Elliptic Curves

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

INTRODUCTION THE BASIC THEORY Weierstrass Equations The Group Law Projective Space and the Point at Infinity Proof of Associativity Other Equations for Elliptic Curves Other Coordinate Systems The j-Invariant Elliptic Curves in Characteristic 2 Endomorphisms Singular Curves Elliptic Curves mod n TORSION POINTS Torsion Points Division Polynomials The Weil Pairing The Tate-Lichtenbaum Pairing Elliptic Curves over Finite Fields Examples The Frobenius Endomorphism Determining the Group Order A Family of Curves Schoof's Algorithm Supersingular Curves The Discrete Logarithm Problem The Index Calculus General Attacks on Discrete Logs Attacks with Pairings Anomalous Curves Other Attacks Elliptic Curve Cryptography The Basic Setup Diffie-Hellman Key Exchange Massey-Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures The Digital Signature Algorithm ECIES A Public Key Scheme Based on Factoring A Cryptosystem Based on the Weil Pairing Other Applications Factoring Using Elliptic Curves Primality Testing Elliptic Curves over Q The Torsion Subgroup: The Lutz-Nagell Theorem Descent and the Weak Mordell-Weil Theorem Heights and the Mordell-Weil Theorem Examples The Height Pairing Fermat's Infinite Descent 2-Selmer Groups Shafarevich-Tate Groups A Nontrivial Shafarevich-Tate Group Galois Cohomology Elliptic Curves over C Doubly Periodic Functions Tori Are Elliptic Curves Elliptic Curves over C Computing Periods Division Polynomials The Torsion Subgroup: Doud's Method Complex Multiplication Elliptic Curves over C Elliptic Curves over Finite Fields Integrality of j-Invariants Numerical Examples Kronecker's Jugendtraum DIVISORS Definitions and Examples The Weil Pairing The Tate-Lichtenbaum Pairing Computation of the Pairings Genus One Curves and Elliptic Curves Equivalence of the Definitions of the Pairings Nondegeneracy of the Tate-Lichtenbaum Pairing ISOGENIES The Complex Theory The Algebraic Theory Velu's Formulas Point Counting Complements Hyperelliptic Curves Basic Definitions Divisors Cantor's Algorithm The Discrete Logarithm Problem Zeta Functions Elliptic Curves over Finite Fields Elliptic Curves over Q Fermat's Last Theorem Overview Galois Representations Sketch of Ribet's Proof Sketch of Wiles's Proof APPENDIX A: NUMBER THEORY APPENDIX B: GROUPS APPENDIX C: FIELDS APPENDIX D: COMPUTER packages REFERENCES INDEX Exercises appear at the end of each chapter.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field. We propose a novel effective method of point counting for both Edwards and elliptic curves. In addition to finding a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, we also find a general formula by which one can determine whether or not a curve Ed[Fp] is supersingular over this field. The method proposed has complexity O ( p log2 2 p ) . This is an improvement over both Schoof’s basic algorithm and the variant which makes use of fast arithmetic (suitable for only the Elkis or Atkin primes numbers) with complexities O(log8 2 pn) and O(log4 2 pn) respectively. The embedding degree of the supersingular curve of Edwards over Fpn in a finite field is additionally investigated. Due existing the birational isomorphism between twisted Edwards curve and elliptic curve in Weierstrass normal form the result about order of curve over finite field is extended on cubic in Weierstrass normal form.

Background/Objectives: Elliptic Curve Cryptography (ECC) is a public-key encryption method that is similar to RSA. ECC uses the mathematical concept of elliptic curves to achieve the same level of security with significantly smaller keys, whereas RSA's security depends on large prime numbers. Elliptic curves and their applications in cryptography will be discussed in this paper. The elliptic curve is formed by the extension of a Diophantine pair of Centered Hexadecagonal numbers to a Diophantine triple with property D(8). Method: The Diffie–Hellman key exchange, named for Whitfield Diffie and Martin Hellman, was developed by Ralph Merkle and is a mathematical technique for safely transferring cryptographic keys over a public channel. Based on the Diffie–Hellman key exchange, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography. The generation of keys, encryption and decryption are the three main operations of the ElGamal cryptosystem. Findings: Given the relative modesty of our objectives, the fundamental algebraic and geometric characteristics of elliptic curves shall be delineated. Then the behaviour of elliptic curves modulo p: ultimately, there is a fairly strong analogy between the structure of the points on an elliptic curve modulo p and the integers modulo n will be studied. In the end, elliptic curve ElGamal encryption analogues of Diffie–Hellman key exchange will be created. Novelty: Elliptic curves are encountered in a multitude of mathematical contexts and have a varied and fascinating history. Elliptic curves are very significant in number theory and are a focus of much recent work. The earlier research works in Elliptic Curve Cryptography has concentrated on computer algorithms and pairing – based algorithms. In this paper, the concept of polygonal numbers and its extension from Diophantine pair to triples is encountered, thus forming an elliptic curve and perform the encryption-decryption process. MSC Classification Number: 11D09, 11D99,11T71,11G05. Keywords: Elliptic curves, Cryptography, Encryption, Decryption, Centered polygonal numbers

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

Proposing an Elliptic Equation for the Symmetrical Sag Vertical Curvesbased on Sight Distance in Highway

At Crypto 2010, Brier et al. proposed the first construction of a hash function into ordinary elliptic curves that was indifferentiable from a random oracle, based on Icart's deterministic encoding from Crypto 2009. Such a hash function can be plugged into essentially any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle model. However, the proof relied on relatively involved tools from algebraic geometry, and only applied to Icart's deterministic encoding from Crypto 2009. In this paper, we present a new, simpler technique based on bounds of character sums to prove the indifferentiability of similar hash function constructions based on any of the known deterministic encodings to elliptic curves or curves of higher genus, such as the algorithms by Shallue, van de Woestijne and Ulas, or the Icart-like encodings recently presented by Kammerer, Lercier and Renault. In particular, we get the first constructions of well-behaved hash functions to Jacobians of hyperelliptic curves. Our technique also provides more precise estimates on the statistical behavior of those deterministic encodings and the hash function constructions based on them. Additionally, we can derive pseudorandomness results for partial bit patterns of such encodings.

Koblitz curves have been proposed to quickly generate random ideal classes and points on hyperelliptic and elliptic curves. To obtain a further speed-up a different way of generating these random elements has recently been proposed. In this paper we give an upper bound on the number of collisions for this alternative approach. For elliptic Koblitz curves we additionally use the same methods to derive a bound for a modified algorithm. These bounds are tight for cyclic subgroups of prime order, which is the case of most practical interest for cryptography.

Until recently, Cryptography has been of interest primarily to the military and diplomatic communities. But the dawning of the information age has revealed an urgent need for cryptography in the private sector too. Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. In this paper, cyclic elliptic curve of the form y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> + ax + b, a, b ∈ GF(p) with order M is considered and key Sequences are derived from random sequence of cyclic elliptic Curve points. Elliptic Curve is a cubic equation in two variables, x and y, with coefficients from a field satisfying certain conditions. For cryptographic applications the coefficients are chosen from finite fields. A point on the Elliptic curve is a pair of (x, y) which satisfies the Elliptic curve equation. The total number of points (x, y) which satisfy the elliptic curve equation along with x=∞, y=∞ is called the Order of the curve `M'. The least integer N for which NP is equal to point at infinity O is called order of the point P. Elliptic curves for which there exists a point P having the same order N, as that of the curve M are called cyclic elliptic curves. A pseudorandom sequence generator based on chaotic function and Elliptic Curve arithmetic over GF(p) is proposed here. The logistic Map is used as a chaotic function which generates a random sequence of real numbers. This random real sequence is converted to binary which drives an Elliptic Curve arithmetic module generating a random sequence of Elliptic Curve points. The sequence of points {P, 2P, ..., NP} is calculated from a base point P, and stored in a file. Every element in this sequence is a point on the cyclic elliptic curve. The Chaotic binary sequence selects x or y-coordinates of elliptic curve points, pre-computed and stored. This forms a random integer sequence. The randomness properties of this sequence have been tested using various techniques like, autocorrelation distribution, crosscorrelation distribution and first return map. It is observed that the sequence generated satisfies the required randomness properties. These sequences find applications in Stream Cipher Systems. An additive Stream Cipher system is designed using this sequence as the key sequence to encrypt images. Results of image encryption and decryption for a medical image is discussed and analyzed in this paper. The results are also compared with the scheme proposed by Lap-Piu Lee and Kwok-Wo Wong [1]. The security analysis of the proposed system is also discussed. It is interesting to observe that, proposed algorithm is superior compared to Lap-Piu Lee scheme [1].

Elliptic curve cryptosystems([19,25]) are based on the elliptic curve discrete logarithm problem (ECDLP). If elliptic curve cryptosystems avoid FR-reduction([11,17]) and anomalous elliptic curve over Fq ([34,3,36]), then with current knowledge we can construct elliptic curve cryptosystems over a smaller definition field. ECDLP has an interesting property that the security deeply depends on elliptic curve traces rather than definition fields, which does not occur in the case of the discrete logarithm problem (DLP). Therefore it is important to characterize elliptic curve traces explicitly from the security point of view. As for FR-reduction, supersingular elliptic curves or elliptic curve E/Fq with trace 2 have been reported to be vulnerable. However unfortunately these have been only results that characterize elliptic curve traces explicitly for FR- or MOV-reductions. More importantly, the secure trace against FR-reduction has not been reported at all. Elliptic curves with the secure trace means that the reduced extension degree is always higher than a certain level.In this paper, we aim at characterizing elliptic curve traces by FR-reduction and investigate explicit conditions of traces vulnerable or secure against FR-reduction. We show new explicit conditions of elliptic curve traces for FR-reduction. We also present algorithms to construct such elliptic curves, which have relation to famous number theory problems.

Pseudo-random number generator is an important mechanism for cryptographic information protection. It can be used independently to generate special data or as the most important element of security of other mechanisms for cryptographic information protection. The application of transformations in a group of points of elliptic and hypereliptic curves is an important direction for the designing of cryptographically stable pseudo-random sequences generators. This approach allows us to build the resistant cryptographic algorithms in which the problem of finding a private key is associated with solving the discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximum period using transformations on the elliptic curves. The maximum sequence period is provided by the use of recurrent transformations with the sequential formation of the elements of the point group of the elliptic curve. In this case, the problem of finding a private key is reduced to solving a theoretically complex discrete logarithm problem. The article also describes the block diagram of the device for generating pseudo-random sequences and the scheme for generating internal states of the generator.

In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R. Pandharipande [8] using Gromov–Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number E_{\mathrm{trop}}(d) of tropical elliptic plane curves of degree d and fixed “tropical j -invariant” interpolating an appropriate number of points in general position and counted with multiplicities. We show that this number is independent of the position of the points and the value of the j -invariant and that it coincides with the number of complex elliptic curves (with j -invariant j ∉ \{ 0, 1728 \} ). The result can be used to simplify G. Mikhalkin's algorithm to count curves via lattice paths (see [6]) in the case of rational plane curves.