arXiv subjects

Ha Thanh Nguyen Tran

Publications and source records attributed to Ha Thanh Nguyen Tran.

10 recordsLinked to original sources

arXiv2023

The size function for imaginary cyclic sextic fields

In this paper, we investigate the size function $h^0$ for number fields. This size function is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors. This conjecture was proved for all number fields with unit group of rank 0 and 1, and also for cyclic cubic fields which have unit group of rank two. We prove the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two.

math.NT↗

arXiv2022

The size function of quadratic extensions of complex quadratic fields

The function $h^0$ for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. In this paper, we prove the conjecture of van der Geer and Schoof about the maximality of $h^0$ at the trivial Arakelov divisor for quadratic extensions of complex quadratic fields.

math.NT↗

arXiv2020

Lattice Blind Signatures with Forward Security

Blind signatures play an important role in both electronic cash and electronic voting systems. Blind signatures should be secure against various attacks (such as signature forgeries). The work puts a special attention to secret key exposure attacks, which totally break digital signatures. Signatures that resist secret key exposure attacks are called forward secure in the sense that disclosure of a current secret key does not compromise past secret keys. This means that forward-secure signatures must include a mechanism for secret-key evolution over time periods. This paper gives a construction of the first blind signature that is forward secure. The construction is based on the SIS assumption in the lattice setting. The core techniques applied are the binary tree data structure for the time periods and the trapdoor delegation for the key-evolution mechanism.

cs.CR↗

arXiv2018

Reduced ideals from the reduction algorithm

The reduction algorithm is used to compute reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among reduced ones. Especially, we represent a sufficient and necessary condition for reduced ideals of real quadratic fields to be obtained from the reduction algorithm.

math.NT↗

arXiv2017

The size function for cyclic cubic fields

The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was proved for many number fields with unit groups of rank one. Our research confirms that the conjecture also holds for cyclic cubic fields, which have unit groups of rank two.

math.NT↗

arXiv2016

Well-Rounded Lattices for Reliability and Security in Rayleigh Fading SISO Channels

For many wiretap channel models asymptotically optimal coding schemes are known, but less effort has been put into actual realizations of wiretap codes for practical parameters. Bounds on the mutual information and error probability when using coset coding on a Rayleigh fading channel were recently established by Oggier and Belfiore, and the results in this paper build on their work. However, instead of using their ultimate inverse norm sum approximation, a more precise expression for the eavesdropper's probability of correct decision is used in order to determine a general class of good coset codes. The code constructions are based on well-rounded lattices arising from simple geometric criteria. In addition to new coset codes and simulation results, novel number-theoretic results on well-rounded ideal lattices are presented.

cs.IT↗

arXiv2016

Well-Rounded Lattices for Coset Coding in MIMO Wiretap Channels

The concept of well-rounded lattices has recently found important applications in the setting of a fading single-input single-output (SISO) wiretap channel. It has been shown that, under this setup, the property of being well-rounded is critical for minimizing the eavesdropper's probability of correct decoding in lower SNR regimes. The superior performance of coset codes constructed from well-rounded lattices has been illustrated in several simulations. In the present article, this work is extended to fading multiple-input multiple-output (MIMO) wiretap channels, and similar design criteria as in the SISO case are derived. Further, explicit coset codes for Rayleigh fading MIMO wiretap channels are designed. In particular, it is shown through extensive simulations that sublattices of the well-known Alamouti code and Golden code which meet our design criteria perform better than scalar multiples of the code lattice for the same parameters.

cs.IT↗

arXiv2016

A generalization of reduced Arakelov divisors of a number field

Let $C \geq 1$. Inspired by the LLL-algorithm, we define strongly $C$-reduced divisors of a number field $F$ which are generalized from the concept of reduced Arakelov divisors. Moreover, we prove that strongly $C$-reduced Arakelov divisors still retain outstanding properties of the reduced ones: they form a finite, regularly distributed set in the Arakelov class group and the oriented Arakelov class group of $F$.

math.NT↗

arXiv2016

On reduced Arakelov divisors of real quadratic fields

We generalize the concept of reduced Arakelov divisors and define $C$-reduced divisors for a given number $C \geq 1$. These $C$-reduced divisors have remarkable properties which are similar to the properties of reduced ones. In this paper, we describe an algorithm to test whether an Arakelov divisor of a real quadratic field $F$ is $C$-reduced in time polynomial in $\log|Δ_F|$ with $Δ_F$ the discriminant of $F$. Moreover, we give an example of a cubic field for which our algorithm does not work.

math.NT↗

arXiv2016

Computing dimensions of spaces of Arakelov divisors of number fields

The function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the "jump algorithm" and Poisson summation formula.

math.NT↗