We introduce an FSB-inspired hash construction over F_p based on an alphabet encoding that assigns both a position and a nonzero field coefficient to each message symbol. This produces a structured sparse representation that is compressed through a public matrix over F_p. We present the main design principles and the current stage of its analysis.

Primality testing is an especially useful topic for public-key cryptography. Thanks to carefully constructed necessary primality conditions, a primality test algorithm based on Pell's cubic can be defined. This test has complexity O(log(n)) and it is deterministic below 2^36.

Canonical liftings were developed to compress group-action-based sigma protocols by replacing a full witness with a compact representative. Under some hypotheses, they seem to preserve the underlying hardness of group-action problems. This abstraction was used to produce a (linkable) ring signature, CE(Le)ReS. Can it be used to construct other primitives, such as blind signatures (e.g., Tanuki, LEAF)? Can they have other roles, as cryptographic primitives? What happens if we instantiate them on, e.g., the group actions underlying lattices and isogenies?

We study the Learning With Errors problem in the bounded-error setting from an algebraic perspective. We present a new algorithm, that solves the resulting polynomial system arising from bounded-error LWE instances. In contrast to previous approaches, whose cost estimates rely on heuristic assumptions such as the degree of regularity or semi-regularity of the system, our analysis yields a precise and provable complexity bound. This provides a more reliable assessment of the algebraic hardness of bounded-error LWE and of the security margins of schemes based on it.

This work introduces a sponge-based hash function using the Grendel permutation as an alternative to Keccak for blockchain signatures. It extends the Legendre symbol and Euler's criterion from prime fields to invertible elements of Z/pqZ and provides an implementation without predefined hashing libraries, together with a security analysis.

We explore the Learning Homomorphism with Noise problem, a group-theoretic formulation of Learning With Errors, one of the hardness assumptions in post-quantum cryptography. We present a generalization inspired by TFHE, a fully homomorphic encryption scheme over the real torus. Finally, we describe an application to functional encryption for inner product. This is joint work with D. Kahrobaei, A. Tortora and M. Tota.

Privacy-preserving proximity testing allows users to determine whether two parties are sufficiently close, without revealing their exact locations. We introduce Geometric Fuzzy Matching (GFM), which generalizes this idea to arbitrary geometric regions. GFM supports both distance-based proximity and complex regions, while keeping both locations private.

We evaluate Dickson polynomials as replacements for monomial S-boxes in a STARK-friendly hash function, leveraging their dense algebraic structure and lack of the multiplicative homomorphic property. Unexpectedly, experimental results show that the solving degree of the resulting polynomial system remains unchanged, despite an increase in time and memory overhead.

We introduce a new model, based on elementary symmetric polynomials, that can be used to solve the exact variant of the Syndrome Decoding Problem in the binary case. We provide an estimate of its computational complexity by showing bounds on the degree of regularity of the ideal associated with the model. Based on joint work with Elisa Gorla.

Arithmetization-oriented (AO) compression functions are a core component of the Merkle tree commitment schemes used in modern ZK-SNARK frameworks. In this talk, we will look at the recently introduced PAX family of AO compression modes, that allow to instantiate a compression function from an underlying cryptographic permutation. The PAX family unifies and generalizes the description of popular approaches in the literature, namely the single-iteration sponge, and the so-called Jive and Trunc modes. In particular, unlike the single-iteration sponge, PAX allows to achieve optimal collision and preimage resistance, allowing a more efficient instantiation in settings, such as Merkle Trees, where Random-Oracle indifferentiability is typically not necessary.