Elliptic curves

Let E be an elliptic curve over Q, let p be an ordinary prime for E, and let K be an imaginary quadratic field. Write K∞/K for the anticyclotomic Zpextension of K and set G∞ = Gal(K∞/K). Following a construction of Section 2 of [BD1] which is recalled in Section 1, one attaches to the data (E,K, p) an anticyclotomic padic Lfunction Lp(E,K) which belongs to the Iwasawa algebra Λ := Zp[[G∞]]. This element, whose construction was inspired by a formula proved in [Gr1], is known, thanks to work of Zhang ([Zh, §1.
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This document includes: Introduction to Elliptic Curves, Elliptic Curve Cryptosystems (ECC), Implementation of ECC in Binary Fields.
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This paper presents an approach related to authenticate mutually a RFID (Radio Frequency Identification) tag from a RFID reader by using the cryptography based on Elliptic curve. Our proposal mutual authentication lies on the Elliptic curve discrete logarithm problem, which is considered the core in order to fight against all of attacks like replay attack, forgery attack and maninthemiddle attack. Scientifically, we prove not only the accuracy and the security of our approach, but also its performance in the mutual authentication between a RFID tag and a reader. ...
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Abstract Elliptic curve cryptosystems oﬀer security comparable to that of traditional asymmetric cryptosystems, such as those based on the RSA encryption and digital signature algorithms, with smaller keys and computationally more eﬃcient algorithms.
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A Hardware Architecture for Elliptic Curve Cryptography and Lossless Data Compression. We present a hardware architecture that combines Elliptic Curve Cryptography (ECC) and lossless data compression in a single chip.
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This work proposes a processor architecture for elliptic curves cryptosystems over ﬁelds GF (2m ). This is a scalable architecture in terms of area and speed that exploits the abilities of reconﬁgurable hardware to deliver optimized circuitry for diﬀerent elliptic curves and ﬁnite ﬁelds.
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The paper gives an introduction to elliptic curve cryptography (ECC) and how it is used in the implementation of digital signature (ECDSA) and key agreement (ECDH) Algorithms. The paper discusses the implementation of ECC on two finite fields, prime field and binary field.
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This paper describes elliptic curve cryptosystems (ECCs), which are expected to become the nextgeneration public key cryptosystems, and also describes Fujitsu Laboratories’ study of ECCs.
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In the underlying ﬁnite ﬁeld arithmetic of an elliptic curve cryptosystem, ﬁeld multiplication is the next computational costly operation other than ﬁeld inversion. We present two novel algorithms for eﬃcient implementation of ﬁeld multiplication and modular reduction used frequently in an elliptic curve cryptosystem deﬁned over GF (2n ).
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This thesis describes how an elliptic curve cryptosystem can be implemented on low cost microprocessors without coprocessors with reasonable performance. We focus in this paper on the Intel 8051 family of microcontrollers popular in smart cards and other costsensitive devices, and on the Motorola Dragonball, found in the Palm Computing Platform.
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Elliptic units, which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincar´e upper halfplane, provide us with a rich source of arithmetic questions and insights. They allow the analytic construction of abelian extensions of imaginary quadratic fields, encode special values of zeta functions through the Kronecker limit formula, and are a prototype for Stark’s conjectural construction of units in abelian extensions of number fields.
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At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a padic Lfunction. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the padic Lfunction is wellbehaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi’s conjecture relates modiﬁed Selmer groups, which he deﬁned, with modiﬁed padic Lfunctions deﬁned by the ﬁrst author.
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Objectives of Chapter 13: To define a digital signature; to define security services provided by a digital signature; to define attacks on digital signatures; to discuss some digital signature schemes, including RSA, ElGamal, Schnorr, DSS, and elliptic curve; to describe some applications of digital signatures.
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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Supporting Symmetric 128bit AES in Networked Embedded Systems: An Elliptic Curve Key Establishment ProtocolonChip
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The paper discusses the implementation of ECC on two finite fields, prime field and binary field. It also gives an overview of ECC implementation on different coordinate systems called the projective coordinate systems.
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In this paper, we present the results of our implementation of elliptic curve cryptography (ECC) over the ﬁeld GF (p) on an 80MHz, 32bit ARM microprocessor. We have produced a practical software library which supports variable length implementation of the elliptic curve digital signature algorithm (ECDSA).
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Implementation of the cryptographic algorisms based on elliptic curves (ECs) over VFFs provides signiﬁcantly higher performance than the implementation of the ECbased algorithms, in which the ECs are deﬁned over the ground ﬁelds and extension ﬁnite ﬁelds of polynomials.
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During the last few years, a considerable effort has been devoted to the development of reconfigurable computers, machines that are based on the close interoperation of traditional microprocessors and Field Programmable Gate Arrays (FPGAs).
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We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables eﬃcient arithmetic. The scaling technique may be used to improve multiplication and inversion in ﬁnite ﬁelds. We present an eﬃcient inversion algorithm that utilizes the structure of scaled modulus.
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The Elliptic Curve Cryptography (ECC) is evolving as an important cryptography, and shows a promise to be an alternative of RSA. Small size, high security and other features characterize ECC. Based on the theory of ECC, this paper analyzes its advantages over other cryptographies and focuses on its principle.
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