Goldfeld's conjecture and congruences between heegner points by Kriz, D, Li, C

“…© The Author(s) 2019. Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). …”
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Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions by Chen, Yongyi

“…A classic early result along this philosophy was achieved by Gross and Zagier, who proved that the derivative of the L-function of an elliptic curve is equal, up to a constant, to the Néron-Tate height pairing of a special point called a Heegner point on the elliptic curve. …”
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On the worst-case side-channel security of ECC point randomization in embedded devices by Azouaoui, Melissa, Durvaux, François, Poussier, Romain, Standaert, François-Xavier, Papagiannopoulos, Kostas, Verneuil, Vincent

“…Point randomization is an important countermeasure to protect Elliptic Curve Cryptography (ECC) implementations against sidechannel attacks. …”
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Expressions for the generating function of the Donaldson invariants for CP² by Malmendier, Andreas

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Simple High-Level Code For Cryptographic Arithmetic With Proofs, Without Compromises by Erbsen, Andres, Philipoom, Jade D., Gross, Jason S., Sloan, Robert Hal, Chlipala, Adam

“…These techniques were used to build an elliptic-curve library that achieves competitive performance for a wide range of prime fields and multiple CPU architectures, showing that implementation and proof effort scales with the number and complexity of conceptually different algorithms, not their use cases. …”
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A reduced τ-adic Naf (RTNAF) representation for an efficient scalar multiplication on Anomalous Binary Curves (ABC) by Yunos, Faridah, Mohd Atan, Kamel Ariffin, Kamel Ariffin, Muhammad Rezal, Md. Said, Mohamad Rushdan

“…Elliptic curve cryptosystems (ECC) provides better security for each bit key utilized compared to the RSA cryptosystem. …”
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Elliptic net scalar multiplication using generalized equivalent elliptic divisibility sequence by Muslim, Norliana, Md. Said, Mohamad Rushdan

“…Combining the new generalization with a few restrictions on the initial value, the paper further proposes and discusses an elliptic net scalar multiplication of rank one for Weistrass equation and non-singular elliptic curve.…”
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Generalizing equivalent elliptic divisibility sequence for elliptic net scalar multiplication by Muslim, Norliana, Md. Said, Mohamad Rushdan

“…Combining the new generalization with a few restrictions on the initial value, the paper further proposes and discusses an elliptic net scalar multiplication of rank one for Weistrass equation and non-singular elliptic curve.…”
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New Lightweight Identity Based Encryption Algorithm for Mobile Device: Preliminary Study by Norhidayah, Muhammad, Jasni, Mohamad Zain, Md Yazid, Mohd Saman

“…One method called elliptic curve will be implemented in IBE. Elliptic curve method has shorter key size, smaller signature length, fast operations and high security working. …”
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Development of a power-efficient low-cost cryptosystem for secured embedded systems by Pua, Yi Cheng.

“…By using a suitable elliptic curve over this field, a cryptosystem that is similar in cryptographic strength to 112-bit Triple-DES can be built. …”
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Improved scalar multiplication algorithm in affine coordinate system using elliptic net by Razali, Zuren, Muslim, Norliana, Kahar, Saliyah, Yunos, Faridah, Mohamed, Kamsiah

“…In elliptic curve encryption, scalar multiplication (SM) is the most expensive and time-consuming operation. …”
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Stark points and p-adic iterated integrals attached to modular forms of weight one by Lauder, A, Darmon, H, Rotger, V

“…Let E be an elliptic curve over Q, and let ϱ♭ and ϱ♯ be odd two-dimensional Artin representations for which ϱ♭⊗ϱ♯ is self-dual. …”

The meta-abelian elliptic KZB associator and periods of Eisenstein series by Matthes, N

“…We compute the image of Enriquez’ elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve.Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne’s computation of the depth one quotient of the Drinfeld associator.We also show how to retrieve Zagier’s extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson–Levin’s elliptic polylogarithms from the meta-abelian elliptic KZB associator.…”

On reduced stable pair invariants by Oberdieck, Georg B

“…Let X = S × E be the product of a K3 surface S and an elliptic curve E. Reduced stable pair invariants of X can be defined via (1) cutting down the reduced virtual class with incidence conditions or (2) the Behrend function weighted Euler characteristic of the quotient of the moduli space by the translation action of E. …”
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Sato–Tate distributions of twists of the Fermat and the Klein quartics by Fité, Francesc, Lorenzo García, Elisa, Sutherland II, Andrew Victor

“…We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is [bar over Q]-isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato–Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. …”
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Overview on elliptic multiple zeta values by Matthes, N

“…Common to both approaches to elliptic multiple zeta values is their representation as iterated integrals on a once-punctured elliptic curve. Having compared the two approaches, we survey various recent results about the algebraic structure of elliptic multiple zeta values, as well as indicating their relation to iterated integrals of Eisenstein series, and to a special algebra of derivations.…”

Enhancement of Security Architecture for Smartcard Based Authentication Protocols by Mohammed, Lawan Ahmed

“…The findings discovered that the proposed Elliptic Curve Cryptograpluc (ECC) method provides greater efficiency than similar method in terms of computational speed. …”
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Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes by Broedel, J, Matthes, N, Richter, G, Schlotterer, O

“…These arise as iterated integrals on an elliptic curve from which a rational lattice has been removed. …”

Elliptic double zeta values by Matthes, N

“…Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. …”

Standardized signature algorithms on ultra-constrained 4-bit MCU by Chen, Chien-Ning, Jacob, Nisha, Kutzner, Sebastian, Ling, San, Poschmann, Axel, Saetang, Sirote

“…In this work, we implement all three digital signature schemes specified in Digital Signature Standard (FIPS 186-3), including DSA and RSA (based on modular exponentiation) as well as ECDSA (based on elliptic curve point multiplication), on an ultra-constrained 4-bit MCU of the EPSON S1C63 family. …”
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