Research Activity

References in the form [year/number] refer to the numbering used on the Publications page.

My research activity lies mainly within the framework of Galois geometries and their close connections with more applied areas of mathematics, including coding theory and cryptography. Galois geometries naturally lead to several interdisciplinary problems, such as the study of rational points on algebraic curves defined over finite fields. In this setting, methods arising from finite geometry interact effectively with combinatorics, group theory, number theory, and algebraic geometry over finite fields.

More specifically, questions concerning remarkable configurations in projective or affine spaces over finite fields—including existence, non-existence, classification, and the construction of infinite families—can often be translated into the study of suitable algebraic varieties over finite fields, or more generally of polynomial functions over finite fields. For these varieties, determining rational points, and especially rational absolutely irreducible components, is often a fundamental step. Hasse–Weil type estimates and their higher-dimensional generalizations, together with the local analysis of singularities and branches, play an important role in this approach.

Other methods based on automorphism groups, Galois groups, and monodromy groups of specific polynomials and function fields have also been applied systematically. I have used these techniques mainly in the study of curves and varieties over finite fields, functions and polynomials with combinatorial or cryptographic properties, finite-geometric configurations, and the codes associated with them.

In recent years, this point of view has been extended systematically to the classification of exceptional phenomena, scattered linear sets and their higher-dimensional generalizations, rank-metric codes, and the interactions among finite geometry, cryptography, and coding theory.

Main research topics

• Algebraic curves, function fields, and varieties over finite fields, with particular emphasis on rational points, automorphism groups, Weierstrass semigroups, and irreducibility problems.

• Permutation polynomials and functions, PN/APN functions, (c)-differential functions, bent functions, and related generalizations.

• Construction and classification of relevant configurations in affine and projective spaces over finite fields, including arcs, caps, saturating sets, blocking sets, semiovals, and ovoids.

• Scattered linear sets, (h)-scattered structures, evasive subspaces, linearized polynomials, and exceptional phenomena.

• Linear codes and their geometric connections, including MDS and covering codes, algebraic-geometric codes, quantum codes, minimal codes, subspace codes, and rank-metric/MRD codes.

• Algebraic-geometric, group-theoretic, and monodromy methods for existence, non-existence, and asymptotic classification problems.

• Cryptographic applications, ranging from resistance to differential attacks to more recent questions concerning privacy in blockchain systems.

Algebraic curves, function fields, and varieties over finite fields

A substantial part of my research is based on the analysis of algebraic varieties over finite fields associated with the objects under investigation. The underlying idea is to translate a combinatorial or algebraic property into a condition on the rational points of a suitable curve or variety.

A central issue is the existence of absolutely irreducible components defined over the base field. To establish this, I use the analysis of singular points and tangent cones, the study of branches through quadratic transformations, hyperplane sections, function-field techniques, birational transformations, and estimates for the number of rational points.

Alongside the local analysis of singularities and branches, an increasing part of my work uses function fields and coverings, automorphism and Galois groups, and monodromy groups. These tools make it possible both to control the number of rational points and to address classification problems in which the property under consideration is required to persist over infinitely many extensions of the base field.

This perspective includes work on automorphism groups of plane curves in positive characteristic [2013/1], maximal and minimal curves and their coverings [2020/7], [2021/12], [2021/14], Weierstrass semigroups of the Suzuki curve [2021/2], and the survey [2021/9], where the role of Hasse–Weil type theorems in the study of relevant classes of polynomial functions is developed systematically.

The same ideas have been used for algebraic-geometric and quantum codes constructed from curves with many rational points [2018/3], [2018/9], [2018/11], [2019/2], [2021/11], for locally recoverable codes obtained from automorphism groups of function fields [2020/11], and for asymptotic classification problems in which the geometry of an associated variety is combined with information on its Galois or monodromy group.

The latter has become one of the main methodological themes of my more recent research.

Permutation polynomials, PN/APN functions, and their generalizations

A broad part of my research concerns functions over finite fields with special algebraic, combinatorial, or cryptographic properties.

The connection with algebraic geometry comes from the fact that one can often associate with a function (f) a curve or variety whose rational points describe collisions of the function. The existence of absolutely irreducible components defined over the base field then allows Hasse–Weil type results to be used to obtain non-existence or classification results.

My work on permutation polynomials, initially developed through complete permutation polynomials and exceptional polynomials [2016/7], [2017/2], has expanded to the systematic study of trinomials, rational functions, and functions with prescribed fibre multiplicities. The construction of two-to-one functions from Galois extensions in [2022/2] also belongs to this direction.

Constructions, classifications, and non-existence results for sparse or structured families of permutation polynomials and rational functions have been obtained in [2018/1], [2018/8], [2018/10], [2020/1], [2021/3], [2021/6], [2021/10], [2021/17], [2025/2], and [2026/2].

In [2018/2], the connection with algebraic curves is formulated explicitly and used as a classification method; related questions for rational functions with small value sets are considered in [2021/1]. The work [2025/6] concerns the asymptotic classification of Kloosterman polynomials, again combining arithmetic over finite fields with structural properties of polynomial functions.

Further conjectural problems concerning special classes of polynomial functions over finite fields are considered in [2024/1].

PN, APN, and (c)-differential functions

For PN/APN functions and their generalizations, early results on planar functions in even characteristic [2019/4], [2020/2], [2020/5] were followed by constructions and non-existence results for (c)-(almost) perfect nonlinear functions [2021/7], infinite families of APN functions [2022/4], generalized APN monomials [2022/6], APN permutations [2022/7], exceptional crooked functions [2022/9], modifications on subfields with low (c)-differential uniformity [2022/12], rational PN/APN functions [2023/5], [2023/6], and, more recently, the non-existence of infinite APN families obtained from patched monomials in odd characteristic [2026/5].

In these works, the term exceptional is used in an asymptotic sense: the goal is to determine which families retain the required property over infinitely many field extensions. The problem can often be reduced to the existence of absolutely irreducible components in an associated curve or variety.

A further development concerns bent functions and differential biases. The work [2026/3] studies a family of bent functions derived from permutations, while other results investigate differential biases and (c)-differential uniformity in relation to resistance against differential attacks.

Results on semifields in characteristic (2) and in odd characteristic [2017/1], [2018/4] provide another manifestation of the same network of connections: planar functions, incidence structures, and finite algebras can be studied through linearized polynomials and geometric properties of the associated varieties.

Relevant configurations in affine and projective spaces

Sets of points satisfying special properties in projective or affine spaces over finite fields form one of the original directions of my research.

Here again, algebraic-geometric techniques often make it possible to translate completeness, covering, or intersection conditions into the study of rational points on suitable curves and varieties.

Arcs

Some of my early work concerns the construction of small complete arcs through computational methods [2012/1], [2013/2], [2013/3], [2013/5], [2013/6], [2014/4], [2015/2], [2016/1].

Lifts of subgroups of rational points of elliptic curves instead provide infinite families of complete arcs in higher-dimensional spaces [2015/1].

For ((n,k))-arcs, families constructed from cubic curves or low-degree rational curves are studied in [2016/6], [2017/4], and [2017/9].

A different method is developed in [2022/11], based on the monodromy groups of polynomials describing the intersections between lines and curves, leading to infinite families of arcs of degree (m).

Saturating sets

The problem of saturating sets is closely connected with covering codes.

Small saturating sets were studied by computational and probabilistic methods in [2013/5], [2013/8], [2014/1], also in connection with NMDS codes.

The extension to multiple coverings, multiple saturating sets, and related structures is considered in [2013/7], [2015/4], and [2016/8].

Caps

Complete caps and the associated quasi-perfect codes form another important theme.

Families and computational constructions are investigated in [2011/1], [2013/4], [2017/5], [2017/7], [2019/6], while probabilistic methods are used in [2017/3], [2017/6].

Bicovering plane arcs and their higher-dimensional generalizations are used to construct complete caps in [2014/2], [2015/3], and [2017/10].

The connection between maximum scattered linear sets and complete caps is studied directly in [2018/5].

Other finite-geometric configurations

Beyond arcs, caps, and saturating sets, my research includes semiovals and (2)-semiarcs [2014/3], [2014/6], [2016/4], blocking semiovals with prescribed automorphism groups [2018/15], Hermitian varieties characterized as codewords [2018/7], and resolving sets in planes, projective spaces, and incidence graphs [2018/13], [2020/4], [2020/9].

More recently, I have studied classification problems for low-degree ovoids in parabolic and hyperbolic quadrics [2022/13], [2024/4], [2024/7], and [2026/8]. These works are based on translating the combinatorial condition into polynomial systems and analysing the varieties parameterizing possible non-classical configurations.

Other recent geometric problems concern minimum-rank saturating linear sets [2024/3], complete ((q+1))-arcs obtained from the Hermitian curve [2025/5], and the non-existence of translation spreads of (H(q^2)) [2025/7].

Scattered linear sets and their generalizations

Scattered linear sets and the associated linearized polynomials form one of the central themes of my recent research.

The connection between scattered linear sets and MRD codes makes it possible to study geometric and coding-theoretic problems simultaneously. Moreover, by associating suitable curves and determinantal varieties with linearized polynomials, classification problems can be approached by algebraic-geometric methods.

Scattered polynomials of indices (0) and (1) are classified through these techniques in [2018/6], and the classification is subsequently extended in [2021/4].

The analysis of varieties associated with minors of Dickson matrices is used in [2020/10] to prove the existence of a new family of maximum scattered linear sets in (PG(1,q^6)), and in [2021/15] to settle a conjecture concerning their classification in (PG(1,q^6)).

The relationship between geometric constructions and scattered polynomials is already explicit in [2018/5]; more recent constructions include scattered trinomials in even characteristic [2024/5].

Exceptional scattered polynomials

Research on scattered objects has progressively moved from individual constructions towards classification and the understanding of asymptotic phenomena.

The asymptotic behaviour of Moore exponent sets is studied in [2020/3]; the classification of exceptional scattered polynomials is developed in [2018/6], [2021/4], [2022/1], and [2026/1].

The general strategy is to associate suitable curves or determinantal varieties with the linearized polynomial and to prove that, except for very rigid forms, these varieties possess absolutely irreducible components defined over the base field. Rational points over sufficiently large extensions then prevent exceptionality.

Higher-scattered structures and evasive subspaces

In parallel, the notion of scatteredness has been extended to more general objects.

Evasive subspaces are introduced and investigated in [2021/16], (r)-fat linearized polynomials in [2022/5], and exceptional scattered sequences in [2024/6].

New higher-dimensional families are constructed in [2025/1], [2025/3], and [2026/4], including (2)-scattered and (h)-scattered subspaces and the corresponding MRD codes.

These results show that the theory of scattered sets can be organized in terms of sequences of linearized polynomials and prescribed intersection conditions with subspaces of fixed dimension, opening classification problems that extend the classical setting of (PG(1,q^n)).

Rank-metric and MRD codes

The connection with rank-metric codes has become particularly important in recent years.

In addition to MRD constructions arising from scattered sets, my work includes Gabidulin codes with non-minimal tensor rank [2022/8], new MRD codes obtained from linear cutting blocking sets [2023/3], exceptional linear MRD codes [2023/4], three-dimensional families of MRD codes [2025/4], and, more recently, linear rank-metric intersecting codes [2026/7].

This direction brings together the classification of linearized polynomials, the geometry of vector spaces over finite extensions, and code-equivalence invariants.

Coding theory and geometric connections

Coding theory appears throughout my research in several metrics and contexts.

Early work concerns quasi-perfect linear codes, NMDS codes, covering codes, and their interpretation through arcs, caps, and saturating sets [2014/1], [2015/1], [2015/4], [2016/8], [2018/12], [2019/5], [2020/6].

The study of functional codes defined by quadrics, Hermitian varieties, and hypersurfaces [2014/5], [2014/7], [2016/5], together with the study of second and third minimum configurations of hyperplanes [2016/3], represents another direction in which the geometry of supports directly determines code weights.

Algebraic-geometric and quantum codes

My work includes multipoint codes on the Giulietti–Korchmáros curve [2018/3], codes from Kummer extensions [2018/9], AG and quantum AG codes from the GGS curve [2018/11], minimum-weight codewords of dual AG codes [2019/2], pure gaps on curves with many rational places [2018/14], locally recoverable codes from automorphism groups [2020/11], and self-orthogonal AG codes with quantum applications [2021/11].

This direction continues earlier work on quaternary quantum caps [2012/2], [2014/8], and [2015/5].

Minimal codes and blocking structures

Starting from minimal linear codes in odd characteristic [2019/3], I have studied weight distributions [2021/8], inductive constructions [2021/13], cutting blocking sets and associated codes [2022/10], strong blocking sets obtained by concatenation [2023/1], minimal codewords arising from point-hyperplane incidence [2023/2], and minimal codewords in Norm–Trace codes [2024/2].

The connection between cutting blocking sets and MRD codes developed in [2023/3] provides a direct bridge between this direction and rank-metric coding theory.

I have also studied equidistant and constant-dimension subspace codes [2016/2], [2017/8], [2021/5], as well as linear codes obtained from maximal arcs [2019/1].

Taken together, these results illustrate how the same geometric language—subspace intersections, incidence varieties, blocking properties, and saturating properties—can be used to address questions concerning distance, covering, minimum weight, and local recovery.

Cross-cutting topics and recent developments

Some works lie outside the main research directions while using similar algebraic tools.

These include the study of a Diophantine equation involving sums of powers [2020/8] and the asymptotic classification of Kloosterman polynomials [2025/6].

More recently, expertise in cryptography and algebra over finite fields has also been applied to privacy problems in distributed systems: [2026/6] introduces ZeroMT+, a scalable privacy-oriented protocol for account-model blockchains.

Although this is a different application area, it continues my interest in cryptographic primitives, finite algebraic structures, and the design of mechanisms with mathematically verifiable guarantees.

A common viewpoint

Overall, the evolution of my research reflects a gradual transition from the study of individual finite configurations to the development of general classification methods.

A central role is played by the possibility of associating a curve, a variety, or a function field with a discrete problem, and then interpreting the relevant combinatorial property through the geometry of its rational points.

Absolute irreducibility, singularities, intersection theory, automorphism groups, Galois theory, monodromy, and computational tools therefore provide a common methodology linking finite geometry, polynomial functions over finite fields, cryptography, and codes in different metrics.