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RSA Problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a ''message'' to an ''exponent'', modulo a composite number ''N'' whose factors are not known. Thus, the task can be neatly described as finding the ''e''th roots of an arbitrary number, modulo N. For large RSA key sizes (in excess of 1024 bits), no efficient method for solving this problem is known; if an efficient method is ever developed, it would threaten the current or eventual security of RSA-based cryptosystems—both for public-key encryption and digital signatures. More specifically, the RSA problem is to efficiently compute ''P'' given an RSA public key (''N'', ''e'') and a ciphertext ''C'' ≡ ''P'' ''e'' (mod ''N''). The structure of the RSA public key requires that ''N'' be a large semiprime (i.e., a product of two large prime numbers), that 2 < ''e'' < ''N'', that ''e'' be |
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] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antoine Joux
Antoine Joux (born 1967) is a French cryptographer,"Antoine Joux, Prix Gödel 2013" Bulletin de la société informatique de France – numéro 1, septembre 2013 one of the three 2013 Gödel Prize laureates., specifically cited for his paper ''A one round protocol for tripartite Diffie-Hellman''. He was at the Université de Versailles Saint-Quentin-en-Yvelines and researcher in the CRYPT team of the laboratory of computer science PRISM of CNRS, currently he is Chair of Cryptology of the ''Fondation partenariale'' of Université Pierre et Marie Curie, UPMC, ''professeur associé'' at the Laboratoire d'informatique de Paris 6, and Senior Crypto-Security Expert at CryptoExperts. References {{DEFAULTSORT:Joux, Antoine 1967 births Living people French cryptographers P ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Group Model
The generic group model is an idealised cryptographic model, where the adversary is only given access to a randomly chosen encoding of a group, instead of efficient encodings, such as those used by the finite field or elliptic curve groups used in practice. The model includes an oracle that executes the group operation. This oracle takes two encodings of group elements as input and outputs an encoding of a third element. If the group should allow for a pairing operation this operation would be modeled as an additional oracle. One of the main uses of the generic group model is to analyse computational hardness assumptions. An analysis in the generic group model can answer the question: "What is the fastest generic algorithm for breaking a cryptographic hardness assumption". A generic algorithm is an algorithm that only makes use of the group operation, and does not consider the encoding of the group. This question was answered for the discrete logarithm problem by Victor Shoup u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straight Line Program
Straight may refer to: Slang * Straight, slang for heterosexual ** Straight-acting, an LGBT person who does not exhibit the appearance or mannerisms of the gay stereotype * Straight, a member of the straight edge subculture Sport and games * Straight, an alternative name for the cross, a type of punch in boxing * Straight, a hand ranking in the card game of poker Places * Straight, Oklahoma, an unincorporated community in Texas County, Oklahoma Media * ''Straight'' (Tobias Regner album), the first album by German singer Tobias Regner * ''Straight'' (2007 film), a German film by Nicolas Flessa * ''Straight'' (2009 film), a Bollywood film starring Vinay Pathak and Gul Panag * "Straight", a song by T-Pain on the 2017 ''Oblivion'' (T-Pain album) * "Straight", a song by A Place to Bury Strangers on the 2015 album ''Transfixiation'' * Straight Records, a record label formed in 1969 * '' The Georgia Straight'' (straight.com), a Canadian weekly newspaper published in Vancouver, B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rabin Cryptosystem
The Rabin cryptosystem is a family of public-key encryption schemes based on a trapdoor function whose security, like that of RSA, is related to the difficulty of integer factorization. The Rabin trapdoor function has the advantage that inverting it has been mathematically proven to be as hard as factoring integers, while there is no such proof known for the RSA trapdoor function. It has the disadvantage that each output of the Rabin function can be generated by any of four possible inputs; if each output is a ciphertext, extra complexity is required on decryption to identify which of the four possible inputs was the true plaintext. Naive attempts to work around this often either enable a chosen-ciphertext attack to recover the secret key or, by encoding redundancy in the plaintext space, invalidate the proof of security relative to factoring. Public-key encryption schemes based on the Rabin trapdoor function are used mainly for examples in textbooks. In contrast, RSA is the ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA Factoring Challenge
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography. They published a list of semiprimes (numbers with exactly two prime factors) known as the RSA numbers, with a cash prize for the successful factorization of some of them. The smallest of them, a 100-decimal digit number called RSA-100 was factored by April 1, 1991. Many of the bigger numbers have still not been factored and are expected to remain unfactored for quite some time, however advances in quantum computers make this prediction uncertain due to Shor's algorithm. In 2001, RSA Laboratories expanded the factoring challenge and offered prizes ranging from $10,000 to $200,000 for factoring numbers from 576 bits up to 2048 bits. The RSA Factoring Challenges ended in 2007. RSA Laboratories stated: "Now that the industry has a cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong RSA Assumption
In cryptography, the strong RSA assumption states that the RSA problem is intractable even when the solver is allowed to choose the public exponent ''e'' (for ''e'' ≥ 3). More specifically, given a modulus ''N'' of unknown factorization, and a ciphertext ''C'', it is infeasible to find any pair (''M'', ''e'') such that ''C'' ≡ ''M'' ''e'' mod ''N''. The strong RSA assumption was first used for constructing signature schemes provably secure against existential forgery without resorting to the random oracle model. See also * Quadratic residuosity problem * Decisional composite residuosity assumption References * Barić N., Pfitzmann B. (1997) Collision-Free Accumulators and Fail-Stop Signature Schemes Without Trees. In: Fumy W. (eds) Advances in Cryptology – EUROCRYPT ’97. EUROCRYPT 1997. Lecture Notes in Computer Science, vol 1233. Springer, Berlin, Heidelberg. * Fujisaki E., Okamoto T. (1997) Statistical zero knowledge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padding (cryptography)
In cryptography, padding is any of a number of distinct practices which all include adding data to the beginning, middle, or end of a message prior to encryption. In classical cryptography, padding may include adding nonsense phrases to a message to obscure the fact that many messages end in predictable ways, e.g. ''sincerely yours''. Classical cryptography Official messages often start and end in predictable ways: ''My dear ambassador, Weather report, Sincerely yours'', etc. The primary use of padding with classical ciphers is to prevent the cryptanalyst from using that predictability to find known plaintext that aids in breaking the encryption. Random length padding also prevents an attacker from knowing the exact length of the plaintext message. A famous example of classical padding which caused a great misunderstanding is " the world wonders" incident, which nearly caused an Allied loss at the WWII Battle off Samar, part of the larger Battle of Leyte Gulf. In that example, A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number ( RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
