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# Elliptic-curve cryptography

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. [1]

95 relations: Abelian group, Algebraic structure, Algebraic variety, Algorithm, Andreas Antonopoulos, Baby-step giant-step, Barrett reduction, Bitwise operation, Bruce Schneier, Classified information in the United States, Computational hardness assumption, Counting points on elliptic curves, Cryptocurrency, Cryptographically secure pseudorandom number generator, Curve25519, Daniel J. Bernstein, Differential fault analysis, Diffie–Hellman key exchange, Digital signature, Digital Signature Algorithm, Discrete logarithm, Divisor (algebraic geometry), DNSCurve, Doubling-oriented Doche–Icart–Kohel curve, Dual EC DRBG, ECC patents, EdDSA, Edward Snowden, Edwards curve, Elliptic curve, Elliptic Curve Digital Signature Algorithm, Elliptic curve point multiplication, Elliptic-curve Diffie–Hellman, Encryption, Federal government of the United States, Finite field, Hessian form of an elliptic curve, Homomorphic signatures for network coding, ID-based encryption, Identity element, IEEE P1363, Implicit certificate, Integer factorization, Integrated Encryption Scheme, Jacobian curve, Key size, Key-agreement protocol, Kleptography, Lagrange's theorem (group theory), Lecture Notes in Computer Science, ... Expand index (45 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Andreas Antonopoulos

Andreas M. Antonopoulos (born 1972) is a Greek-British bitcoin advocate.

## Baby-step giant-step

In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm.

## Barrett reduction

In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett.

## Bitwise operation

In digital computer programming, a bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits.

## Bruce Schneier

Bruce Schneier (born January 15, 1963, is an American cryptographer, computer security professional, privacy specialist and writer. He is the author of several books on general security topics, computer security and cryptography. Schneier is a fellow at the Berkman Center for Internet & Society at Harvard Law School, a program fellow at the New America Foundation's Open Technology Institute. He has been working for IBM since they acquired Resilient Systems where Schneier was CTO. He is also a contributing writer for The Guardian news organization.

## Classified information in the United States

The United States government classification system is established under Executive Order 13526, the latest in a long series of executive orders on the topic.

## Computational hardness assumption

In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where efficiently typically means "in polynomial time").

## Counting points on elliptic curves

An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.

## Cryptocurrency

A cryptocurrency (or crypto currency) is digital asset designed to work as a medium of exchange that uses strong cryptography to secure financial transactions, control the creation of additional units, and verify the transfer of assets.

## Cryptographically secure pseudorandom number generator

A cryptographically secure pseudo-random number generator (CSPRNG) or cryptographic pseudo-random number generator (CPRNG) is a pseudo-random number generator (PRNG) with properties that make it suitable for use in cryptography.

## Curve25519

In cryptography, Curve25519 is an elliptic curve offering 128 bits of security and designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme.

## Daniel J. Bernstein

Daniel Julius Bernstein (sometimes known simply as djb; born October 29, 1971) is a German-American mathematician, cryptologist, and programmer.

## Differential fault analysis

Differential fault analysis (DFA) is a type of side channel attack in the field of cryptography, specifically cryptanalysis.

## Diffie–Hellman key exchange

Diffie–Hellman key exchange (DH)Synonyms of Diffie–Hellman key exchange include.

## Digital signature

A digital signature is a mathematical scheme for presenting the authenticity of digital messages or documents.

## Digital Signature Algorithm

The Digital Signature Algorithm (DSA) is a Federal Information Processing Standard for digital signatures.

## Discrete logarithm

In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.

## Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties.

## DNSCurve

DNSCurve is a proposed new secure protocol for the Domain Name System (DNS), designed by Daniel J. Bernstein.

## Doubling-oriented Doche–Icart–Kohel curve

In mathematics, the doubling-oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written.

## Dual EC DRBG

Dual_EC_DRBG (Dual Elliptic Curve Deterministic Random Bit Generator) is an algorithm that was presented as a cryptographically secure pseudorandom number generator (CSPRNG) using methods in elliptic curve cryptography.

## ECC patents

Patent-related uncertainty around elliptic curve cryptography (ECC), or ECC patents, is one of the main factors limiting its wide acceptance.

## EdDSA

In public-key cryptography, Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based on Twisted Edwards curves.

## Edward Snowden

Edward Joseph Snowden (born June 21, 1983) is an American computer professional, former Central Intelligence Agency (CIA) employee, and former contractor for the United States government who copied and leaked classified information from the National Security Agency (NSA) in 2013 without authorization.

x^2+y^2.

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Elliptic Curve Digital Signature Algorithm

In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.

## Elliptic curve point multiplication

Elliptic curve point multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly.

## Elliptic-curve Diffie–Hellman

Elliptic-curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel.

## Encryption

In cryptography, encryption is the process of encoding a message or information in such a way that only authorized parties can access it and those who are not authorized cannot.

## Federal government of the United States

The federal government of the United States (U.S. federal government) is the national government of the United States, a constitutional republic in North America, composed of 50 states, one district, Washington, D.C. (the nation's capital), and several territories.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Hessian form of an elliptic curve

In geometry, the Hessian curve is a plane curve similar to folium of Descartes.

## Homomorphic signatures for network coding

Network coding has been shown to optimally use bandwidth in a network, maximizing information flow but the scheme is very inherently vulnerable to pollution attacks by malicious nodes in the network.

## ID-based encryption

ID-based encryption, or identity-based encryption (IBE), is an important primitive of ID-based cryptography.

## Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

## IEEE P1363

IEEE P1363 is an Institute of Electrical and Electronics Engineers (IEEE) standardization project for public-key cryptography.

## Implicit certificate

In cryptography, implicit certificates are a variant of public key certificate, such that a public key can be reconstructed from any implicit certificate, and is said then to be implicitly verified, in the sense that the only party who can know the associated private key is the party identified in the implicit certificate.

## Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

## Integrated Encryption Scheme

Integrated Encryption Scheme (IES) is a hybrid encryption scheme which provides semantic security against an adversary who is allowed to use chosen-plaintext and chosen-ciphertext attacks.

## Jacobian curve

In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation).

## Key size

In cryptography, key size or key length is the number of bits in a key used by a cryptographic algorithm (such as a cipher).

## Key-agreement protocol

In cryptography, a key-agreement protocol is a protocol whereby two or more parties can agree on a key in such a way that both influence the outcome.

## Kleptography

Kleptography is the study of stealing information securely and subliminally and it was introduced by Adam Young and Moti Yung in the Proceedings of Advances in Cryptology—Crypto '96.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Lecture Notes in Computer Science

Springer Lecture Notes in Computer Science (LNCS) is a series of computer science books published by Springer Science+Business Media (formerly Springer-Verlag) since 1973.

## Lenstra elliptic-curve factorization

The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.

## Localization of a category

In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms.

## Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two.

## Montgomery curve

In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987.

## MQV

MQV (Menezes–Qu–Vanstone) is an authenticated protocol for key agreement based on the Diffie–Hellman scheme.

## National Institute of Standards and Technology

The National Institute of Standards and Technology (NIST) is one of the oldest physical science laboratories in the United States.

## National Security Agency

The National Security Agency (NSA) is a national-level intelligence agency of the United States Department of Defense, under the authority of the Director of National Intelligence.

## Neal Koblitz

Neal I. Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington.

## NSA Suite B Cryptography

NSA Suite B Cryptography is a set of cryptographic algorithms promulgated by the National Security Agency as part of its Cryptographic Modernization Program.

## Object identifier

In computing, object identifiers or OIDs are an identifier mechanism standardized by the International Telecommunications Union (ITU) and ISO/IEC for naming any object, concept, or "thing" with a globally unambiguous persistent name.

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

## Pairing-based cryptography

Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group with a mapping e:G_1 \times G_2 \to G_T to construct or analyze cryptographic systems.

## Plane curve

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane.

## PlayStation 3

The PlayStation 3 (PS3) is a home video game console developed by Sony Computer Entertainment.

## Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

## Pollard's rho algorithm for logarithms

Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem.

## Power analysis

In cryptography, power analysis is a form of side channel attack in which the attacker studies the power consumption of a cryptographic hardware device (such as a smart card, tamper-resistant "black box", or integrated circuit).

## Proxy re-encryption

Proxy re-encryption schemes are cryptosystems which allow third parties (proxies) to alter a ciphertext which has been encrypted for one party, so that it may be decrypted by another.

## Public-key cryptography

Public-key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner.

## Quantum computing

Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement.

## Quantum cryptography

Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks.

## RSA (cryptosystem)

RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.

## RSA Security

RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company.

## Schnorr signature

In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm.

## Schoof's algorithm

Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.

## Schoof–Elkies–Atkin algorithm

The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field.

## SECG

In cryptography, the Standards for Efficient Cryptography Group (SECG) is an international consortium founded by Certicom in 1998.

## Security level

In cryptography, security level is a measure of the strength that a cryptographic primitive &mdash; such as a cipher or hash function &mdash; achieves.

## Shor's algorithm

Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization formulated in 1994.

## Side-channel attack

In computer security, a side-channel attack is any attack based on information gained from the implementation of a computer system, rather than weaknesses in the implemented algorithm itself (e.g. cryptanalysis and software bugs).

## Signcryption

In cryptography, signcryption is a public-key primitive that simultaneously performs the functions of both digital signature and encryption.

## Slashdot

Slashdot (sometimes abbreviated as /.) is a social news website that originally billed itself as "News for Nerds.

## Smart card

A smart card, chip card, or integrated circuit card (ICC), is any pocket-sized card that has embedded integrated circuits.

## Supersingular isogeny key exchange

Supersingular isogeny Diffie–Hellman key exchange (SIDH) is a post-quantum cryptographic algorithm used to establish a secret key between two parties over an otherwise insecure communications channel.

## Tate pairing

In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by and extended by.

## The New York Times

The New York Times (sometimes abbreviated as The NYT or The Times) is an American newspaper based in New York City with worldwide influence and readership.

## Toffoli gate

In logic circuits, the Toffoli gate (also CCNOT gate), invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any reversible circuit can be constructed from Toffoli gates.

## Tripling-oriented Doche–Icart–Kohel curve

The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve.

## Twisted Edwards curve

In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008.

## Twisted Hessian curves

In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic.

## Twists of curves

In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny.

## Victor S. Miller

Victor Saul Miller (born 3 March 1947 in Brooklyn, New York) is an American mathematician at the Center for Communications Research (CCR) of the Institute for Defense Analyses in Princeton, New Jersey, U.S. He received his A.B. in mathematics from Columbia University in 1968, and his Ph.D. in mathematics from Harvard University in 1975.

## Weil pairing

In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity.

## Weil restriction

In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

## Wired (magazine)

Wired is a monthly American magazine, published in print and online editions, that focuses on how emerging technologies affect culture, the economy, and politics.

## References

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