Ordinary Elliptic Curves Research Articles

On ordinary isogeny graphs with level structures

Let ℓ and p be two distinct prime numbers. We study ℓ-isogeny graphs of ordinary elliptic curves defined over a finite field of characteristic p, together with a level structure. Firstly, we show that as the level varies over all p-powers, the graphs form an Iwasawa-theoretic abelian p-tower, which can be regarded as a graph-theoretical analogue of the Igusa tower of modular curves. Secondly, we study the structure of the crater of these graphs, generalizing previous results on volcano graphs. Finally, we solve an inverse problem of graphs arising from the crater of ℓ-isogeny graphs with level structures, partially generalizing a recent result of Bambury, Campagna and Pazuki.

Denominators of the Weierstrass coefficients of the canonical lifting

Given an ordinary elliptic curve E/k:y02=x03+a0x0+b0 over a field k of characteristic p≥5 with j-invariant j0, the j-invariant of its canonical lifting E/W(k):y2=x3+ax+b is j=(j0,J1(j0),J2(j0),…), for some Ji∈Fp(X). Thus the Weierstrass coefficients of E can be given by a=λ4⋅27j/(6912−4j), b=λ6⋅27j/(6912−4j), where λ=((b0/a0)1/2,0,0,…), and therefore can be seen as functions on (a0,b0). Here we study the denominators of the coordinates of these a and b. We show that the only possible factors for these denominators are powers of a0, b0, and the Hasse invariant h. Upper bounds for these powers are given for each one of them.

Correction to: Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields

Correction to: Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields

Classification of Du Val del Pezzo surfaces of Picard rank one in characteristic two and three

In this paper, we classify Du Val del Pezzo surfaces of Picard rank one in characteristic two and three. We also show that if a Du Val del Pezzo surface is Frobenius split, then a general anti-canonical member is smooth. Furthermore, in characteristic two, it is an ordinary elliptic curve.

Differential addition on binary elliptic curves

This paper presents new and fast differential addition (i.e., the addition of two points with the known difference) and doubling formulas, as the core step in Montgomery scalar multiplication, for various forms of elliptic curves over binary fields. The formulas are provided for binary Edwards, binary Hessian and binary Huff elliptic curves with cost of 5M+4S+1D when the given difference point is in affine form. Here, M,S,D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively. In this paper, we show that every ordinary elliptic curve E over binary field F2n has Lopez and Dahab differential addition and doubling formulas with suitable function. This paper also presents, new complete differential addition formulas for binary Edwards curves with cost of 5M+4S+2D. Also, new complete Montgomery ladder and scalar multiplication algorithms are presented using the complete differential addition formulas.

On the decisional Diffie–Hellman problem for class group actions on oriented elliptic curves

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order {mathcal {O}} in an unknown ideal class [{mathfrak {a}}] in {{,textrm{cl},}}({mathcal {O}}) that connects two given {mathcal {O}}-oriented elliptic curves (E, iota ) and (E', iota ') = [{mathfrak {a}}](E, iota ). When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sotáková and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie–Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski’s recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time.

Spanning the isogeny class of a power of an elliptic curve

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

Hashing to Elliptic Curves of $$j=0$$ and Mordell–Weil Groups

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product $E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E_b$. More precisely, we propose to search such a curve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surface of $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove that it is just isomorphic to $\mathbb{Z}/3$.

A subexponential-time, polynomial quantum space algorithm for inverting the CM group action

AbstractWe present a quantum algorithm which computes group action inverses of the complex multiplication group action on isogenous ordinary elliptic curves, using subexponential time, but only polynomial quantum space. One application of this algorithm is that it can be used to find the private key from the public key in the isogeny-based CRS and CSIDH cryptosystems. Prior claims by Childs, Jao, and Soukharev of such a polynomial quantum space algorithm for this problem are false; our algorithm (along with contemporaneous, independent work by Biasse, Iezzi, and Jacobson) is the first such result.

Weierstrass coefficients of the canonical lifting

Given an ordinary elliptic curve [Formula: see text] over a field of characteristic [Formula: see text], there are functions [Formula: see text] and [Formula: see text] such that the curve [Formula: see text] where [Formula: see text] and [Formula: see text] is the canonical lifting of [Formula: see text]. Although these functions are not uniquely determined, we prove that they can be taken to be in [Formula: see text], defined for all ordinary elliptic curves of the given characteristic, and modular, with [Formula: see text] and [Formula: see text].

Beta Weil pairing revisited

In this paper, we extend the $$\beta $$ -Weil pairing, initially introduced in the setting of ordinary elliptic curves with even embedding degree by Aranha et al. [2], to ordinary elliptic curves of any embedding degree. We also propose a new optimal pairing which is the product of some rational functions with the same Miller loop and having a simple final exponentiation. The new pairing is appropriated for using the multi-pairing technique for an efficient implementation. We focus our computation at high security level. Exploiting the fact that the $$\beta $$ -Weil pairing is suitable for parallel execution, we first show that calculating the extended $$\beta $$ -Weil pairing over pairing-friendly elliptic curves with embedding degree 27 is more efficient than calculating the optimal ate pairing. Finally we show that calculating our new pairing over Barreto–Lynn–Scott curves with embedding degree 12 (BLS12) and pairing-friendly elliptic curves with embedding degree 15 is more efficient than calculating the optimal ate pairing.

Arithmetic properties of signed Selmer groups at non-ordinary primes

We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for $p$-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.

Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three

Abstract We study distance functions on the set of ordinary (or non-supersingular) elliptic curves in short Weierstrass form (or simplified Weierstrass form) over a finite field of characteristic three. Mishra and Gupta (2008) firstly construct distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Afterward, Vetro (2011) constructs some other distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Recently, Hakuta (2015) has proposed distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic two. However, to our knowledge, no analogous result is known in the characteristic three case. In this paper, we shall prove that one can construct distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic three. A cryptographic application of our distance functions is also discussed.

Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields

In this paper we study the dynamics of rational maps induced by endomorphisms of ordinary elliptic curves defined over finite fields.

Elliptic curves generation for isogeny-based cryptosystems

Methods of generating supersingular and ordinary elliptic curves for isogeny-based cryptosystems have been studied. The influence of the class field polynomial on the time of generating ordinary elliptic curves has been analyzed and the comparative time of generating curves using Weber and Hilbert polynomials have been presented. Parameters that influence on the cryptographic security of isogeny-based cryptosystems have been considered.

CANONICAL LIFTS OF FAMILIES OF ELLIPTIC CURVES

We show that the canonical lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti and Erdoğan.

On the construction of irreducible polynomials over finite fields via odd prime degree endomorphisms of elliptic curves

In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.

Elliptic curves with isomorphic groups of points over finite field extensions

Consider a pair of ordinary elliptic curves E and E′ defined over the same finite field Fq. Suppose they have the same number of Fq-rational points, i.e. |E(Fq)|=|E′(Fq)|. In this paper we characterise for which finite field extensions Fqk, k≥1 (if any) the corresponding groups of Fqk-rational points are isomorphic, i.e. E(Fqk)≅E′(Fqk).

SOME REMARKS ON A DISTANCE BETWEEN TWO ORDINARY ELLIPTIC CURVES OVER THE FINITE FIELD F_{2^5}

Rishivarman and Parthasarathy (2013) have constructed a map from the direct product of two copies of the set of ordinary elliptic curves with short Weierstrass equation over F25 to non-negative real numbers (plus infinity), and they have claimed that the map is a metric. In this paper we point out that the proof of the claim contains several flaws. Even worse, we shall show that the map is not a metric.

An old letter from J. Tate to B. Dwork, as viewed recently by J. Tate and B. Mazur

In this paper, the constant , which is defined in the letter for a fixed ordinary elliptic curve, is viewed as a special value of formal function on a vast ind-profinite ``Galois cover of the `Hasse Domain' '' parametrizing all ordinary elliptic curves, viewed as a formal scheme.