Research

During Ph.D., my research interests are been focused upon two specific topics: representations of finite-dimensional algebras and quiver representations. In particular, I  studied the behavior of the invariants for the actions of products of classical groups on the variety of symmetric quiver representations. Moreover I went into more depth the theory of Cluster algebras and the theory of Cluster Categories.

Then I worked with Alessandro D'Andrea ("Sapienza" University of Rome) and Francesco Vaccarino (Politecnico di Torino) about the connection of Representation theory with Algebraic Statistics (applied to Phylogenetics), Computer Science and Dynamical Systems (applied to modeling and simulations of natural, biological and social systems). In particular, we studied Sequential Dynamical Systems (SDS), using  techniques regarding Representation Theory and Semigroup Theory. Together with Alessandro D'Andrea we then continued these studies.

Currently I am dealing with some aspects of Algebraic Cryptography, I am mainly focusing on the application of Group Theory to Symmetric Cryptography; in particular, I am interested on the following topics:

Group Theory and Symmetric Cryptography. We study the groups generated by the encryption functions of block ciphers, as AES and PRESENT, and other block ciphers in which the action of Key Space is different to translation, as GOST. Moreover, we describe some properties of the components, both linear and non-linear, acting within block ciphers useful to prevent some their weaknesses. We have also designed a new block cipher structure (Wave-shaped ciphers) within which it is possible to insert a non-linear component  optimal from the point of view of differential cryptanalysis and then we proved to be resistant to some algebraic attacks based on Group Theory. We also showed that the group generated by the round functions of the AES cipher cannot be embedded into a linear group acting on a vector space W, unless the dimension of W is huge, making this embedding useless in practice. Recently, we studied the relationships between the elementary abelian regular subgroups and the Sylow 2-subgroups of their normalisers in the symmetric group Sym(F_n^2), in view of the interest for their applications in symmetric cryptography.

Integer Partitions. Starting from the relationship between a certain normalizer chain in the symmetric group Sym(F_n^2) and unrefinable partitions into distinct parts previously studied, we show complete classification of maximal unrefinable partitions for triangular numbers and we design two algorithm in to test whether a given partition is unre nable or not and to enumerate all such partitions whose sum is a given number.

Fully Homomorphic Encryption (FHE). We computed two bounds on the size of the secret key for the FHE scheme over the integers of van Dijk et al. (DGHV scheme) to decrypt correctly a ciphertext after a fixed number of additions and a fixed number of multiplication. Moreover we improve the original bound on the dimension of the secret key for a general circuit.

Cryptanalysis on block ciphers. We introduce a slight improvement to the Partial Sum Attack (one of the most powerful attacks, independent of the key schedule, against reduced-round versions of AES) which lowers the number of chosen plaintexts needed to successfully mount it and which can be carried out completely in practice.

Asymmetric Cryptography. We provided a rigorous analysis of the RSA cryptographic keys employed in the Certification Authority (CA) to certify the keys exchange during some financial transactions. In particular, we provide an attacker model useful to determine the optimal length and cryptoperiod of RSA moduli used by such CAs. In detail, we base our analysis on the execution times of the attacks known in literature, which depend also on the computational power of the attacker.

Attribute Based Encryption. We proposed a new key-policy revocable-storage attribute- based encryption (RS-ABE) scheme, i.e. a public-key encryption scheme which employs some attributes to manage the access to a certain document using a  keys depending on these attributes and in which we introduce user revocation. Moreover, we prove its security in term of indistinguishability under a chosen-plaintext attack (IND-CPA).

Tokenization. We proposed a tokenization algorithm and we provide some formal proofs of security for it, which imply our algorithm satisfies the most signicant security requirements described in  tokenization guide-lines of Payment Card Industry Security Standard Council (PCI SSC).

Information Theory. We derived from the entropy theorem a simple proof of a pointwise inequality first stated by Ornstein and Shields.