Efficient Dense Gaussian Elimination over the Finite Field with Two Elements

In this work we describe an efficient implementation of a hierarchy of algorithms for Gaussian elimination upon dense matrices over the field with two elements. We discuss both well-known and new algorithms as well as our implementations in the M4RI library, which has been adopted into Sage. The focus of our discussion is a block iterative algorithm for PLE decomposition which is inspired by the M4RI algorithm. The implementation presented in this work provides considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the alacrity of our approach

Algebraic Cryptanalysis of Deterministic Symmetric Encryption

Deterministic symmetric encryption is widely used in many cryptographic applications. The security of deterministic block and stream ciphers is evaluated using cryptanalysis. Cryptanalysis is divided into two main categories: statistical cryptanalysis and algebraic cryptanalysis. Statistical cryptanalysis is a powerful tool for evaluating the security but it often requires a large number of plaintext/ciphertext pairs which is not always available in real life scenario. Algebraic cryptanalysis requires a smaller number of plaintext/ciphertext pairs but the attacks are often underestimated compared to statistical methods. In algebraic cryptanalysis, we consider a polynomial system representing the cipher and a solution of this system reveals the secret key used in the encryption. The contribution of this thesis is twofold. Firstly, we evaluate the performance of existing algebraic techniques with respect to number of plaintext/ciphertext pairs and their selection. We introduce a new strategy for selection of samples. We build this strategy based on cube attacks, which is a well-known technique in algebraic cryptanalysis. We use cube attacks as a fast heuristic to determine sets of plaintexts for which standard algebraic methods, such as Groebner basis techniques or SAT solvers, are more efficient. Secondly, we develop a~new technique for algebraic cryptanalysis which allows us to speed-up existing Groebner basis techniques. This is achieved by efficient finding special polynomials called mutants. Using these mutants in Groebner basis computations and SAT solvers reduces the computational cost to solve the system. Hence, both our methods are designed as tools for building polynomial system representing a cipher. Both tools can be combined and they lead to a significant speedup, even for very simple algebraic solvers