scholarly journals An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1431
Author(s):  
Yuri Borissov ◽  
Miroslav Markov
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Explicit Formula ◽  
Point Counting ◽  
Prime Field ◽  
Algorithmic Solution ◽  
The Family ◽  
Order Of Magnitude ◽  
Modulo P ◽  
Quantities Of Interest

Here, we elaborate an approach for determining the number of points on elliptic curves from the family Ep={Ea:y2=x3+a(modp),a≠0}, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the six cardinalities associated with the family Ep, for p≡1(mod3), whose complexity is O˜(log2p), thus improving the best-known algorithmic solution with almost an order of magnitude.

scholarly journals Moments of the Critical Values of Families of Elliptic Curves, with Applications

2010 ◽  
Vol 62 (5) ◽  
pp. 1155-1181 ◽  
Author(s):  
Matthew P. Young
Keyword(s):  
Elliptic Curves ◽  
Large Family ◽  
Critical Values ◽  
Central Values ◽  
The Family ◽  
Order Of Magnitude ◽  
Rank 2 ◽  
Family Based ◽  
First Moment ◽  
Positive Rank

AbstractWe make conjectures on the moments of the central values of the family of all elliptic curves and on themoments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.Furthermore, as arithmetical applications, we make a conjecture on the distribution of ap's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

scholarly journals Amicable pairs and aliquot cycles on average

2015 ◽  
Vol 11 (06) ◽  
pp. 1751-1790 ◽  
Author(s):  
James Parks
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
The Family ◽  
Order Of Magnitude

Silverman and Stange defined the notion of an aliquot cycle of length L for a fixed elliptic curve E/ℚ, and conjectured an order of magnitude for the function that counts such aliquot cycles. We show that the conjectured upper bound holds for the number of aliquot cycles on average over the family of all elliptic curves with short bounds on the size of the parameters in the family.

scholarly journals An asymptotic for the average number of amicable pairs for elliptic curves

2017 ◽  
Vol 166 (1) ◽  
pp. 33-59 ◽  
Author(s):  
JAMES PARKS
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Asymptotic Constant ◽  
The Family ◽  
Order Of Magnitude ◽  
Precise Asymptotic

AbstractAmicable pairs for a fixed elliptic curve defined over ℚ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves.

scholarly journals On the discrete logarithm problem for prime-field elliptic curves

2018 ◽  
Vol 51 ◽  
pp. 168-182 ◽  
Author(s):  
Alessandro Amadori ◽  
Federico Pintore ◽  
Massimiliano Sala
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Prime Field

scholarly journals On the number of prime divisors of the order of elliptic curves modulo p

2005 ◽  
Vol 117 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Jörn Steuding ◽  
Annegret Weng
Keyword(s):  
Elliptic Curves ◽  
Prime Divisors ◽  
Modulo P ◽  
Number Of Prime Divisors

scholarly journals Generator of PRN's on the norm group

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 26-39
Author(s):  
P. Fugelo ◽  
S. Varbanets
Keyword(s):  
Prime Number ◽  
Quadratic Field ◽  
Exponential Sum ◽  
Random Numbers ◽  
Gaussian Integers ◽  
New Family ◽  
Serial Test ◽  
Polynomial Representations ◽  
Modulo P ◽  
Norm Group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

scholarly journals Analysis and combinatorics of partition zeta functions

2020 ◽  
pp. 1-10
Author(s):  
Robert Schneider ◽  
Andrew V. Sills
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Complete Information ◽  
Combinatorial Proof ◽  
Fixed Length ◽  
The Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

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