Construction of High Rank Elliptic Curves

We construct an elliptic curve over the field of rational functions with torsion group$\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$and rank equal to four, and an elliptic curve over$\mathbb{Q}$with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- (respectively two-) parameter subfamilies of high rank over the function field $\mathbb{Q}(t)$ (respectively $\mathbb{Q}(t,k)$). Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}(t,k)$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}(t)$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}(t)$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank (8, 9, 10 and 11).

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e. Z/2Z x Z/2Z, Z/2Z x Z/4Z, Z/2Z x Z/6Z or Z/2Z x Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.

Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.

A public-key encryption technique akin to RSA can be referred to as elliptic curve cryptography (ECC). While RSA's security relies on huge prime numbers, ECC leverages the mathematical idea of elliptic curves to offer the same level of security with much smaller keys. In this paper, we will discuss elliptic curves and examine their applications in cryptography. A Diophantine pair of Hex numbers and Pronic numbers is extended to a Diophantine triple with appropriate property, that generates the elliptic curve and perform the encryption-decryption process.

By representing a genus one curve as a plane curve with five double points, we are able to construct a 3- parameter family of genus one curves over Q with Jacobians having a torsion subgroup isomorphic to Z5. This leads, by specializing the parameters, to elliptic curves over Q of the Mordell-Weil group with high rank and with a torsion subgroup isomorphic to Z5. We also show this family contains as a subfamily the principal homogeneous space parameterizing elliptic curves with a rational point of order 5, namely X1(5). We explicitly describe these families by equations in the Weierstrass form.

The problem of the extendibility of Diophantine triples is closely connected with the Mordell-Weil group of the associated elliptic curve. In this paper, we examine Diophantine triples $\\{k-1,k+1,c_l(k)\\}$ and prove that the torsion group of the associated curves is $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$ for $l=3,4$ and $l\\equiv 1$ or $2 \\pmod{4}$. Additionally, we prove that the rank is greater than or equal to 2 for all $l\\ge2$. This represents an improvement of previous results by Dujella, Peth\\H{o} and Najman, where cases $k=2$ and $l\\le3$ were considered.

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

We establish the analogue for D5 of the theory of algebraic equation of type Er (T. SHIODA: Construction of elliptic curves with high rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43, 1991, No. 4, 673-719), which is one of the results of the theory of Mordell-Weil lattices. As an application, we give a method of constructing an elliptic curve over Q(t) with rank 5, together with explicit generators of the Mordell-Weil group.

The construction of an efficient cryptographic system, based on the combination of the ElGamal elliptic curve algorithm and convolutional codes using the Viterbi decoding algorithm over the Gaussian channel, is proposed. The originality is based on the construction of the mapping of encryption and coding at the channel level and the constraints imposed on the construction of the elliptic curve. When using elliptic curves and codes for cryptography it is necessary to construct elliptic curves with a given or known number of points over a given finite field with a range of constraint lengths, in order to represent the input alphabet and produce channel gain, respectively. The results show that the benefit of coding with encryption increases the security and coding gain and reduces the expansion factor, but at the expense of higher complexity.

Finding suitable elliptic curves for pairing-based cryptosystems is a crucial step for their actual deployment. Miyaji, Nakabayashi and Takano [12] (MNT) were the first to produce ordinary pairing-friendly elliptic curves of prime order with embedding degree \( k \in \lbrace 3, 4, 6 \rbrace \). Scott and Barreto [16] as well as Galbraith et al. [10] extended this method by allowing the group order to be non-prime. The advantage of this idea is the construction of much more suitable elliptic curves, which we will call generalized MNT curves. A necessary step for the construction of such elliptic curves is finding the solutions of a generalized Pell equation. However, these equations are not always solvable and this fact considerably affects the efficiency of the curve construction. In this paper we discuss a way to construct generalized MNT curves through Pell equations which are always solvable and thus considerably improve the efficiency of the whole generation process. We provide analytic tables with all polynomial families that lead to non-prime pairing-friendly elliptic curves with embedding degree \( k \in \lbrace 3, 4, 6 \rbrace \) and discuss the efficiency of our method through extensive experimental assessments.

We develop an algorithm for bounding the rank of elliptic curves in the family $y^2=x^3-B x$, all of them with torsion group $\mathbb {Z} /(2 \mathbb {Z})$ and modular invariant $j=1728$. We use it to look for curves of high rank in this family and present four such curves of rank $13$ and $22$ of rank $12$.