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Modular Exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie–Hellman key exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer (the base) is raised to the power (the exponent), and divided by a positive integer (the modulus); that is, . From the definition of division, it follows that . For example, given , and , dividing by leaves a remainder of . Modular exponentiation can be performed with a ''negative'' exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and . Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce Schneier
Bruce Schneier (; born January 15, 1963) is an American cryptographer, computer security professional, privacy specialist, and writer. Schneier is an Adjunct Lecturer in Public Policy at the Harvard Kennedy School and a Fellow at the Berkman Klein Center for Internet & Society as of November, 2013. He is a board member of the Electronic Frontier Foundation, Access Now, and The Tor Project; and an advisory board member of Electronic Privacy Information Center and VerifiedVoting.org. He is the author of several books on general security topics, computer security and cryptography and is a squid enthusiast. Early life and education Bruce Schneier is the son of Martin Schneier, a Brooklyn Supreme Court judge. He grew up in the Flatbush neighborhood of Brooklyn, New York, attending P.S. 139 and Hunter College High School. After receiving a physics bachelor's degree from the University of Rochester in 1984, he went to American University in Washington, D.C., and got his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reversible Computing
Reversible computing is any model of computation where every step of the process is time-reversible. This means that, given the output of a computation, it's possible to perfectly reconstruct the input. In systems that progress deterministically from one state to another, a key requirement for reversibility is a one-to-one correspondence between each state and its successor. Reversible computing is considered an unconventional approach to computation and is closely linked to quantum computing, where the principles of quantum mechanics inherently ensure reversibility (as long as quantum states are not measured or " collapsed"). Reversibility There are two major, closely related types of reversibility that are of particular interest for this purpose: physical reversibility and logical reversibility. A process is said to be ''physically reversible'' if it results in no increase in physical entropy; it is isentropic. There is a style of circuit design ideally exhibiting thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shor's Algorithm
Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. It is one of the few known quantum algorithms with compelling potential applications and strong evidence of superpolynomial speedup compared to best known classical (non-quantum) algorithms. On the other hand, factoring numbers of practical significance requires far more qubits than available in the near future. Another concern is that noise in quantum circuits may undermine results, requiring additional qubits for quantum error correction. Shor proposed multiple similar algorithms for solving the factoring problem, the discrete logarithm problem, and the period-finding problem. "Shor's algorithm" usually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations Exponential growth, exponentially faster than any modern "classical" computer. Theoretically a large-scale quantum computer could post-quantum cryptography, break some widely used encryption schemes and aid physicists in performing quantum simulator, physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications. The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in classical computing. However, unlike a classical bit, which can be in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Companion Matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial p(x)=c_0 + c_1 x + \cdots + c_x^ + x^n is the square matrix defined as C(p)=\begin 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_ \end. Some authors use the transpose of this matrix, C(p)^T , which is more convenient for some purposes such as linear recurrence relations ( see below). C(p) is defined from the coefficients of p(x), while the characteristic polynomial as well as the minimal polynomial of C(p) are equal to p(x) . In this sense, the matrix C(p) and the polynomial p(x) are "companions". Similarity to companion matrix Any matrix with entries in a field has characteristic polynomial p(x) = \det(xI - A) , which in turn has companion matrix C(p) . These matrices are related as follows. The following statements are equivalent: * ''A'' is similar over ''F'' to C(p) , i.e. ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perrin Numbers
In mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation . The Perrin numbers, named after the French engineer , bear the same relationship to the Padovan sequence as the Lucas numbers do to the Fibonacci sequence. Definition The Perrin numbers are defined by the recurrence relation :\begin P(0)&=3, \\ P(1)&=0, \\ P(2)&=2, \\ P(n)&=P(n-2) +P(n-3) \mboxn>2, \end and the reverse :P(n) =P(n+3) -P(n+1) \mboxn<0. The first few terms in both directions are Perrin numbers can be expressed as sums of the three initial terms : The first fourteen prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Numbers
In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the '' Fibonacci Quarterly''. Applications of Fibon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant-recursive Sequence
In mathematics, an sequence, infinite sequence of numbers s_0, s_1, s_2, s_3, \ldots is called constant-recursive if it satisfies an equation of the form :s_n = c_1 s_ + c_2 s_ + \dots + c_d s_, for all n \ge d, where c_i are constant (mathematics), constants. The equation is called a linear recurrence with constant coefficients, linear recurrence relation. The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence. For example, the Fibonacci sequence :0, 1, 1, 2, 3, 5, 8, 13, \ldots, is constant-recursive because it satisfies the linear recurrence F_n = F_ + F_: each number in the sequence is the sum of the previous two. Other examples include the power of two sequence 1, 2, 4, 8, 16, \ldots, where each number is the sum of twice the previous number, and the square number sequence 0, 1, 4, 9, 16, 25, \ldots. All arithmetic progressions, all geometric progressions, and all polynomials are constant-recu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". Knuth is the author of the multi-volume work '' The Art of Computer Programming''. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive multi-volume monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. it consists of published volumes 1, 2, 3, 4A, and 4B, with more expected to be released in the future. The Volumes 1–5 are intended to represent the central core of computer programming for sequential machines; the subjects of Volumes 6 and 7 are important but more specialized. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
