NuTMiC 2026: Posters

NuTMiC 2026 is the first conference in the NuTMiC series hosting the poster session. We have two posters accepted for the session:

Author: Dorota Wedmann
Title: Number theory in group-based post-quantum cryptography
Abstract: Group-based post-quantum cryptography is an alternative to lattice-based cryptography, being equally versatile in its applications. By using well-known problems from combinatorial group theory, it is possible to construct a wide spectrum of cryptographic schemes, ranging from basic identification schemes to much more complex fully homomorphic encryption schemes. This presentation focuses on the intersection between combinatorial group theory and number theory in the context of post-quantum cryptography. Particular attention will be given to Diophantine equations over solvable groups and Right-Angled Artin Groups (RAAGs), which are frequently considered as platform groups for cryptographic schemes. Furthermore, I will introduce a novel identification scheme based on the exponential equations problem in groups, alongside its reinterpretation through the lens of solving systems of Diophantine equations.

Author: Grzegorz Bernaś and Sebastian Borowy
Title: Exceptional numbers in factoring with elliptic curve and Jacobi symbol signature methods
Abstract: The integer factorization problem is a difficult computational problem. It lies at the foundation of the most popular cryptosystem, RSA. In our work, we showcase the results that come from studying square-free integer factorization using elliptic curve methods. The three complementary methods that we have chosen are: 2-adic approach, factoring with known difference, and Jacobi symbol signature approach. All three methods use elliptic curves in residue rings and their properties, such as order and Frobenius trace. The first method we use, the 2-adic approach, was described by Dąbrowski, Pomykała and Shparlinski. It uses a double-and-add approach in order to find divisors during elliptic curve addition. The second method, which can be seen in the article by Morain, Renault and Smith implements LLL and Coppersmith methods that can be combined with elliptic curves. Lastly, the third method was formulated by Urroz and Pomykała based on the work of Pomykała and Dryło. It exploits elliptic curve twists, affecting Jacobi symbol signature. In our research, we define and count square-free numbers, with divisors that cannot be found in deterministic polynomial time using methods described above - such integers will be called exceptional. We describe conditions necessary for an integer to be factorized. We show upper estimates for exceptional integers and their precise count up to the chosen numbers; the goal of our research is to show that there is an asymptotically low number of the exceptional integers and to provide their examples.