ISSN:

1930-5346

eISSN:

1930-5338

## Advances in Mathematics of Communications

2014 , Volume 8 , Issue 4

Special issue on complex geometrical optics (CGO) solutions

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2014, 8(4): i-ii
doi: 10.3934/amc.2014.8.4i

*+*[Abstract](105)*+*[PDF](106.0KB)**Abstract:**

Every second year, the GEOCRYPT conference brings together researchers from arithmetic geometry and cryptography. After Guadeloupe in 2009 and Corsica in 2011, French Polynesia was the host of its 2013 edition which took place in Punaauia, on the island of Tahiti, October 7-11.

The main topic of GEOCRYPT has always been the application of pure mathematical techniques to the safety and efficiency of modern communication systems, with particular interest in the fields of arithmetic and algebraic geometry over finite fields, algorithms for finite fields, error correcting codes, cryptology, boolean functions, discrete dynamical systems, and their interactions. The 2013 edition had the honour to feature six invited and eleven contributed talks from renowned international experts presenting strong results recently obtained on topics ranging from pure mathematics to cryptographic algorithms. The GEOCRYPT 2013 Program Committee carefully evaluated the submitted abstracts and selected the best contributions for presentation at the conference.

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2014, 8(4): 375-387
doi: 10.3934/amc.2014.8.375

*+*[Abstract](438)*+*[PDF](352.7KB)**Abstract:**

By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus $2$ curves over finite fields of characteristic $2$. With our technique we obtain new results for these Jacobians: we show how to find their $3$-torsion subgroup, we prove there is none with $3$-torsion subgroup of rank $3$ and we prove that the maximal $3$-power order subgroup is isomorphic to either $\mathbb{Z}/3^{v}\mathbb{Z}$ or $(\mathbb{Z}/3^{\frac v2}\mathbb{Z})^2$ or $(\mathbb{Z}/3^{\frac v4}\mathbb{Z})^4$, where $v$ is the $3$-adic valuation $v_{3}$(#Jac(C)$(\mathbb{F}_{2^m})$). Ours are the first trisection formulae available in literature.

2014, 8(4): 389-406
doi: 10.3934/amc.2014.8.389

*+*[Abstract](171)*+*[PDF](450.5KB)**Abstract:**

Real hyperelliptic curves admit two structures suitable for cryptography --- the Jacobian (a finite abelian group) and the infrastructure. Mireles Morales described precisely the relationship between these two structures, and made the assertion that when implemented with balanced divisor arithmetic, the Jacobian generically yields more efficient arithmetic than the infrastructure for cryptographic applications. We confirm that this assertion holds for genus two curves, through rigorous analysis and the first detailed numerical performance comparisons, showing that cryptographic key agreement can be performed in the Jacobian without any extra operations beyond those required for basic scalar multiplication. We also present a modified version of Mireles Morales' map that more clearly reveals the algorithmic relationship between the two structures.

2014, 8(4): 407-425
doi: 10.3934/amc.2014.8.407

*+*[Abstract](172)*+*[PDF](475.7KB)**Abstract:**

Hafner and McCurley described a subexponential time algorithm to compute the ideal class group of a quadratic field, which was generalized to families of fixed degree number fields by Buchman. The main ingredient of this method is a subexponential time algorithm to derive relations between primes of norm bounded by a subexponential value. Besides ideal class group computation, this was successfully used to evaluate isogenies, compute endomorphism rings, solve the discrete logarithm problem in the class group and find a generator of a principal ideal. In this paper, we present a generalization of the relation search to classes of number fields with degree growing to infinity.

2014, 8(4): 427-436
doi: 10.3934/amc.2014.8.427

*+*[Abstract](175)*+*[PDF](368.2KB)**Abstract:**

We give an overview of a method for using elliptic curves with complex multiplication to give efficient deterministic polynomial time primality tests for the integers in sequences of a special form. This technique has been used to find the largest proven primes $N$ for which there was no known significant partial factorization of $N-1$ or $N+1$.

2014, 8(4): 437-458
doi: 10.3934/amc.2014.8.437

*+*[Abstract](178)*+*[PDF](446.2KB)**Abstract:**

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $l$-radicals for small odd integers $l$, and make it explicit for $l=5$.

2014, 8(4): 459-477
doi: 10.3934/amc.2014.8.459

*+*[Abstract](178)*+*[PDF](386.6KB)**Abstract:**

We present an analysis of Bernstein's batch integer smoothness test when applied to the case of polynomials over a finite field $\mathbb{F}_q.$ We compare the performance of our algorithm with the standard method based on distinct degree factorization from both an analytical and a practical point of view. Our results show that although the batch test is asymptotically better as a function of the degree of the polynomials to test for smoothness, it is unlikely to offer significant practical improvements for cases of practical interest.

2014, 8(4): 479-495
doi: 10.3934/amc.2014.8.479

*+*[Abstract](175)*+*[PDF](287.4KB)**Abstract:**

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational $2$-torsion point on their Jacobian. We extend their observation to curves given by Laurent polynomials with a fixed Newton polygon, provided that the polygon satisfies a certain combinatorial property. We also show that in each of these cases, if the curve is ordinary, then there is no need for the words ``sufficiently general''. Our treatment includes many classical families, such as hyperelliptic curves of odd genus and $C_{a,b}$ curves. In the hyperelliptic case, we provide alternative proofs using an explicit description of the $2$-torsion subgroup.

2014, 8(4): 497-509
doi: 10.3934/amc.2014.8.497

*+*[Abstract](129)*+*[PDF](373.2KB)**Abstract:**

Let $t$ be an integer $\ge 5$. The absolute irreducibility of the polynomial $\phi_t(x, y) = \frac{x^t + y^t + 1 + (x + y + 1)^t}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb{F}_2$) plays an important role in the study of APN functions. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.

2014, 8(4): 511-519
doi: 10.3934/amc.2014.8.511

*+*[Abstract](183)*+*[PDF](324.0KB)**Abstract:**

It is possible that a simple (or absolutely simple) Abelian variety defined over a number field splits modulo every prime of good reduction. This is a new problem that arises in designing crypto systems using Abelian varieties of dimension larger than $1$. We discuss what is behind this phenomenon. In particular, we discuss the question of given an absolutely simple abelian variety over a number field, whether it has simple specializations at a set of places of positive Dirichlet density? A conjectural answer to this question was given by Murty and Patankar, and we explain some recent progress towards proving the conjecture. Our result ([14], Theorem 1.1) is based on the classification of pairs $(G,V)$ consisting of a semi-simple algebraic group $G$ over a non-archimedean local field and an absolutely irreducible representation $V$ of $G$ such that $G$ admits a maximal torus acting irreducibly on $V$.

2016 Impact Factor: 0.8

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