Giacomo Micheli

Giacomo Micheli, Mihran Papikian

We establish a connection between Drinfeld modules and rank-metric codes, focusing on the case of semifield codes. Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. We show that Sheekey's construction [She20] fits naturally into this framework, yielding a short conceptual proof of one of his main results. We then give a new construction of infinite families of semifield codes arising from Drinfeld modules defined over finite fields.

Karan Khathuria, Giacomo Micheli, Violetta Weger

In this paper we show that the $\mathbb Z/p^{m}\mathbb Z$-module structure of the ring $E_p^{(m)}$ is isomorphic to a $\mathbb Z/p^{m}\mathbb Z$-submodule of the matrix ring over $\mathbb Z/p^{m}\mathbb Z$. Using this intrinsic structure of $E_p^{(m)}$, solving a linear system over $E_p^{(m)}$ becomes computationally equivalent to solving a linear system over $\mathbb Z/p^{m}\mathbb Z$. As an application we break the protocol based on the Diffie-Hellman Decomposition problem and ElGamal Decomposition problem over $E_p^{(m)}$. Our algorithm terminates in a provable running time of $O(m^{6})$ $\mathbb Z/p^{m}\mathbb Z$-operations.

Andrea Ferraguti, Giacomo Micheli

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and $\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree 3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$ if and only if it permutes infinitely many of its extension fields. As another corollary, we derive the well-known classification of permutation polynomials of degree 3. As a consequence of our classification, we can also show that there is no complete permutation rational function of degree $3$ unless $3\mid q$ and $\varphi$ is a polynomial.

Giacomo Micheli

In this paper we characterize the set of polynomials $f\in\mathbb F_q[X]$ satisfying the following property: there exists a positive integer $d$ such that for any positive integer $\ell$ less or equal than the degree of $f$, there exists $t_0$ in $\mathbb F_{q^d}$ such that the polynomial $f-t_0$ has an irreducible factor of degree $\ell$ over $\mathbb F_{q^d}[X]$. This result is then used to progress in the last step which is needed to remove the heuristic from one of the quasi-polynomial time algorithms for discrete logarithm problems (DLP) in small characteristic. Our characterization allows a construction of polynomials satisfying the wanted property. The method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.

Giacomo Micheli, Michele Schiavina

We present a generalized version of the knapsack protocol proposed by D. Naccache and J. Stern at the Proceedings of Eurocrypt (1997). Our new framework will allow the construction of other knapsack protocols having similar security features. We will outline a very concrete example of a new protocol using extension fields of a finite field of small characteristic instead of the prime field Z/pZ, but more efficient in terms of computational costs for asymptotically equal information rate and similar key size.

Giacomo Micheli

In the papers by Alvarez et al. and Pathak and Sanghi a non-commutative based public key exchange is described. A similiar version of it has also been patented (US7184551). In this paper we present a polynomial time attack that breaks the variants of the protocol presented in the two papers. Moreover we show that breaking the patented cryptosystem US7184551 can be easily reduced to factoring. We also give some examples to show how efficiently the attack works.