Aleksandr Aleksandrovich Razborov (Russian: Алекса́ндр Алекса́ндрович Разбо́ров; born February 16, 1963), sometimes known as Sasha Razborov, is a Soviet and Russian mathematician and computational theorist who won the Nevanlinna Prize in 1990 for introducing the "approximation method" in proving Boolean circuit lower bounds of some essential algorithmic problems, and the Gödel Prize with Steven Rudich in 2007 for their paper "Natural Proofs. " His advisor was Sergei Adian.
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Natural proof
In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one. While these proofs are in some sense "natural", it can be shown (assuming a widely believed conjecture on the existence of one-way functions) that no such proof can possibly be used to solve the P vs. NP problem.
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Circuit complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of Boolean circuits that compute them. A Boolean circuit with n input bits is a directed acyclic graph in which every node (usually called gates in this context) is either an input node of in-degree 0 labeled by one of the n input bits, an AND gate, an OR or a NOT gate. One of these gates is designated as the output gate.
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Majority function
In Boolean logic, the majority function (also called the median operator) is a function from n inputs to one output. The value of the operation is false when n/2 or more arguments are false, and true otherwise. Alternatively, representing true values as 1 and false values as 0, we may use the formula The "−1/2" in the formula serves to break ties in favor of zeros when n is even; a similar formula can be used for a function that breaks ties in favor of ones.
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Pseudorandom function family
In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in the following way: no efficient algorithm can distinguish between a function chosen randomly from the PRF family and a random oracle (a function whose outputs are fixed completely at random). Pseudorandom functions are vital tools in the construction of cryptographic primitives, especially secure encryption schemes.
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Average-case complexity
Average-case complexity is a subfield of computational complexity theory that studies the complexity of algorithms on random inputs. The study of average-case complexity has applications in the theory of cryptography. Leonid Levin presented the motivation for studying average-case complexity as follows:: "Many combinatorial problems (called search or NP problems) have easy methods of checking solutions for correctness.
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Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation, or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g. , in their processing time or working space requirements) to changes in input size.
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Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties. More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various aspects in information security such as data confidentiality, data integrity, and authentication. Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering.
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