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SWIFFT
In cryptography, SWIFFT is a collection of provably secure hash functions. It is based on the concept of the fast Fourier transform (FFT). SWIFFT is not the first hash function based on FFT, but it sets itself apart by providing a mathematical proof of its security. It also uses the LLL basis reduction algorithm. It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/ ideal lattices in the ''worst case''. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other cryptographic hash functions. Unlike many other provably secure hash functions, the algorithm is quite fast, yielding a throughput of 40Mbit/s on a 3.2 GHz Intel Pentium 4. Although SWIFFT satisfies many desirable cryptographic and statistical properties, it was not designed to be an "all-purpose" cryptographic hash function. For example, it is not a pseudorandom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Lattice Cryptography
In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Vadim Lyubashevsky.Lattice-Based Identification Schemes Secure Under Active Attacks In ''Proceedings of the Practice and theory in public key cryptography , 11th international conference on Public key cryptography'', 2008. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient. Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example, cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign. Ideal lattices also form the basis for quantum computer attack resistant cryptography b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provably Secure Cryptographic Hash Function
In cryptography, cryptographic hash functions can be divided into two main categories. In the first category are those functions whose designs are based on mathematical problems, and whose security thus follows from rigorous mathematical proofs, complexity theory and formal reduction. These functions are called Provably Secure Cryptographic Hash Functions. To construct these is very difficult, and few examples have been introduced. Their practical use is limited. In the second category are functions which are not based on mathematical problems, but on an ad-hoc constructions, in which the bits of the message are mixed to produce the hash. These are then believed to be hard to break, but no formal proof is given. Almost all hash functions in widespread use reside in this category. Some of these functions are already broken, and are no longer in use. ''See'' Hash function security summary. Types of security of hash functions Generally, the ''basic'' security of cryptographic hash f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provably Secure Cryptographic Hash Function
In cryptography, cryptographic hash functions can be divided into two main categories. In the first category are those functions whose designs are based on mathematical problems, and whose security thus follows from rigorous mathematical proofs, complexity theory and formal reduction. These functions are called Provably Secure Cryptographic Hash Functions. To construct these is very difficult, and few examples have been introduced. Their practical use is limited. In the second category are functions which are not based on mathematical problems, but on an ad-hoc constructions, in which the bits of the message are mixed to produce the hash. These are then believed to be hard to break, but no formal proof is given. Almost all hash functions in widespread use reside in this category. Some of these functions are already broken, and are no longer in use. ''See'' Hash function security summary. Types of security of hash functions Generally, the ''basic'' security of cryptographic hash f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptographic Hash Function
A cryptographic hash function (CHF) is a hash algorithm (a map of an arbitrary binary string to a binary string with fixed size of n bits) that has special properties desirable for cryptography: * the probability of a particular n-bit output result (hash value) for a random input string ("message") is 2^ (like for any good hash), so the hash value can be used as a representative of the message; * finding an input string that matches a given hash value (a ''pre-image'') is unfeasible, unless the value is selected from a known pre-calculated dictionary (" rainbow table"). The ''resistance'' to such search is quantified as security strength, a cryptographic hash with n bits of hash value is expected to have a ''preimage resistance'' strength of n bits. A ''second preimage'' resistance strength, with the same expectations, refers to a similar problem of finding a second message that matches the given hash value when one message is already known; * finding any pair of different messa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NIST Hash Function Competition
The NIST hash function competition was an open competition held by the US National Institute of Standards and Technology (NIST) to develop a new hash function called SHA-3 to complement the older SHA-1 and SHA-2. The competition was formally announced in the ''Federal Register'' on November 2, 2007. "NIST is initiating an effort to develop one or more additional hash algorithms through a public competition, similar to the development process for the Advanced Encryption Standard (AES)." The competition ended on October 2, 2012 when NIST announced that Keccak would be the new SHA-3 hash algorithm. The winning hash function has been published as NIST FIPS 202 the "SHA-3 Standard", to complement FIPS 180-4, the ''Secure Hash Standard''. The NIST competition has inspired other competitions such as the Password Hashing Competition. Process Submissions were due October 31, 2008 and the list of candidates accepted for the first round was published on December 9, 2008. NIST held a conf ... [...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] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Coefficients
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number-theoretic Transform
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Definition Let R be any ring, let n\geq 1 be an integer, and let \alpha \in R be a principal ''n''th root of unity, defined by:Martin Fürer,Faster Integer Multiplication, STOC 2007 Proceedings, pp. 57–66. Section 2: The Discrete Fourier Transform. : \begin & \alpha^n = 1 \\ & \sum_^ \alpha^ = 0 \text 1 \leq k < n \qquad (1) \end The discrete Fourier transform maps an ''n''-tuple of elements of to another ''n''-tuple of elements of according to the following formula: : By convention, the tuple is said to be in the ''time domain'' and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness Extractor
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random information entropy, entropy source, together with a short, uniformly random seed, generates a highly random output that appears Independent and identically distributed random variables, independent from the source and Uniform distribution (discrete), uniformly distributed. Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (Hardware_random_number_generator, TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source. Sometimes the term "bias" is used to denote a weakly random sou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Hashing
In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. Many universal families are known (for hashing integers, vectors, strings), and their evaluation is often very efficient. Universal hashing has numerous uses in computer science, for example in implementations of hash tables, randomized algorithms, and cryptography. Introduction Assume we want to map keys from some universe U into m bins (labelled = \). The algorithm will have to handle some data set S \subseteq U of , S, =n keys, which is not known in advance. Usually, the goal of hashing is to obtain a low number of collisions (keys from S that land in the same bin). A deterministic hash function cannot offer any guarantee in an adversarial sett ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
