Cryptography is the study of hidden message passing. It is also the story of Alice and Bob, their shady friends, their numerous and crafty enemies, and their dubious relationship. One uses cryptography to mangle a message su ciently such that only intended recipients of that message can \unmangle the message and read it Elliptic Curves The Equation of an Elliptic Curve An Elliptic Curve is a curve given by an equation of the form y2 = x3 +Ax+B There is also a requirement that the discriminant ¢ = 4A3 +27B2 is nonzero. Equivalently, the polynomial x3 +Ax+B has distinct roots. This ensures that the curve is nonsingular. For reasons to be explained later, we also toss in a In 1985 Neal Koblitz and Victor Miller independently proposed elliptic curve cryptography. The security of this scheme would rest on the diﬃculty of the dis-crete logarithm problem in the group formed from the points on an elliptic curve over a ﬁnite ﬁeld. To date the best method for computing elliptic logarithms is fully exponential. This translates into much smaller key sizes permitting on The main advantage of elliptic curves cryptography is that to achieve a certain level of security shorter keys are su cient than in case of \usual cryptography. Using shorter keys can result in a considerable savings in hardware implementations. The second advantage of the elliptic curves cryptography is that quite a few of attack electronic crypto-currency, and elliptic curve cryptography is central to its operation: Bitcoin addresses are directly derived from elliptic-curve public keys, and transactions are authenticated using digital signatures. The public keys and signatures are published as part of the publicly available and auditable block chain to prevent double-spending
in this guide for a level of understanding of Elliptic Curve cryptography that is suﬃcient to be able to explain the entire process to a computer. This is guide is mainly aimed at computer scientists with some mathematical background who are interested in learning more about Elliptic Curve cryptography. It is an introduction to the world of Elliptic Cryptography and should be supplemented by. Speeding up elliptic curve cryptography can be done by speeding up point arithmetic algorithms and by improving scalar multiplication algorithms. This thesis provides a speed up of some point arithmetic algorithms. The study of addition chains has been shown to be useful in improving scalar multiplication algorithms, when the scalar is xed. A special form of an addition chain called a Lucas. 3.2 Attacks on the Elliptic Curve Discrete Logarithm Prob lem In cryptography, an attack is a method of solving a problem. Speciﬁcally, the aim of an attack is to ﬁnd a fast method of solving a problem on which an encryption algorithm depends. The known methods of attack on the elliptic curve (EC) discrete log problem that work for all curves are slow arbitrary algebraically closed elds, while chapter 3 will deal with elliptic curves over nite elds. In section 4 an algorithm will be given that computes the most important quantity of elliptic curves over nite elds, i.e., its number of rational points. Part 3: In the last part I will focus on the role of elliptic curves in cryptography. First, in chapter 5, Elliptic curve cryptog- raphy (ECC) was discovered in 1985 by Neal Koblitz and Victor Miller. Elliptic curve cryptographic schemes are public-key mechanisms that provide the same functional- ity as RSA schemes. However, their security is based on the hardness of a different problem, namely the elliptic curve discrete logarithm problem (ECDLP)
Elliptic Curve Public Key Cryptography. The curve is intersected by lines in 0, 1, 2, or 3 places Touching in 1 place, a line is tangent to the curve If (x,y) is on the curve, so is (x,y) Restriction ensures right side/left side do not meet at origin Any two points generate a third point on the curve Elliptic Curve Cryptography - An Implementation Tutorial 1 Elliptic Curve Cryptography An Implementation Guide Anoop MS email@example.com Abstract: 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 th Implementing Curve25519/X25519: A Tutorial on Elliptic Curve Cryptography MARTIN KLEPPMANN, University of Cambridge, United Kingdom Many textbooks cover the concepts behind Elliptic Curve Cryptography, but few explain how to go from the equations to a working, fast, and secure implementation. On the other hand, while the code of many cryptographic libraries is available as open source, it ca ELLIPTIC CURVE CRYPTOGRAPHY: THE SERPENTINE COURSE OF A PARADIGM SHIFT ANN HIBNER KOBLITZ, NEAL KOBLITZ, AND ALFRED MENEZES Abstract. Over a period of sixteen years elliptic curve cryptography went from being an approach that many people mistrusted or misunderstood to being a public key technology that enjoys almost unquestioned acceptance. W Elliptic curves in cryptography. In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes
cryptography. Factoring using elliptic curves Primality testing using elliptic curves Showing equivalence between Discrete Logarithm Problem and Di e Hell-man Problem for some special classes of groups. Chapter X - Hyperelliptic Cryptosystems The nal chapter gives a brief introduction to hyperelliptic curves and its ap-plication to cryptography. It discusses the arithmetic on hyperelliptic curves Cryptography and Elliptic Curves A Beginner's Guide Thomas R. Shemanske STUDENT MATHEMATICAL LIBRARY Volume 83. Modern Cryptography and Elliptic Curves A Beginner's Guide Thomas R. Shemanske STUDENT MATHEMATICAL LIBRARY Volume 83 American Mathematical Society Providence, Rhode Island 10.1090/stml/083. Editorial Board SatyanL.Devadoss EricaFlapan JohnStillwell(Chair) SergeTabachnikov. 3 Elliptic curve cryptography In order to encrypt messages using elliptic curves we mimic the scheme in Example 2. First of all Alice and Bob agree on an elliptic curve E over F q and a point P 2E(F q). As the discrete logarithm problem is easier to solve for groups whose order is composite, they will choose their curve such that n := jE(F q)j is prime. Suppose Alice wants to send a message M. elliptic curve cryptography included in the implementation. It is envisioned that implementations choosing to comply with this document will typically choose also to comply with its companion document, SEC 1 . It is intended to make a validation system available so that implementors can check compliance with this document - see the SECG website, www.secg.org, for further information. 1.3. Elliptic Curves: Number Theory and Cryptography. 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.
Elliptic Curves and Cryptography CHRIS ROHLICEK May 2, 2018 Introduction The National Institute of Standards and Technology (NIST) is an agency of the U.S. Department of Commerce whose job today includes the estab-lishment of standards for such practices as the encryption of government information. After Edward Snowden leaked a number of classiﬁed docu- ments from the NSA, the means by which. Unter Elliptic Curve Cryptography oder deutsch Elliptische-Kurven-Kryptografie versteht man asymmetrische Kryptosysteme, die Operationen auf elliptischen Kurven über endlichen Körpern verwenden. Diese Verfahren sind nur sicher, wenn diskrete Logarithmen in der Gruppe der Punkte der elliptischen Kurve nicht effizient berechnet werden können. Jedes Verfahren, das auf dem diskreten Logarithmus in endlichen Körpern basiert, wie z. B. der Digital Signature Algorithm, das Elgamal. Elliptic Curve Cryptography 5 3.1. Elliptic Curve Fundamentals 5 3.2. Elliptic Curves over the Reals 5 3.3. Elliptic Curves over Finite Fields 8 3.4. Computing Large Multiples of a Point 9 3.5. Elliptic Curve Discrete Logarithm Problem 10 3.6. Elliptic Curve Di e-Hellman (ECDH) 10 3.7. ElGamal System on Elliptic Curves 11 3.8. Elliptic Curve Digital Signature Algorithm 11 3.9. Attacks on ECC. Introduction What is an elliptic curve Cryptography Real world An elliptic curve y2 = x3 + 2x2 − 3x Two points P = (−3,0) and Q = (−1,2). give a new point R = (3,6)
Elliptic curve cryptography largely relies on the algebraic structure of elliptic curves, usually over nite elds, and they are de ned in the following way. De nition 1.1 An elliptic curve Eis a curve (usually) of the form y2 = x3 + Ax+ B, where Aand Bare constant. This equation is called the Weierstrass equation, and we will use it through- out the paper . Let K be a eld. If A;B 2K, we say. Download Free PDF. Download Free PDF. A Tutorial on Elliptic Curve Cryptography (ECC) A Tutorial on Elliptic Curve Cryptography 2. Dinesh Dhadi. Download PDF. Download Full PDF Package . This paper. A short summary of this paper. 37 Full PDFs related to this paper. READ PAPER. A Tutorial on Elliptic Curve Cryptography (ECC) A Tutorial on Elliptic Curve Cryptography 2. Download. A Tutorial on. Elliptic curve cryptography (ECC) is an approach used for public key encryption that utilizes the mathematics behind elliptic curves in order to generate security between key pairs. Equations based on elliptic curves are relatively easy to perform but extremely difficult to reverse. In cryptography, this is a very valuable characteristic since it offers greater security while requiring less. tests, and in public-key cryptography. • Elliptic curve systems as applied to cryptography were first proposed in 1985 independently by Neal Koblitz from the University of Washington, and Victor Miller, who was then at IBM, Yorktown Heights. Códigos y Criptografía Francisco Rodríguez Henríquez Elliptic Curves • An elliptic curve over real numbers is defined as the set of points (x,y.
. Despite three NIST curves having been standardized, at the 128-bit security level or higher, the smallest curve size, secp256r1, is by far the most commonly used. Many servers seem to prefer the curves de ned over smaller elds. Weak keys. We observed signi cant numbers of non-related. Part 3: In the last part I will focus on the role of elliptic curves in cryptography. First, in chapter 5, I will give a few explicit examples of how elliptic curves can be used in cryptography. After that I will explain the most important attacks on the discrete logarithm problem. These include attacks on the discrete logarithm problem for general groups in chapter 6 and three attacks on this.
Elliptic Curve Cryptography and Government Backdoors Ben Schwennesen Duke University Math 89S (Mathematics of the Universe) Professor Hubert Bray April 24, 2016. Introduction For as long as humans have roamed the Earth, they have kept secrets. Further still, as long as secrets have been withheld, there have been people attempting to expose them. Continual advancements in technology have had. RFC 5639 ECC Brainpool Standard Curves & Curve Generation March 2010 over GF(p) together with a neutral element O and well-defined laws for addition and inversion define a group E(GF(p)) -- the group of GF(p) rational points on E. Typically, for cryptographic applications, an element G of prime order q is chosen in E(GF(p)). A comprehensive introduction to elliptic curve cryptography can be.
in cryptography and elliptic curve techniques were developed for factorization and primality testing. In the 1980s and 1990s, elliptic curves played an impor-tant role in the proof of Fermat's Last Theorem. The goal of the present book is to develop the theory of elliptic curves assuming only modest backgrounds in elementary number theory and in groups and ﬁelds, approximately what would. Workshop on Elliptic Curve Cryptography (ECC) About ECC. ECC is an annual workshops dedicated to the study of elliptic curve cryptography and related areas. Since the first ECC workshop, held 1997 in Waterloo, the ECC conference series has broadened its scope beyond elliptic curve cryptography and now covers a wide range of areas within modern cryptography. For instance, past ECC conferences. ELLIPTIC CURVE CRYPTOGRAPHY From the very beginning, you need to know better about Elliptic curve cryptography (ECC). So, Elliptic curve cryptography is a helpful strategy for cryptography and an alternative method from the well-known RSA method for securities. It is a wonderful way that people have been using for past years for public-key encryption by utilizing the mathematics behind. This is a graph of secp256k1's elliptic curve y 2 = x 3 + 7 over the real numbers. Note that because secp256k1 is actually defined over the field Z p, its graph will in reality look like random scattered points, not anything like this. secp256k1 refers to the parameters of the elliptic curve used in Bitcoin's public-key cryptography, and is defined in Standards for Efficient Cryptography (SEC. E cient Algorithms for Generating Elliptic Curves over Finite Fields Suitable for Use in Cryptography Vom Fachbereich Informatik der Technischen Universit at Darmstadt genehmigte Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr.rer.nat.) von Harald Baier aus Fulda (Hessen) Referenten: Prof. Dr. J. Buchmann Prof. Dr. G. K ohler Tag der Einreichung: 26.03.2002 Tag.
Elliptic Curve Cryptography (ECC) oﬀers smaller key sizes, faster computation, as well as memory, energy and bandwidth savings and is thus better suited for small devices. While RSA and ECC can be accelerated with dedicated cryp-tographic coprocessors such as those used in smart cards, coprocessors require additional hardware adding to the size and complexity of the devices. Therefore, they. AnIntroductiontoPairing-Based Cryptography Alfred Menezes Abstract. Bilinear pairings have been used to design ingenious protocols for such tasks as one-round three-party key agreement, identity-based encryption, and aggregate signatures. Suitable bilinear pairings can be constructed from the Tate pairing for specially chosen elliptic curves. This article gives an introduction to the protocols.
Elliptic-curve cryptography (ECC) is cryptography based on the algebraic structure of elliptic curves over finite fields. ECC requires smaller keys compared to non-ECC cryptography. Why should I consider ECC? One of the main benfits of ECC is a smaller key size which reduces storage and transmission requirements. With this reduced size, you increase the speed in using ECC. ECC can provide the. Elliptic Curve Cryptography is particularly useful in solving such problems. There are existing protocols, called key exchange protocols, which successfully do this, but not all key exchange protocols are made equal. Table 1 [NIS05] shows one of the most notable diﬀerences between elliptic curve protocols and protocols based on factoring or ﬁnite ﬁelds. The middle and right column give. Elliptic curve cryptography is the backbone behind bitcoin technology and other crypto currencies, especially when it comes to to protecting your digital ass.. Elliptic Curves An elliptic curve over a finite field has a finite number of points with coordinates in that finite field Given a finite field, an elliptic curve is defined to be a group of points (x,y) with x,y GF, that satisfy the following generalized Weierstrass equation: y2 + a 1 xy + a Elliptic curve cryptography, in essence, entails using the group of points on an elliptic curve as the underlying number system for public key cryptography. There are two main reasons for using elliptic curves as a basis for public key cryptosystems. The first reason is that elliptic curve based cryptosystems appear to provide better security than traditional cryptosystems for a given key size.
Since the appearance of the authors first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular Elliptic Curve Cryptography wird von modernen Windows-Betriebssystemen (ab Vista) unterstützt. Produkte der Mozilla Foundation (u. a. Firefox, Thunderbird) unterstützen ECC mit min. 256 Bit Key-Länge (P-256 aufwärts).. Die in Österreich gängigen Bürgerkarten (e-card, Bankomat- oder a-sign Premium Karte) verwenden ECC seit ihrer Einführung 2004/2005, womit Österreich zu den Vorreitern. Elliptic Curves in Cryptography Fall 2011 Textbook. Required: Elliptic Curves: Number Theory and Cryptography, 2nd edition by L. Washington. Online edition of Washington (available from on-campus computers; click here to set up proxies for off-campus access).; There is a problem with the Chapter 2 PDF in the online edition of Washington: most of the lemmas and theorems don't display correctly Read the latest articles of Journal of Number Theory at ScienceDirect.com, Elsevier's leading platform of peer-reviewed scholarly literatur Elliptic Curve Cryptography Discrete Logarithm Problem [ ECCDLP ] • Addition is simple P + P = 2P Multiplication is faster , it takes only 8 steps to compute 100P, using point doubling and add 1. P * 2 = 2P 2. P + 2P = 3P 3. 3P * 2 = 6P 4. 6P *2 = 12P 5. 12P * 2 =24 P 6. P + 24 P = 25 P 7. 25P * 2 = 50 P 8. 50P *2 = 100 P CYSINFO CYBER SECURITY MEETUP - 17TH SEPTEMBER 2016 13. Elliptic.
In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes Elliptic Curve Cryptography: ECDH and ECDSA. This post is the third in the series ECC: a gentle introduction. In the previous posts, we have seen what an elliptic curve is and we have defined a group law in order to do some math with the points of elliptic curves. Then we have restricted elliptic curves to finite fields of integers modulo a prime Elliptic Curves and Cryptography by Ian Blake, Gadiel Seroussi and Nigel Smart. This book is useful resource for those readers who have already understood the basic ideas of elliptic curve cryptography. This book discusses many important implementation details, for instance finite field arithmetic and efficient methods for elliptic curve operations. The book also gives a description of the. Risparmia su Cryptography. Spedizione gratis (vedi condizioni Elliptic Curves over the Reals TIplain theme Elliptic curves over the reals I Simple graphical representation I of the curve as well as I addition (and doubling) of points. f(x;y) = y2 (x3 + ax+ b) = g(y) h(x)= 0; a;b;x;y 2R I Preliminaries I The curve is symmetric to x-axis I Interesting: Nulls of cubic curve h(x) I Known: if for thediscriminant = 4 a3 + 27b2 of h holds
Elliptic curves over nite elds and applications to cryptography Erik Wallace May 29, 2018 1 Introduction These notes are not as complete or self contained as I would like. For further reading on elliptic curves, the following books are are recommended: \Rational points on elliptic curves by Silverman and Tate, \The arithmetic of elliptic curves by Silverman, \Elliptic curves by Husem oller. Cryptography and Elliptic curves Inna Lukyanenko March 26, 2007 1 / 38. Introduction to Cryptography Digital Signatures Finite ﬁelds Elliptic curves ECDSA Outline 1 Introduction to Cryptography 2.
RSA and Elliptic Curve Cryptography, are generally considered the most powerful cryptosystems that could provide a high level of security. However, RSA involves very intensive computational arithmetic with a key size of 1024-2048 bits. Therefore, ECC could be a feasible solution to provide a similar level of security with a smaller key size and lesser arithmetic computations. However, the. Elliptic Curve Cryptography has been a recent research area in the field of Cryptography. It provides higher level of security with lesser key size compared to other Cryptographic techniques. This paper provides an overview of ellipticcurves and their use in cryptography. The focus is on the performance advantages to be obtained by using elliptic curve cryptography instead of a traditional. 1.4 Elliptic Curve Encryption Elliptic curve cryptography can be used to encrypt plaintext messages, M, into ciphertexts.The plaintext message M is encoded into a point P M from the ﬁnite set of points in the elliptic group, E p(a,b).The ﬁrst step consists in choosing a generator point, G ∈ E p(a,b), such that the smallest value of n for which nG = O is a very large prime number
Elliptic curve cryptography in TLS, as speci ed in RFC 4492 , includes elliptic curve Di e-Hellman (ECDH) key exchange in two avours: xed-key key exchange with ECDH certi cates; and ephemeral ECDH key exchange using an RSA or ECDSA certi cate for authentication. While we focus our discussion on the ephemeral cipher suites providing perfect forward secrecy, our implementa-tion results are. Since Elliptic Curve cryptography is a relatively new phenomenon, research is still ongoing. There are many open questions which are currently being studied. Active areas of research include developing algorithms and/or modifying known ones to break current elliptic curve cryptosystems. One such algorithm is the Pollard-Rho algorithm that solves the Elliptic Curve Discrete Logarithm. Cryptography Based on Groups 2 1.2. What Types of Group are Used 6 1.3. What it Means in Practice 8 Chapter II. Finite Field Arithmetic 11 II. 1. Fields of Odd Characteristic 11 II.2. Fields of Characteristic Two 19 Chapter III. Arithmetic on an Elliptic Curve 29 111.1. General Elliptic Curves 30 111.2. The Group Law 31 111.3. Elliptic Curves over Finite Fields 34 111.4. The Division. Elliptic Curve Cryptography (ECC) is a public key cryptography developed independently by Victor Miller and Neal Koblitz in the year 1985. In Elliptic Curve Cryptography we will be using the curve equation of the form . y2 = x3 + ax + b (1) which is known as Weierstrass equation, where a and b are the constant with. 4a3 + 27b2 = 0 (2) 1.1 Mathematics in elliptic curve cryptography over finite.
Hyper-and-elliptic-curve cryptography Citation for published version (APA): Bernstein, D. J., & Lange, T. (2014). Hyper-and-elliptic-curve cryptography. (Cryptology ePrint Archive; Vol. 2014/379). IACR. Document status and date: Published: 01/01/2014 Document Version: Publisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document. In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all.
best methods for elliptic curve point multiplication on general curves. Keywords: elliptic curves, point multiplication, GLV method, isogenies. 1 Introduction Let E be an elliptic curve over a nite eld F q and let P;Q 2E(F q) have order r. The fundamental operations in elliptic curve cryptography are poin Elliptic curve cryptography Suppose we ﬁx an elliptic curve E a,b and choose a point g ∈ E a,b. Then we can consider the subgroup hgi (which is possibly the entire group if it happens to be cyclic). Using exponentiation in this group (which is simply multiplication in our additive notation), we can deﬁne analogs of Diﬃe-Hellman key exchange and related cryptosystems such as ElGamal. Elliptic curve cryptography (ECC) was discovered in 1985 by Neal Koblitz and Victor Miller. Elliptic curve cryptographic schemes are public-key mechanisms that provide the same functionality as RSA schemes. However, their security is based on the hardness of a different problem, namely the elliptic curve discrete logarithm problem (ECDLP). Currently the best algorithms known to solve the ECDLP.
In elliptic curve cryptography, the group used is the group of rational points on a given elliptic curve. This is how elliptic curve public key cryptography works. For Alice and Bob to communicate securely over an insecure network they can exchange a private key over this network in the following way: 1. A particular rational base point P is published in a public domain for use with a. Elliptic curve cryptography An elliptic curve E over a ﬁeld K is the set of solutions (x,y) ∈K ×K which satisfy the Weierstrass equation y2 +a 1xy +a3y = x3 +a2x2 +a4x +a6 where a1,a2,a3,a4,a6 ∈K and the curve discriminant is ∆ 6= 0; together with a point at inﬁnity denoted by O. If K is a ﬁeld of characteristic 2, then the curve is called a binary elliptic curve and there are two.
Elliptic curves actually received their names from their relation to so called elliptic integrals Z x2 x1 dx p x3 + ax + b Z x2 x1 xdx p x3 + ax + b that arise in the computation of the arc-length of ellipses Elliptic Curve Cryptography - An Implementation Tutorial 5 s = (3x J 2 + a) / (2y J) mod p, s is the tangent at point J and a is one of the parameters chosen with the elliptic curve If y. curves, elliptic curve cryptography, and two of the three types of Edwards curves. Then in Chapter 3, we build on the theoretical foundation of the previous Chapter to discuss binary Edwards curves. Next, in Chapter 4, we examine the practical considerations involved in applying that theory in cryptographic practice, including a discussion of the shortcomings of four recently proposed normal. Elliptic curve cryptography (ECC) [32,37] is increasingly used in practice to instantiate public-key cryptography protocols, for example implementing digital signatures and key agreement. More than 25 years after their introduction to cryptography, the practical bene ts of using elliptic curves are well-understood: they o er smaller key sizes  and more e cient implementations  at the. View elliptic curve cryptography.pdf from STEI 4097 at Institut Teknologi Bandung. Elliptic Curve Public Key Cryptography Why? ECC offers greater security for a given key size. Elliptic Curve View Elliptic Curve Cryptography(ECC).pdf from CSE 1011 at Vellore Institute of Technology. Inroduction Elliptic Curve Cryptography ECC • ECC was introduced by Victor Miller and Neal Koblitz i
Elliptic Curve Cryptography Public Key Algorithm Identifiers The algorithm field in the SubjectPublicKeyInfo structure [PKI] indicates the algorithm and any associated parameters for the ECC public key (see Section 2.2). Three algorithm identifiers are defined in this document: Turner, et al. Standards Track [Page 3] RFC 5480 ECC. Springer Professional Computing: Guide to Elliptic Curve Cryptography (PDF) Alfred J. Menezes, Darrel Hankerson, Scott Vanstone 0 Sterne. eBook Statt 160. 49 € 19. 135. 23 € Erschienen am 01.06.2006. Leider schon ausverkauft-80%. Das Praxisbuch zu QuarkXPress für Windows und Mac. Elliptic-curve cryptography ( ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC requires smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.  Elliptic curves are applicable for key agreement, digital signatures.