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Feige–Fiat–Shamir Identification Scheme
In cryptography, the Feige–Fiat–Shamir identification scheme is a type of parallel zero-knowledge proof developed by Uriel Feige, Amos Fiat, and Adi Shamir in 1988. Like all zero-knowledge proofs, it allows one party, the Prover, to prove to another party, the Verifier, that they possess secret information without revealing to Verifier what that secret information is. The Feige–Fiat–Shamir identification scheme, however, uses modular arithmetic and a parallel verification process that limits the number of communications between Prover and Verifier. Setup Following a common convention, call the prover Peggy and the verifier Victor. Choose two large prime integers ''p'' and ''q'' and compute the product ''n = pq''. Create secret numbers s_1, \cdots, s_k 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 divi ... [...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] |
Zero-knowledge Proof
In cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true while the prover avoids conveying any additional information apart from the fact that the statement is indeed true. The essence of zero-knowledge proofs is that it is trivial to prove that one possesses knowledge of certain information by simply revealing it; the challenge is to prove such possession without revealing the information itself or any additional information. If proving a statement requires that the prover possess some secret information, then the verifier will not be able to prove the statement to anyone else without possessing the secret information. The statement being proved must include the assertion that the prover has such knowledge, but without including or transmitting the knowledge itself in the assertion. Otherwise, the statement would not be proved in zero-knowledge because it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uriel Feige
Uriel Feige ( he, אוריאל פייגה) is an Israeli computer scientist who was a doctoral student of Adi Shamir. Life Uriel Feige currently holds the post of Professor at the Department of Computer Science and Applied Mathematics, the Weizmann Institute of Science, Rehovot in Israel. Work He is notable for co-inventing the Feige–Fiat–Shamir identification scheme along with Amos Fiat and Adi Shamir. Honors and awards He won the Gödel Prize in 2001 "for the PCP theorem and its applications to hardness of approximation In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. Scope Hardness of approximation complements the study of approximation algorithms by prov ...". References {{DEFAULTSORT:Feige, Uriel Living people Gödel Prize laureates Theoretical computer scientists Modern cryptographers Public-key cryptographers 20th-century Israeli mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amos Fiat
Amos Fiat (born December 1, 1956) is an Israeli computer scientist, a professor of computer science at Tel Aviv University. He is known for his work in cryptography, online algorithms, and algorithmic game theory. Biography Fiat earned his Ph.D. in 1987 from the Weizmann Institute of Science under the supervision of Adi Shamir. After postdoctoral studies with Richard Karp and Manuel Blum at the University of California, Berkeley, he returned to Israel, taking a faculty position at Tel Aviv University. Research Many of Fiat's most highly cited publications concern cryptography, including his work with Adi Shamir on digital signatures (leading to the Fiat–Shamir heuristic for turning interactive identification protocols into signature schemes) and his work with David Chaum and Moni Naor on electronic money, used as the basis for the ecash system. With Shamir and Uriel Feige in 1988, Fiat invented the Feige–Fiat–Shamir identification scheme, a method for using public- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adi Shamir
Adi Shamir ( he, עדי שמיר; born July 6, 1952) is an Israeli cryptographer. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the Feige–Fiat–Shamir identification scheme (along with Uriel Feige and Amos Fiat), one of the inventors of differential cryptanalysis and has made numerous contributions to the fields of cryptography and computer science. Education Born in Tel Aviv, Shamir received a Bachelor of Science (BSc) degree in mathematics from Tel Aviv University in 1973 and obtained his Master of Science (MSc) and Doctor of Philosophy (PhD) degrees in Computer Science from the Weizmann Institute in 1975 and 1977 respectively. Career and research After a year as a postdoctoral researcher at the University of Warwick, he did research at Massachusetts Institute of Technology (MIT) from 1977 to 1980 before returning to be a member of the faculty of Mathematics and Computer Science at the Weizma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, 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 use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice And Bob
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptographic systems and protocols, and in other science and engineering literature where there are several participants in a thought experiment. The Alice and Bob characters were invented by Ron Rivest, Adi Shamir, and Leonard Adleman in their 1978 paper "A Method for Obtaining Digital Signatures and Public-key Cryptosystems". Subsequently, they have become common archetypes in many scientific and engineering fields, such as quantum cryptography, game theory and physics. As the use of Alice and Bob became more widespread, additional characters were added, sometimes each with a particular meaning. These characters do not have to refer to people; they refer to generic agents which might be different computers or even different programs running on a single computer. Overview Alice and Bob are the names of fictional characters used for convenience and to aid comprehension. For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
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 (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Square Root
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic nonresidue modulo ''n''. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers. History, conventions, and elementary facts Fermat, Euler, Lagrange, 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 states that if the context makes it clear, the adjective "quadratic" may be dropped. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
