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Vicentiu Pasol

Publications and source records attributed to Vicentiu Pasol.

8 recordsLinked to original sources
arXiv2026

Bridges connecting Encryption Schemes

The present work investigates a type of morphisms between encryption schemes, called bridges. By associating an encryption scheme to every such bridge, we define and examine their security. Inspired by the bootstrapping procedure used by Gentry to produce fully homomorphic encryption schemes, we exhibit a general recipe for the construction of bridges. Our main theorem asserts that the security of a bridge reduces to the security of the first encryption scheme together with a technical additional assumption.

cs.CR↗
arXiv2026

Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography

In this work we study quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell--Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. We use geometric techniques to show that asymptotically the Jacobians of almost all integral models with two rational points at infinity have rank $r \geq 1$. Since there are $\asymp X^{\frac{13}{2}}$ such models among the $X^7$ curves $y^2=f(x)$ of height at most $X$, this yields a lower bound of logarithmic density $13/14$ for the subset of such curves whose Jacobians have rank at least $1$. We further present a large explicit subfamily of genus-$2$ curves, ordered by height as above, for which the Jacobians have rank $r \geq 2$, yielding an unconditional logarithmic density of at least $5/7$. Independently, we give a construction of genus-$2$ curves with split Jacobian and rank at least $2$, producing a subfamily of logarithmic density at least $2/21$. Finally, we analyze quadratic and biquadratic twist families in the split-Jacobian setting, obtaining a positive proportion of rank-$2$ twists. These results have implications for Regev's quantum algorithm in hyperelliptic curve cryptography.

math.NT↗
arXiv2013

Pair correlation of angles between reciprocal geodesics on the modular surface

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

math.NT↗
arXiv2013

On the Petersson scalar product of arbitrary modular forms

We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $Γ_1(N)$ and $k>2$. For $k=2$ we give examples when it is degenerate, and when it is nondegenerate.

math.NT↗
arXiv2013

An algebraic property of Hecke operators and two indefinite theta series

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.

math.NT↗
arXiv2013

Modular forms and period polynomials

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $Γ_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

math.NT↗
arXiv2009

Fusion rings arising from normal Hopf subalgebras

For any normal commutative Hopf subalgebra $K=k^G$ of a semisimple Hopf algebra we describe the ring inside $kG$ obtained by the restriction of $H$-modules. If $G=\Z_p$ this ring determines a fusion ring and we give a complete description for it. The case $G=\Z_{p^n}$ and some other applications are presented.

math.RT↗