NationMaster - Encyclopedia: 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. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz^[1] and Victor S. Miller^[2] in 1985. A big random number is used to make a public-key pair. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... Neal Koblitz is a Professor of Mathematics in the University of Washington in the Department of Mathematics. ... Victor S. Miller (b. ... This article is about the year. ...

Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as, for instance, Lenstra elliptic curve factorization, but this use of elliptic curves is not usually referred to as "elliptic curve cryptography." Prime decomposition redirects here. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... 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. ...

Introduction

Public key cryptography is based on the creation of mathematical puzzles that are difficult to solve without certain knowledge about how they were created. The creator keeps that knowledge secret (the private key) and publishes the puzzle (the public key). The puzzle can then be used to scramble a message in a way that only the creator can unscramble. Early public key systems, such as the RSA algorithm, used products of two large prime numbers as the puzzle: a user picks two large random primes as his private key, and publishes their product as his public key. The difficulty of factoring ensures that no one else can derive the private key (i.e., the two prime factors) from the public one within a reasonable amount of time. However, due to recent progress in factoring, RSA public keys must now be thousands^{[citation needed]} of bits long to provide adequate security. SWAdair | Talk 06:28, 11 May 2005 (UTC) Categories: Possible copyright violations ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... ...

Another class of puzzle involves solving the equation a^b = c for b when a and c are known. Such equations involving real or complex numbers are easily solved using logarithms. However, in some large finite groups, finding solutions to such equations is quite difficult and is known as the discrete logarithm problem. Logarithms to various bases: is to base e, is to base , and is to base . ... In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. ...

An elliptic curve is a plane curve defined by an equation of the form In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

The set of points on such a curve (i.e., all solutions of the equation together with a point at infinity) can be shown to form an abelian group (with the point at infinity as identity element). If the coordinates x and y are chosen from a large finite field, the solutions form a finite abelian group. The discrete logarithm problem on such elliptic curve groups is believed to be more difficult than the corresponding problem in (the multiplicative group of nonzero elements of) the underlying finite field. Thus keys in elliptic curve cryptography can be chosen to be much shorter for a comparable level of security. (See: cryptographic key length) The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... In cryptography, the key size (alternatively key length) is a measure of the number of possible keys which can be used in a cipher. ...

As for other popular public key cryptosystems, no mathematical proof of difficulty has been published for ECC as of 2006. However, the U.S. National Security Agency has endorsed ECC technology by including it in its Suite B set of recommended algorithms. Although the RSA patent has expired, there are patents in force covering some aspects of ECC. 2006 is a common year starting on Sunday of the Gregorian calendar. ... â€œNSAâ€ redirects here. ... Suite B is a set of public key cryptography algorithms based on elliptic curve cryptography promulgated by the US National Security Agency as part of a US Government standard for securing sensitive-but-unclassified (SBU) information. ...

Mathematical introduction

Elliptic curves used in cryptography are typically defined over two types of finite fields: fields of odd characteristic ( $mathbb{F}_p$ , where

p > 3

is a large prime number) and fields of characteristic two ( $mathbb{F}_{2^m}$ ). When the distinction is not important we denote both of them as $mathbb{F}_q$ , where

q = p

q = 2 m

. In $mathbb{F}_p$ the elements are integers ( $0 le x < p$ ) which are combined using modular arithmetic. The case of $mathbb{F}_{2^m}$ is slightly more complicated (see finite field arithmetic for details): one obtains different representations of the field elements as bitstrings for each choice of irreducible binary polynomial

f (x)

of degree

m

. In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... Arithmetic in a finite field is different from standard integer arithmetic. ...

The set of all pairs of affine coordinates

(x, y)

for $x, y in mathbb{F}_q$ form the affine plane $mathbb{F}_q times mathbb{F}_q$ . An elliptic curve is the locus of points in the affine plane whose coordinates satisfy a certain cubic equation together with a point at infinity

O

(the point at which the locus in the projective plane intersects the line at infinity). In the case of characteristic p > 3 the defining equation of $E(mathbb{F}_p)$ can be written: In mathematics, an affine combination of vectors x1, ..., xn is a linear combination in which the sum of the coefficients is 1, thus: . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients are scalars in K. This concept is important...

where $a in mathbb{F}_p$ and $b in mathbb{F}_p$ are constants such that $4 a^3 + 27 b^2 ne 0$ . In the binary case the defining equation of $E(mathbb{F}_{2^m})$ can be written:

where $a in mathbb{F}_{2^m}$ and $b in mathbb{F}_{2^m}$ are constants and $b ne 0$ . Although the point at infinity

O

has no affine coordinates, it is convenient to represent it using a pair of coordinates which do not satisfy the defining equation, for example,

O = (0,0)

if $b ne 0$ and

O = (0,1)

otherwise. According to Hasse's theorem on elliptic curves the number of points on a curve is close to the size of the underlying field; more precisely: $(sqrt q - 1)^2 leq |E(mathbb{F}_q)| leq (sqrt q + 1)^2$ . In mathematics, Hasses theorem on elliptic curves bounds the number of points on an elliptic curve over a finite field. ...

The points on an elliptic curve form an abelian group $(E(mathbb{F}), +)$ with

O

, the distinguished point at infinity, playing the role of additive identity. In other words, given two points $P, Q in E(mathbb{F}_q)$ , there is a third point, denoted by $P+Q ,$ on $E(mathbb{F}_q)$ , and the following relations hold for all $P, Q, R in E(mathbb{F}_q)$ In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

We already specified how

O

is defined. If we define the negative of a point

P = (x, y)

to be

- P = (x, - y)

for $P in E(mathbb{F}_p)$ and

- P = (x, x + y)

for $P in E(mathbb{F}_{2^m})$ , we can define the addition operation as follows:

(Geometrically,

P + Q

is the inverse of the third point of intersection of the cubic with the line through

P

and

Q

(Geometrically,

2 P

is the inverse of the third point of intersection of the cubic with its tangent line at

P

Certicom's Online ECC Tutorial contains a Java applet that can be used to experiment with addition in different EC groups.

We already described the underlying field $mathbb{F}_q$ and the group of points of elliptic curve $E(mathbb{F}_q)$ but there is yet another mathematical structure commonly used in cryptography — a cyclic subgroup of $E(mathbb{F}_q)$ . For any point

G

the set In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

is a cyclic group. It is convenient to use the following notation:

0 G = O

1 G = G

2 G = G + G

3 G = G + G + G

, etc. The calculation of

k G

, where

k

is an integer and

G

is a point, is called scalar multiplication.

Cryptographic schemes

Since the (additive) cyclic group described above can be considered similar to the (multiplicative) group of powers of an integer

g

modulo prime

p

: $(g^0, g, g^2, g^3, g^4, ldots)$ , the problem of finding

k

given points

k G

and

G

is called the elliptic curve discrete logarithm problem (ECDLP). The assumed hardness of several problems related to the discrete logarithm in the subgroup of $E(mathbb{F}_q)$ allows cryptographic use of elliptic curves. Most of the elliptic curve cryptographic schemes are related to the discrete logarithm schemes which were originally formulated for usual modular arithmetic: In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. ...

Not all the DLP schemes should be ported to the elliptic curve domain. For example, the well known ElGamal encryption scheme was never standardized by official bodies and should not be directly used over an elliptic curve (the standard encryption scheme for ECC is called Elliptic Curve Integrated Encryption Scheme). The main reason is that although it is straightforward to convert an arbitrary message (of limited length) to an integer modulo

p

, it is not that simple to convert a bitstring to a point of a curve (it is not true that for every

x

there is a

y

such that $(x,y) in E(mathbb{F}_q)$ ). (Another factor is that ElGamal scheme is vulnerable to chosen-ciphertext attacks.) Elliptic Curve Diffie-Hellman (ECDH) is a key agreement protocol that allows two parties to estabilish a shared secret key over an insecure channel. ... Diffie-Hellman key exchange is a cryptographic protocol which allows two parties to agree on a secret key over an insecure communication channel. ... Elliptic Curve DSA (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which operates on elliptic curve groups. ... The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures. ... MQV (Menezes-Qu-Vanstone) is an authenticated protocol for key agreement based on the Diffie-Hellman scheme. ... MQV (Menezes-Qu-Vanstone) is an authenticated protocol for key agreement based on the Diffie-Hellman scheme. ... The ElGamal algorithm is an asymmetric key encryption algorithm for public key cryptography which is based on Diffie-Hellman key agreement. ... Integrated Encryption Scheme (IES) is a public-key encryption scheme which provides semantic security against an adversary who is allowed to use chosen-plaintext and chosen-ciphertext attacks. ...

Some believe that ECDLP-based cryptography is going to replace cryptography based on integer factorization (e.g., RSA) and finite-field cryptography (e.g., DSA). At the RSA Conference 2005, the National Security Agency (NSA) announced Suite B which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information. The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures. ... â€œNSAâ€ redirects here. ... Suite B is a set of public key cryptography algorithms based on elliptic curve cryptography promulgated by the US National Security Agency as part of a US Government standard for securing sensitive-but-unclassified (SBU) information. ...

Another major source of cryptographic applications of elliptic curves is bilinear operator (based on the Weil pairing or the Tate pairing) which allows, for example, to make efficient ID-based cryptography (see also The Pairing-Based Crypto Lounge). In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing (bilinear form, though with multiplicative notation) on the torsion subgroup of E. The name is for André Weil, who gave... ID-based cryptography (or identity based cryptography or identity based encryption) is a key authentication system in which the public key of a user is some unique information about the identity of the user (e. ...

Implementation considerations

Although the details of each particular elliptic curve scheme are described in the article referenced above some common implementation considerations are discussed here.

Domain parameters

To use ECC all parties must agree on all the elements defining the elliptic curve, that is domain parameters of the scheme. The field is defined by

p

in the prime case and the pair of

m

and

f

in the binary case. The elliptic curve is defined by the constants

a

and

b

used in its defining equation. Finally, the cyclic subgroup is defined by its generator (aka. base point)

G

. For cryptographic application the order of

G

, that is the smallest non-negative number

n

such that

n G = O

, must be prime. Since

n

is the size of a subgroup of $E(mathbb{F}_q)$ it follows from the Lagrange's theorem that the number $h = frac{|E|}{n}$ is integer. In cryptographic applications this number

h

, called cofactor, at least must be small ( $h le 4$ ) and, preferably,

h = 1

. Let us summarize: in the prime case the domain parameters are

(p, a, b, G, n, h)

and in the binary case they are

(m, f, a, b, G, n, h)

. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ...

Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use.

The generation of domain parameters is not usually done by each participant since this involves counting the number of points on a curve which is time-consuming and troublesome to implement. As a result several standard bodies published domain parameters of elliptic curves for several common field sizes:

If one (despite the said above) wants to build his own domain parameters he should select the underlying field and then use one of the following strategies to find a curve with appropriate (i.e., near prime) number of points using one of the following methods:

Several classes of curves are weak and shall be avoided: Schoofs algorithm, first described by R. Schoof in 1985, allows one to calculate the number of points on an elliptic curve over a finite field and is used mostly in elliptic curve cryptography. ... The Schoof-Elkies-Atkin algorithm (SEA) is an algorithm used for finding the order of or calculate the number of points on an elliptic curve over a finite field. ...

Key sizes

Since all the fastest known algorithms that allow to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need $O(sqrt{n})$ steps, it follows that the size of the underlying field shall be roughly twice the security parameter. For example, for 128-bit security one needs a curve over $mathbb{F}_q$ , where $q approx 2^{256}$ . This can be contrasted with finite-field cryptography (e.g., DSA) which requires^[12] 3072-bit public keys and 256-bit private keys, and integer factorization cryptography (e.g., RSA) which requires 3072-bit public and private keys. The hardest ECC scheme (publicly) broken to date had a 109-bit key (that is about 55 bits of security). For the prime field case, it was broken near the beginning of 2003 using over 10,000 Pentium class PCs running continuously for over 540 days (see [2]). For the binary field case, it was broken in April 2004 using 2600 computers for 17 months (see [3]). This does not adequately cite its references or sources. ... Pollards rho algorithm for logarithms is an algorithm for solving the discrete logarithm problem analogous to Pollards rho algorithm for solving the Integer factorization problem. ... The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures. ... In cryptography, RSA is an algorithm for public-key encryption. ... Year 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... This article does not cite any references or sources. ...

Projective coordinates

A close examination of the addition rules shows that in order to add two points one needs not only several additions and multiplications in $mathbb{F}_q$ but also an inversion operation. The inversion (for given $x in mathbb{F}_q$ find $y in mathbb{F}_q$ such that

x y = 1

) is one to two orders of magnitude slower^[13] than multiplication. Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates

(X, Y, Z)

using the following relation: $x = frac{X}{Z}$ , $y = frac{Y}{Z}$ ; in the Jacobian system a point is also represented with three coordinates

(X, Y, Z)

, but a different relation is used: $x = frac{X}{Z^2}$ , $y = frac{Y}{Z^3}$ ; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations

(X, Y, Z, a Z 4)

; and in the Chudnovsky Jacobian system five coordinates are used

(X, Y, Z, Z 2, Z 3)

. Note that there are may be different naming conventions, for example, IEEE P1363-2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used.^[14] IEEE P1363 is an Institute of Electrical and Electronics Engineers (IEEE) standardization project for public key cryptography. ...

Fast reduction (NIST curves)

Reduction modulo

p

(which is needed for addition and multiplication) can be executed much faster if the prime

p

is a pseudo-Mersenne prime that is $p approx 2^d$ , for example,

p = 2 521 - 1

p = 2 256 - 2 32 - 2 9 - 2 8 - 2 7 - 2 6 - 2 4 - 1

. Compared to Barrett reduction there can be an order of magnitude speed-up.^[15] The curves over $mathbb{F}_p$ with pseudo-Mersenne

p

are recommended by NIST. Yet another advantage of the NIST curves is the fact that they use

a = - 3

which improves addition in Jacobian coordinates. In mathematics, a Mersenne number is a number that is one less than a power of two. ...

NIST-Recommended Elliptic Curves

NIST recommends 15 elliptic curves. Specifically, FIPS 186-2 has 10 recommended finite fields. There are 5 prime fields $mathbb{F}_p$ for

p 192

p 224

p 256

p 284

and

p 521

. For each of the prime fields one randomly selected elliptic curve is recommended. There five binary fields $mathbb{F}_2^m$ for

2163

2233

2283

2409

, and

2571

. For each of the binary fields one randomly selected elliptic curve and one Koblitz curve was selected. Thus 5 prime curves and 10 binary curves. The curves were chosen for optimal security and implementation efficiency.^[16]

Side-channel attacks

Unlike DLP systems (where it is possible to use the same procedure for squaring and multiplication) the EC addition is significantly different for doubling (

P = Q

) and general addition ( $P ne Q$ ). Consequently, it is important to counteract side channel attacks (e.g., timing and simple power analysis attacks) using, for example, fixed pattern window (aka. comb) methods^[17] (note that this does not increase the computation time). In cryptography, a side channel attack is any attack based on information gained from the physical implementation of a cryptosystem, rather than theoretical weaknesses in the algorithms (compare cryptanalysis). ...

Patents

At least one ECC scheme (ECMQV) and some implementation techniques are covered by patents. Uncertainty about the availability of unencumbered ECC has limited the acceptance of ECC. Patent-related uncertainty around the Elliptic Curve Cryptography (ECC) is one of the main factors limiting its wide acceptance, for example, the OpenSSL team has accepted the ECC patch only in 2005 (version 0. ...

Example of Useability

See also Diffie-Hellman key exchange. Diffie-Hellman (D-H) key exchange is a cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. ...

Alice and Bob agrees on using a public elliptic curve $E(a,b,p)~$ (meaning they both choose to use the same $y^2~mod~p = (x^3+ax+b)~mod~p~$ elliptic curve). They also agrees on using one point $P~$ of the curve. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as cryptography and physics. ...

Privately, Alice choose an integer $d_A~$ and Bob an integer $d_B~$ . Alice sends to Bob the point $d_A P~$ and Bob sends Alice $d_B P~$ . Each one can then find out $d_A(d_B P)=(d_A d_B) P~$ which is also a point on the curve, and it is their privately-shared key, used to encrypt data.

If Eve has been listening to their communications, she knows $E(a,b,p), P, d_A P, d_B P~$ . To be able to find out what $d_A d_B P~$ is, she would have to know what $d_A~$ is, knowing $P~$ and $d_A P~$ . But if the numbers are very large, we do not know any efficient method to do that in a reasonable time.

Implementations

Open source

Proprietary/commercial

The Cryptographic Application Programming Interface (also known variously as CryptoAPI, Microsoft Cryptography API, or simply CAPI) is an application programming interface included with Microsoft Windows operating systems that provides services to enable developers to secure Windows-based applications using cryptography. ... Windows Vista is a line of graphical operating systems used on personal computers, including home and business desktops, notebook computers, Tablet PCs, and media centers. ... Windows Server 2008 is the name of the next server operating system from Microsoft. ... The Microsoft . ... Sun Javaâ„¢ System Web Server (formerly Sun ONE Web Server, before that iPlanet Web Server, and before that Netscape Enterprise Server) is a web server designed for medium and large business applications. ... Java Platform, Standard Edition or Java SE (formerly known up to version 5. ... Java Card refers to a technology that allows small Java-based applications (applets) to be run securely on smart cards and similar devices. ...

References

See also

Categories: All articles with unsourced statements | Articles with unsourced statements since September 2007 | Cryptography | Asymmetric-key cryptosystems | Finite fields In cryptography, the Standards for Efficient Cryptography Group (SECG) is an international consortium founded by Certicom in 1998. ... A big random number is used to make a public-key pair. ... Céilí (Irish reformed spelling), or Ceilidh (Scottish and older Gaelic spelling), pronounced Kay-Lee in either case, is the traditional Gaelic social dance in Ireland and Scotland. ... The Cramer-Shoup system is an asymmetric key encryption algorithm for public key cryptography, and was the first efficient scheme proven to be secure against adaptive chosen ciphertext attack using standard cryptographic assumptions. ... Diffie-Hellman (D-H) key exchange is a cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. ... The Digital Signature Algorithm (DSA) is a United States Federal Government standard or FIPS for digital signatures. ... Elliptic Curve Diffie-Hellman (ECDH) is a key agreement protocol that allows two parties to estabilish a shared secret key over an insecure channel. ... Elliptic Curve DSA (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which operates on elliptic curve groups. ... Encrypted Key Exchange is an authentication protocol which uses two messages to establish a key, and then verifies a match in two transmissions. ... The ElGamal algorithm is an asymmetric key encryption algorithm for public key cryptography which is based on Diffie-Hellman key agreement. ... The ElGamal Signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms. ... In cryptography, GMR is a digital signature algorithm named after its inventors Shafi Goldwasser, Silvio Micali and Ron Rivest. ... Integrated Encryption Scheme (IES) is a public-key encryption scheme which provides semantic security against an adversary who is allowed to use chosen-plaintext and chosen-ciphertext attacks. ... In cryptography, a Lamport signature or Lamport one-time signature scheme is a method for constructing a digital signature. ... MQV (Menezes-Qu-Vanstone) is an authenticated protocol for key agreement based on the Diffie-Hellman scheme. ... NTRUEncrypt, also known as the NTRU encryption algorithm, is an asymmetric key encryption algorithm for public key cryptography. ... NTRUSign, also known as the NTRU Signature Algorithm, is a public key cryptography digital signature algorithm based on the GGH signature scheme. ... The Paillier cryptosystem is an asymmetric algorithm for public key cryptography, invented by Pascal Paillier in 1999. ... The Rabin cryptosystem is an asymmetric cryptographic technique, which like RSA is based on the difficulty of factorization. ... This article is about an algorithm for public-key encryption. ... A Schnorr signature is a digital signature scheme based on discrete logarithms. ... SPEKE (Simple Password Exponential Key Exchange) is a cryptographic method for password-authenticated key agreement. ... The Secure Remote Password Protocol (SRP) is a password-authenticated key agreement protocol which allows a user to authenticate herself to a server, which is resistant to dictionary attacks mounted by an eavesdropper, and does not require a trusted third party. ... In cryptography, XTR is an algorithm for public-key encryption. ... In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. ... In cryptography, the RSA problem is the task of finding eth roots modulo a composite number N whose factors are not known. ... CRYPTREC is the Cryptography Research and Evaluation Committee set up by the Japanese Government to evaluate and recommend cryptographic techniques for government and industrial use. ... IEEE P1363 is an Institute of Electrical and Electronics Engineers (IEEE) standardization project for public key cryptography. ... NESSIE (New European Schemes for Signatures, Integrity and Encryption) was a European research project funded from 2000–2003 to identify secure cryptographic primitives. ... Suite B is a set of public key cryptography algorithms based on elliptic curve cryptography promulgated by the US National Security Agency as part of a US Government standard for securing sensitive-but-unclassified (SBU) information. ... A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a signature in digital, rather than written, form. ... In public-key cryptography, a public key fingerprint is a string of bytes used to identify or validate a longer public key. ... Diagram of a public key infrastructure In cryptography, a public key infrastructure (PKI) is an arrangement that binds public keys with respective user identities by means of a certificate authority (CA). ... In cryptography, a web of trust is a concept used in PGP, GnuPG, and other OpenPGP-compatible systems to establish the authenticity of the binding between a public key and a user. ... In cryptography, the key size (alternatively key length) is the size of the digits used to create an encrypted text; it is therefore also a measure of the number of possible keys which can be used in a cipher, and the number of keys which must be tested to break... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek ÎºÏÏ…Ï€Ï„ÏŒÏ‚ kryptÃ³s hidden, and the verb Î³ÏÎ¬Ï†Ï‰ grÃ¡fo write or Î»ÎµÎ³ÎµÎ¹Î½ legein to speak) is the study of message secrecy. ... The history of cryptography begins thousands of years ago. ... Cryptanalysis (from the Greek kryptÃ³s, hidden, and analÃ½ein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ... This article is intended to be an analytic glossary, or alternatively, an organized collection of annotated pointers. ... This article does not cite any references or sources. ... Encryption Decryption In cryptography, a block cipher is a symmetric key cipher which operates on fixed-length groups of bits, termed blocks, with an unvarying transformation. ... The operation of the keystream generator in A5/1, a LFSR-based stream cipher used to encrypt mobile phone conversations. ... A big random number is used to make a public-key pair. ... In cryptography, a cryptographic hash function is a hash function with certain additional security properties to make it suitable for use as a primitive in various information security applications, such as authentication and message integrity. ... A cryptographic message authentication code (MAC) is a short piece of information used to authenticate a message. ... A cryptographically secure pseudo-random number generator (CSPRNG) is a pseudo-random number generator (PRNG) with properties that make it suitable for use in cryptography. ...

Contents

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