scholarly journals Average normalisations of elliptic curves

2002 ◽  
Vol 66 (3) ◽  
pp. 353-358 ◽  
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Almost All

Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.

scholarly journals Finding elliptic curves with a subgroup of prescribed size

2016 ◽  
Vol 13 (01) ◽  
pp. 133-152
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Time Complexity ◽  
Rational Functions ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithm ◽  
Almost All

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime [Formula: see text] and positive integer [Formula: see text], outputs an elliptic curve [Formula: see text] over the finite field [Formula: see text] for which the cardinality of [Formula: see text] is divisible by [Formula: see text]. The running time of the algorithm is [Formula: see text], and this leads to more efficient constructions of rational functions over [Formula: see text] whose image is small relative to [Formula: see text]. We also give an unconditional version of the algorithm that works for almost all primes [Formula: see text], and give a probabilistic algorithm with subexponential time complexity.

scholarly journals The Frequency of Elliptic Curve Groups over Prime Finite Fields

2016 ◽  
Vol 68 (4) ◽  
pp. 721-761 ◽  
Author(s):  
Vorrapan Chandee ◽  
Chantal David ◽  
Dimitris Koukoulopoulos ◽  
Ethan Smith
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Arithmetic Progressions ◽  
Constant Multiple ◽  
Group Order ◽  
Almost All ◽  
Candidate Group

AbstractLetting p vary over all primes and E vary over all elliptic curves over the finite field 𝔽p, we study the frequency to which a given group G arises as a group of points E(𝔽p). It is well known that the only permissible groups are of the form Gm,k:=ℤ/mℤ×ℤ/mkℤ. Given such a candidate group, we let M(Gm,k) be the frequency to which the group Gm,karises in this way. Previously, C.David and E. Smith determined an asymptotic formula for M(Gm,k) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(Gm,k), pointwise and on average. In particular, we show thatM(Gm,k) is bounded above by a constant multiple of the expected quantity when m ≤ kA and that the conjectured asymptotic for M(Gm,k) holds for almost all groups Gm,k when m ≤ k1/4-∈. We also apply our methods to study the frequency to which a given integer N arises as a group order #E(𝔽p).

scholarly journals Finding the group structure of elliptic curves over finite fields

2005 ◽  
Vol 72 (2) ◽  
pp. 251-263 ◽  
Author(s):  
John B. Friedlander ◽  
Carl Pomerance ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Polynomial Time ◽  
Group Structure ◽  
Curves Over Finite Fields ◽  
Divisor Sums ◽  
Almost All ◽  
Probabilistic Polynomial Time

We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some divisor sums.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

Bilinear character sums and sum-product problems on elliptic curves

2010 ◽  
Vol 53 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Omran Ahmadi ◽  
Igor Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Multiplicative Character ◽  
Additive Character ◽  
Product Problem ◽  
Multiplicative Character Sums

AbstractLet E be an ordinary elliptic curve over a finite field q of q elements. We improve a bound on bilinear additive character sums over points on E, and obtain its analogue for bilinear multiplicative character sums. We apply these bounds to some variants of the sum-product problem on E.

scholarly journals On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

2015 ◽  
Vol 18 (1) ◽  
pp. 308-322 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Log P ◽  
Point Counting ◽  
Generalized Riemann Hypothesis ◽  
Expected Time ◽  
Almost All ◽  
Counting Algorithm

For an elliptic curve$E/\mathbb{Q}$without complex multiplication we study the distribution of Atkin and Elkies primes$\ell$, on average, over all good reductions of$E$modulo primes$p$. We show that, under the generalized Riemann hypothesis, for almost all primes$p$there are enough small Elkies primes$\ell$to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in$(\log p)^{4+o(1)}$expected time.

scholarly journals Elliptic curves over the ring R

2019 ◽  
Vol 38 (3) ◽  
pp. 193-201 ◽  
Author(s):  
A. Boulbot ◽  
Abdelhakim Chillali ◽  
A. Mouhib
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Canonical Projection ◽  
First Time ◽  
Weierstrass Equation

Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

scholarly journals The Order of Edwards and Montgomery Curves

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Log P ◽  
Edwards Curves ◽  
Digital Signature Algorithm ◽  
Supersingular Curves ◽  
Edwards Curve

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA) [2]. It is well known that the problem of discrete logarithm is NP-hard on group on elliptic curve (EC) [5]. The orders of groups of an algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn is studied by us. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve [F ] d p E over a finite field Fp . It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. The method we have proposed has much less complexity 22 O p log p at not large values p in comparison with the best Schoof basic algorithm with complexity 8 2 O(log pn ) , as well as a variant of the Schoof algorithm that uses fast arithmetic, which has complexity 42O(log pn ) , but works only for Elkis or Atkin primes. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve [F ] d p E is supersingular over this field or not. The symmetric of the Edwards curve form and the parity of all degrees made it possible to represent the shape curves and apply the method of calculating the residual coincidences. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A oneto- one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over F pn .

scholarly journals EMBEDDING FINITE FIELDS INTO ELLIPTIC CURVES

2019 ◽  
pp. 889
Author(s):  
Amirmehdi Yazdani Kashani ◽  
Hassan Daghigh
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Elliptic Curve Cryptosystems ◽  
General Elliptic ◽  
Genus 2

Many elliptic curve cryptosystems require an encoding function from a finite field Fq into Fq-rational points of an elliptic curve. We propose a uniform encoding to general elliptic curves over Fq. We also discuss about an injective case of SWU encoing for hyperelliptic curves of genus 2. Moreover we discuss about an injective encoding for elliptic curves with a point of order two over a finite field and present a description for these elliptic curves.

scholarly journals ON THE RANK OF QUADRATIC TWISTS OF ELLIPTIC CURVES OVER FUNCTION FIELDS

2006 ◽  
Vol 02 (02) ◽  
pp. 267-288 ◽  
Author(s):  
E. KOWALSKI
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Upper Bounds ◽  
Density Theorem ◽  
Chebotarev Density Theorem ◽  
Quadratic Twists ◽  
Rank 2

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.

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