has better asymptotic complexity, along with a record computation of a discrete logarithm in a 521-bit finite field with TNFS. This variant had never been implemented before due to the difficulty of sieving in higher dimensions, i.e., dimensions greater than two. Finally, the security of deployed protocols not only relies on the hardness of the underlying mathematical problem but also on the implementation of the algorithms involved. Many fast modular exponentiation algorithms have piled up over the years and some implementations have brought vulnerabilities that are exploitable by side-channel attacks, in particular cache attacks. The second aspect of this thesis thus considers key recover methods when partial information is recovered from a side channel.
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Submitted on : Monday, September 6, 2021 - 10:48:26 AM
Last modification on : Friday, February 4, 2022 - 3:29:52 AM
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Gabrielle de Micheli. Discrete Logarithm Cryptanalyses : Number Field Sieve and Lattice Tools for Side-Channel Attacks. Computer Science [cs]. Université de Lorraine, 2021. English. ⟨NNT : 2021LORR0104⟩. ⟨tel-03335360⟩



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