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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 ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, 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 (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ars Technica
''Ars Technica'' is a website covering news and opinions in technology, science, politics, and society, created by Ken Fisher and Jon Stokes in 1998. It publishes news, reviews, and guides on issues such as computer hardware and software, science, technology policy, and video games. ''Ars Technica'' was privately owned until May 2008, when it was sold to Condé Nast Digital, the online division of Condé Nast Publications. Condé Nast purchased the site, along with two others, for $25 million and added it to the company's ''Wired'' Digital group, which also includes '' Wired'' and, formerly, Reddit. The staff mostly works from home and has offices in Boston, Chicago, London, New York City, and San Francisco. The operations of ''Ars Technica'' are funded primarily by advertising, and it has offered a paid subscription service since 2001. History Ken Fisher, who serves as the website's current editor-in-chief, and Jon Stokes created ''Ars Technica'' in 1998. Its purpose was t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worst-case Complexity
In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as in asymptotic notation). It gives an upper bound on the resources required by the algorithm. In the case of running time, the worst-case time complexity indicates the longest running time performed by an algorithm given ''any'' input of size , and thus guarantees that the algorithm will finish in the indicated period of time. The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms. The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input. Definition Given a model of computation and an algorithm \mathsf that halts on each input s, the mapping ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Very Smooth Hash
In cryptography, Very Smooth Hash (VSH) is a secure cryptographic hash function invented in 2005 by Scott Contini, Arjen Lenstra, and Ron Steinfeld. Provably secure means that finding collisions is as difficult as some known hard mathematical problem. Unlike other secure collision-resistant hashes, VSH is efficient and usable in practice. Asymptotically, it only requires a single multiplication per message-bits and uses RSA-type arithmetic. Therefore, VSH can be useful in embedded environments where code space is limited. Two major variants of VSH were proposed. For one, finding a collision is as difficult as finding a nontrivial modular square root of a very smooth number modulo . The other one uses a prime modulus (with no trapdoor), and its security proof relies on the hardness of finding discrete logarithms of very smooth numbers modulo . Both versions have similar efficiency. VSH is not suitable as a substitute for a random oracle, but can be used to build ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Residue
In number theory, an integer ''q'' is a quadratic residue modulo operation, modulo ''n'' if it is Congruence relation, congruent to a Square number, perfect square modulo ''n''; that is, if there exists an integer ''x'' such that :x^2\equiv q \pmod. Otherwise, ''q'' is a quadratic nonresidue modulo ''n''. Quadratic residues are used in applications ranging from acoustical engineering to cryptography and the Integer factorization, factoring of large numbers. History, conventions, and elementary facts Fermat, Euler, Joseph Louis Lagrange, Lagrange, Adrien-Marie Legendre, Legendre, and other number theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's ''Disquisitiones Arithmeticae'' (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and says that if the context makes it clear, the adjective "quadratic" may be dropped. For a giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm
In mathematics, for given real numbers a and b, the logarithm \log_b(a) is a number x such that b^x=a. Analogously, in any group G, powers b^k can be defined for all integers k, and the discrete logarithm \log_b(a) is an integer k such that b^k=a. In arithmetic modulo an integer m, the more commonly used term is index: One can write k=\mathbb_b a \pmod (read "the index of a to the base b modulo m") for b^k \equiv a \pmod if b is a primitive root of m and \gcd(a,m)=1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. In cryptography, the computational complexity of the discrete logarithm problem, along with its application, was first proposed in the Diffie–Hellman problem. Several important algorithms in public-key cryptography, such as ElGamal, base their security on the hardness assumption that the discrete logarithm problem (DLP) over carefully chosen groups has no efficient solution. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Reduction
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second. A polynomial-time reduction proves that the first problem is no more difficult than the second one, because whenever an efficient algorithm exists for the second problem, one exists for the first problem as well. By contraposition, if no efficient algorithm exists for the first problem, none exists for the second either. Polynomial-time reductions are frequently used in complexity theory for defining both complexity classes and complete problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SHA-1
In cryptography, SHA-1 (Secure Hash Algorithm 1) is a hash function which takes an input and produces a 160-bit (20-byte) hash value known as a message digest – typically rendered as 40 hexadecimal digits. It was designed by the United States National Security Agency, and is a U.S. Federal Information Processing Standard. The algorithm has been cryptographically broken but is still widely used. Since 2005, SHA-1 has not been considered secure against well-funded opponents; as of 2010 many organizations have recommended its replacement. NIST formally deprecated use of SHA-1 in 2011 and disallowed its use for digital signatures in 2013, and declared that it should be phased out by 2030. , chosen-prefix attacks against SHA-1 are practical. As such, it is recommended to remove SHA-1 from products as soon as possible and instead use SHA-2 or SHA-3. Replacing SHA-1 is urgent where it is used for digital signatures. All major web browser vendors ceased acceptance of SHA-1 SSL certifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-way Compression Function
In cryptography, a one-way compression function is a function that transforms two fixed-length inputs into a fixed-length output.Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Fifth Printing (August 2001) page 328. The transformation is one-way function, "one-way", meaning that it is difficult given a particular output to compute inputs which compress to that output. One-way compression functions are not related to conventional data compression algorithms, which instead can be inverted exactly (lossless compression) or approximately (lossy compression) to the original data. One-way compression functions are for instance used in the Merkle–Damgård construction inside cryptographic hash functions. One-way compression functions are often built from block ciphers. Some methods to turn any normal block cipher into a one-way compression function are Davies–Meyer, Matyas–Meyer–Oseas, Miyaguchi–Preneel (single-block-length compr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
