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Curve448
In cryptography, Curve448 or Curve448-Goldilocks is an elliptic curve cryptography, elliptic curve potentially offering 224 bits of security and designed for use with the elliptic-curve Diffie–Hellman (ECDH) key agreement scheme. Developed by Mike Hamburg of Rambus Cryptography Research, Curve448 allows fast performance compared with other proposed curves with comparable security. The reference implementation is available under an MIT license. The curve was favored by the Internet Research Task Force Crypto Forum Research Group (IRTF CFRG) for inclusion in Transport Layer Security (TLS) standards along with Curve25519. In 2017, NIST announced that Curve25519 and Curve448 would be added to "Special Publication 800-186", which specifies approved elliptic curves for use by the US Federal Government. A 2019 draft oFIPS 186-5confirms this claim. Both are described in . The name X448 is used for the DH function. Mathematical properties Hamburg chose the Solinas prime, Solinas trinom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve25519
In cryptography, Curve25519 is an elliptic curve used in elliptic-curve cryptography (ECC) offering 128 bits of security (256-bit key size) and designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme. It is one of the fastest curves in ECC, and is not covered by any known patents. The reference implementation is public domain software. The original Curve25519 paper defined it as a Diffie–Hellman (DH) function. Daniel J. Bernstein has since proposed that the name Curve25519 be used for the underlying curve, and the name X25519 for the DH function. Mathematical properties The curve used is y^2 = x^3 + 486662x^2 + x, a Montgomery curve, over the prime field defined by the prime number 2^ - 19, and it uses the base point x = 9. This point generates a cyclic subgroup whose order is the prime 2^ + 27742317777372353535851937790883648493, this subgroup has a co-factor of 8, meaning the number of elements in the subgroup is 1/8 that of the elliptic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solinas Prime
In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form f(2^m), where f(x) is a low-degree polynomial with small integer coefficients. These primes allow fast modular reduction algorithms and are widely used in cryptography. They are named after Jerome Solinas. This class of numbers encompasses a few other categories of prime numbers: * Mersenne primes, which have the form 2^k-1, * Crandall or pseudo-Mersenne primes, which have the form 2^k-c for small odd c. Modular Reduction Algorithm Let f(t) = t^d - c_t^ - ... - c_0 be a monic polynomial of degree d with coefficients in \mathbb and suppose that p = f(2^m) is a Solinas prime. Given a number n , shift right one position, injecting 0 on the left and adding (component-wise) the output value times the vector _0,...,c_/math> at each step (see for details). Let X_ be the integer in the jth register on the ith step and note that the first row of X is given by (X_) = _0,...,c_/math>. Then if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security ( data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synonymo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
US Federal Government
The federal government of the United States (U.S. federal government or U.S. government) is the national government of the United States, a federal republic located primarily in North America, composed of 50 states, a city within a federal district (the city of Washington in the District of Columbia, where most of the federal government is based), five major self-governing territories and several island possessions. The federal government, sometimes simply referred to as Washington, is composed of three distinct branches: legislative, executive, and judicial, whose powers are vested by the U.S. Constitution in the Congress, the president and the federal courts, respectively. The powers and duties of these branches are further defined by acts of Congress, including the creation of executive departments and courts inferior to the Supreme Court. Naming The full name of the republic is "United States of America". No other name appears in the Constitution, and this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Software
Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work. At the lowest programming level, executable code consists of machine language instructions supported by an individual processor—typically a central processing unit (CPU) or a graphics processing unit (GPU). Machine language consists of groups of binary values signifying processor instructions that change the state of the computer from its preceding state. For example, an instruction may change the value stored in a particular storage location in the computer—an effect that is not directly observable to the user. An instruction may also invoke one of many input or output operations, for example displaying some text on a computer screen; causing state changes which should be visible to the user. The processor executes the instructions in the order they are provided, unless it is instructed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nothing-up-my-sleeve Number
In cryptography, nothing-up-my-sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. They are used in creating cryptographic functions such as hashes and ciphers. These algorithms often need randomized constants for mixing or initialization purposes. The cryptographer may wish to pick these values in a way that demonstrates the constants were not selected for a nefarious purpose, for example, to create a backdoor to the algorithm. These fears can be allayed by using numbers created in a way that leaves little room for adjustment. An example would be the use of initial digits from the number as the constants. Using digits of millions of places after the decimal point would not be considered trustworthy because the algorithm designer might have selected that starting point because it created a secret weakness the designer could later exploit. Digits in the positional representations of real numbers such as , ''e'', and irration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwards Curve
In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form. Definition The equation of an Edwards curve over a field ''K'' which does not have characteristic 2 is: : x^2 + y^2 = 1 + d x^2 y^2 \, for some scalar d\in K\setminus\. Also the following form with parameters ''c'' and ''d'' is called an Edwards curve: : x^2 + y^2 = c^2(1 + dx^2 y^2) \, where ''c'', ''d'' ∈ ''K'' with ''cd''(1 − ''c''4·''d'') ≠ 0. Every Edwards curve is birationally equivalent to an elliptic curve in Montgomery form, and thus admits an algebraic group law once one chooses a point to serve as a neutral el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karatsuba Multiplication
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. Knuth D.E. (1969) ''The Art of Computer Programming. v.2.'' Addison-Wesley Publ.Co., 724 pp. It is a divide-and-conquer algorithm that reduces the multiplication of two ''n''-digit numbers to three multiplications of ''n''/2-digit numbers and, by repeating this reduction, to at most n^\approx n^ single-digit multiplications. It is therefore asymptotically faster than the traditional algorithm, which performs n^2 single-digit products. For example, the traditional algorithm requires (210)2 = 1,048,576 single-digit multiplications to multiply two 1024-digit numbers (''n'' = 1024 = 210), whereas the Karatsuba algorithm requires 310 = 59,049 thus being ~17.758 times faster. The Karatsuba algorithm was the first multiplication algorithm asymptotically faster than the quadratic "grade school" algorithm. The Toom–Cook algorithm (1963) is a fas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden-ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curves
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.Commercial National Security Algorithm Suite and Quantum Computing FAQ U.S. National Security Agency, January 2016. Elliptic curves are applicable for , s, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transport Layer Security
Transport Layer Security (TLS) is a cryptographic protocol designed to provide communications security over a computer network. The protocol is widely used in applications such as email, instant messaging, and voice over IP, but its use in securing HTTPS remains the most publicly visible. The TLS protocol aims primarily to provide security, including privacy (confidentiality), integrity, and authenticity through the use of cryptography, such as the use of certificates, between two or more communicating computer applications. It runs in the presentation layer and is itself composed of two layers: the TLS record and the TLS handshake protocols. The closely related Datagram Transport Layer Security (DTLS) is a communications protocol providing security to datagram-based applications. In technical writing you often you will see references to (D)TLS when it applies to both versions. TLS is a proposed Internet Engineering Task Force (IETF) standard, first defined in 1999, and the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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