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Next-bit Test
In cryptography and the theory of computation, the next-bit testAndrew Chi-Chih YaoTheory and applications of trapdoor functions In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982. is a test against pseudo-random number generators. We say that a sequence of bits passes the next bit test for at any position i in the sequence, if any attacker who knows the i first bits (but not the seed) cannot predict the (i+1)st with reasonable computational power. Precise statement(s) Let P be a polynomial, and S=\ be a collection of sets such that S_k contains P(k)-bit long sequences. Moreover, let \mu_k be the probability distribution of the strings in S_k. We now define the next-bit test in two different ways. Boolean circuit formulation A predicting collectionManuel Blum and Silvio Micali, How to generate cryptographically strong sequences of pseudo-random bits, in SIAM J. COMPUT., Vol. 13, No. 4, November 1984 C=\ is a collection of boolean circuits, such ... [...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] |
Theory Of Computation
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: ''"What are the fundamental capabilities and limitations of computers?".'' In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several models in use, but the most commonly examined is the Turing machine. Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible "reasonable" mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Chi-Chih Yao
Andrew Chi-Chih Yao (; born December 24, 1946) is a Chinese computer scientist and computational theorist. He is currently a professor and the dean of Institute for Interdisciplinary Information Sciences (IIIS) at Tsinghua University. Yao used the minimax theorem to prove what is now known as Yao's Principle. Yao was a naturalized U.S. citizen, and worked for many years in the U.S. In 2015, together with Yang Chen-Ning, he renounced his U.S. citizenship and became an academician of the Chinese Academy of Sciences. Early life Yao was born in Shanghai, China. He completed his undergraduate education in physics at the National Taiwan University, before completing a Doctor of Philosophy in physics at Harvard University in 1972, and then a second PhD in computer science from the University of Illinois at Urbana–Champaign in 1975. Academic career Yao was an assistant professor at Massachusetts Institute of Technology (1975–1976), assistant professor at Stanford University (1976 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-random
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, people use pseudorandom numbers, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process. In many applications, the deterministic process is a computer algorithm called a pseudorandom number generator, which must first be provided wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuel Blum
Manuel Blum (born 26 April 1938) is a Venezuelan-American computer scientist who received the Turing Award in 1995 "In recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking". Education Blum was born to a Jewish family in Venezuela. Blum was educated at MIT, where he received his bachelor's degree and his master's degree in electrical engineering in 1959 and 1961 respectively, and his Ph.D. in mathematics in 1964 supervised by Marvin Minsky.. Career Blum worked as a professor of computer science at the University of California, Berkeley until 2001. From 2001 to 2018, he was the Bruce Nelson Professor of Computer Science at Carnegie Mellon University, where his wife, Lenore Blum, was also a professor of Computer Science. In 2002, he was elected to the United States National Academy of Sciences. In 2006, he was elected a member of the National Academy of Engineering for contributions to abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silvio Micali
Silvio Micali (born October 13, 1954) is an Italian computer scientist, professor at the Massachusetts Institute of Technology and the founder of Algorand. Micali's research centers on cryptography and information security. In 2012, he received the Turing Award for his work in cryptography. Personal life Micali graduated in mathematics at La Sapienza University of Rome in 1978 and earned a PhD degree in computer science from the University of California, Berkeley in 1982; for research supervised by Manuel Blum. Micali has been on the faculty at MIT, Electrical Engineering and Computer Science Department, since 1983. His research interests are cryptography, zero knowledge, pseudorandom generation, secure protocols, and mechanism design. Career Micali is best known for some of his fundamental early work on public-key cryptosystems, pseudorandom functions, digital signatures, oblivious transfer, secure multiparty computation, and is one of the co-inventors of zero-knowledge proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Circuits
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are defined in terms of the logic gates they contain. For example, a circuit might contain binary AND and OR gates and unary NOT gates, or be entirely described by binary NAND gates. Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit. Boolean circuits provide a model for many digital components used in computer engineering, including multiplexers, adders, and arithmetic logic units, but they exclude sequential logic. They are an abstraction that omits many aspects relevant to designing real digital logic circuits, such as metastability, fanout, glitches, power consumption, and propagation delay variability. Formal definition In giving a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Turing Machines
In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turing machine can—unlike a deterministic Turing Machine—have stochastic results; that is, on a given input and instruction state machine, it may have different run times, or it may not halt at all; furthermore, it may accept an input in one execution and reject the same input in another execution. In the case of equal probabilities for the transitions, probabilistic Turing machines can be defined as deterministic Turing machines having an additional "write" instruction where the value of the write is uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a "1" or a "0" on to the tape). Another common reformulation is simply a deterministic Turing machine with an added tape full of random bits ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P/poly
In computational complexity theory, P/poly is a complexity class representing problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. It can also be defined equivalently in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend on the input length but not on the input itself. In this formulation, P/poly is the class of decision problems that can be solved by a polynomial-time Turing machine with advice strings of length polynomial in the input size. These two different definitions make P/poly central to circuit complexity and non-uniform complexity. For example, the popular Miller–Rabin primality test can be formulated as a P/poly algorithm: the "advice" is a list of candidate values to test. It is possible to precompute a list of O(n) values such that every composite n-bit number will be certain to have a witness a in the list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yao's Test
In cryptography and the theory of computation, Yao's test is a test defined by Andrew Chi-Chih Yao in 1982,Andrew Chi-Chih YaoTheory and applications of trapdoor functions In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982. against pseudo-random sequences. A sequence of words passes Yao's test if an attacker with reasonable computational power cannot distinguish it from a sequence generated uniformly at random. Formal statement Boolean circuits Let P be a polynomial, and S=\_k be a collection of sets S_k of P(k)-bit long sequences, and for each k, let \mu_k be a probability distribution on S_k, and P_C be a polynomial. A predicting collection C=\ is a collection of boolean circuits of size less than P_C(k). Let p_^C be the probability that on input s, a string randomly selected in S_k with probability \mu(s), C_k(s)=1, i.e. p_^C= s\in S_k\text\mu_k(s)/math> Moreover, let p_^C be the probability that C_k(s)=1 on input s a P(k)-bit long sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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