On the Mordell-Weil group of elliptic curves induced by families of Diophantine triples

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.

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.

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.

We list a number of strategies for construction of elliptic curves having high rank with special emphasis on those curves induced by Diophantine triples, in which we have contributed more. These strategies have been developed by many authors. In particular we present a new example of a curve, induced by a Diophantine triple, with torsion $$\mathbb {Z}/ 2 \mathbb {Z}\times \mathbb {Z}/ 4\mathbb {Z}$$ and with rank 9 over $$\mathbb {Q}$$. This is the present record for this kind of curves.

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.

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).

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.

Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field F, we show that for a large set of number fields L, whose Galois group of their normal closure over F has certain properties, it will hold that E(L)tors = E(F)tors for all elliptic curves E defined over F. Our methods turn out to be particularly useful in studying the possible torsion groups E(K)tors, where K is a number field and E is a base change of an elliptic curve defined over Q. Suppose that E is a base change of an elliptic curve over Q for the remainder of the abstract. We prove that E(K)tors = E(Q)tors for all elliptic curves E defined over Q and all number fields K of degree d, where d is not divisible by a prime ≤ 7. Using this fact, we determine all the possible torsion groups E(K)tors over number fields K of prime degree p ≥ 7. We determineall thepossibledegrees of [Q(P):Q], where P is a point of prime order p for all p such that p ≢ 8 (mod 9) or for any D ∈1, 2, 7,11,19,43, 67,163; this is true for a set of density of all primes and in particular for all p< 3167. Using this result, we determine all the possible prime orders of a point P ∈ E(K)tors, where [K: Q] = d for all d ≤ 3342296. Finally, we determine all the possible groups E(K)tors, where K is a quartic number field and E is an elliptic curve defined over Q and show that no quartic sporadic point on a modular curve X1(m, n) comes from an elliptic curve defined over Q

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient sloppy reduction. The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units.

The local-global principle for divisibility in CM elliptic curves

We study the growth and stability of the Mordell–Weil group and Tate–Shafarevich group of an elliptic curve defined over the rationals, in various cyclic Galois extensions of prime power order. Mazur and Rubin introduced the notion of diophantine stability for the Mordell–Weil group an elliptic curve $$E_{/{{\mathbb {Q}}}}$$ at a given prime p. Inspired by their definition of stability for the Mordell–Weil group, we introduce an analogous notion of stability for the Tate–Shafarevich group, called -stability. From the perspective of Iwasawa theory, it benefits us to introduce a stronger notion of diophantine stability for the Mordell–Weil group. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is strongly diophantine stable and -stable at $$100\%$$ of primes p. Next, we show that standard conjectures on rank distribution give lower bounds for the proportion of rational elliptic curves E that are strongly diophantine stable at a fixed prime $$p\ge 11$$ . Related questions are studied for rank jumps and growth of ranks Tate–Shafarevich groups on average in prime power cyclic extensions.

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic \({\mathbb {Z}}_p\)-extension of K does not admit any proper \(\Lambda \)-submodule of finite index, where \(\Lambda \) is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have \(\Lambda \)-corank one, so they are not \(\Lambda \)-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg–Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley–Lei.