On zeros of Eichler integrals

We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields.

By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2 + b^2 = c^2 + d^2$. We use this notion to construct infinite families of elliptic curves of higher rank as far as possible. Furthermore, we give particular examples of rank eight.

In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch–Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over Q without complex multiplication.

Following Lang and Trotter, we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over $$\mathbb {Q}$$ . In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the $$\ell $$ -adic limits determining the $$\ell $$ -factors of the universal constant, unlike the Lang–Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239–268, 2004), following the work of David et al. (Math Ann 368(1–2):685–752, 2017).

An elliptic curve cryptosystem (ECC) is one of public key cryptosystem, whose security is based on elliptic curve discrete logarithm problem (ECDLP). An elliptic curve is uniquely determined by mathematical parameters such as j-invariant of an elliptic curve. By giving trace of elliptic curve, t, a definition field \(\mathbb F_{p}\), and discriminant D, an elliptic curve with order \(\sharp {E(\mathbb F_{p})} = n\) is determined. Therefore it is an open problem to determine explicit relations between the mathematical parameters and the embedding degrees k. Hirasawa and Miyaji presented concrete relations between the mathematical parameters and the embedding degrees. In this research, a new explicit relation between elliptic-curve parameters and embedding degrees is investigated by generalizing their research.

This article presents a software and laboratory suite for studying the mathematical foundations and practical applications of elliptic curve cryptography (ECC). The suite is implemented in Python using the PyQt6 framework and the sympy library for cryptographic computations. The program provides an interactive interface for entering elliptic curve parameters, visualizing points on the curve, constructing Cayley tables for point addition, and checking group properties. Key features of the suite include the implementation of the Tonelli–Shanks algorithm for finding absolute square roots, the ability to work with curves over finite fields of large order, and a bilingual interface (Russian/English). The developed suite can be used in educational settings to teach the fundamentals of elliptic curve cryptography.

In this paper, we present a new fault attack on elliptic curve scalar product algorithms. This attack is tailored to work on the classical Montgomery ladder method when the y-coordinate is not used. No weakness has been reported so far on such implementations, which are very efficient and were promoted by several authors. But taking into account the twist of the elliptic curves, we show how, with few faults (around one or two faults), we can retrieve the full secret exponent even if classical countermeasures are employed to prevent fault attacks. It turns out that this attack has not been anticipated as the security of the elliptic curve parameters in most standards can be strongly reduced. Especially, the attack is meaningful on some NIST or SECG parameters.

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo N and occasionally taking GCDs with N. These began with Pollard’s p − 1 p - 1 and Monte Carlo methods. More recently, Williams published a p + 1 p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ± 1 p \pm 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n / 2 + o ( n ) n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 log 2 {\log _2} n multiplications for arbitrary P.

We exhibit several families of elliptic curves with torsion group isomorphic to $ \Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusi\'c and Tadi\'c and we find the exact rank over $\Q(t)$ to be 3 and we also determine free generators of the Mordell-Weil group for each family. By suitable specializations, we obtain the known and new examples of curves over $\Q$ with torsion $ \Z/6\Z$ and rank $8$, which is the current record.

Given an elliptic curve E and a finite Abelian group G, we consider the problem of counting the number of primes p for which the group of points modulo p is isomorphic to G. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves.

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

Wireless Sensor Networks (WSNs) pose a number of unique security challenges that demand innovation in several areas including the design of cryptographic primitives and protocols. Despite recent progress, the efficient implementation of Elliptic Curve Cryptography (ECC) for WSNs is still a very active research topic and techniques to further reduce the time and energy cost of ECC are eagerly sought. This paper presents an optimized ECC implementation that we developed from scratch to comply with the severe resource constraints of 8-bit sensor nodes such as the MICAz and IRIS motes. Our ECC software uses Optimal Prime Fields (OPFs) as underlying algebraic structure and supports two different families of elliptic curves, namely Weierstraß-form and twisted Edwards-form curves. Due to the combination of efficient field arithmetic and fast group operations, we achieve an execution time of 5.8 · 106 clock cycles for a full 158-bit scalar multiplication on an 8-bit ATmega128 micro-controller, which is 2.78 times faster than the widely-used TinyECC library. Our implementation also shows that the energy cost of scalar multiplication on a MICAz (or IRIS) mote amounts to just 19~mJ when using a twisted Edwards curve over a 160-bit OPF. This result compares fairly well with the energy figures of two recently-presented hardware designs of ECC based on twisted Edwards curves.

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$.

We show that the configuration space of four unordered points in C \mathbb {C} with barycenter 0 is isomorphic to the space of triples ( E , Q , ω ) (E,Q,\omega ) , where E E is an elliptic curve, Q ∈ E ∘ Q\in E^\circ a nonzero point, and ω \omega a nonzero holomorphic differential on E E . At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group B 4 B_4 : namely, the sequence 1 → F 2 → B 4 → B 3 → 1 1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1 , where F 2 F_2 is a free group of rank 2, related to Ferrari’s solution of the quartic, and the sequence 1 → Z → B 4 → Aut + ⁡ ( F 2 ) → 1 1\rightarrow \mathbb {Z} \rightarrow B_4\rightarrow \operatorname {Aut}^+(F_2)\rightarrow 1 of Dyer-Formanek-Grossman [Arch. Math. (Basel) 38 (1982), pp. 404–409].

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves