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RSA-640
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007. RSA Laboratories (which is an initialism of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. , the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA-150
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Security, RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of integer factorization, factoring large integers. The challenge was ended in 2007. RSA Laboratories (which is an initialism of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to United States dollar, US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. , the smallest 23 of the 54 listed numbers have been facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA-140
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007. RSA Laboratories (which is an initialism of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. , the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, ... [...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] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form : \begin & \exp\left(\left((64/9)^+o(1)\right)\left(\log n\right)^ \left(\log\log n\right)^\right) \\ pt= & L_n\left /3,(64/9)^\right\end in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number , it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order . The size of these values is exponentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple-precision Arithmetic
In computing, an arithmetic logic unit (ALU) is a combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers. This is in contrast to a floating-point unit (FPU), which operates on floating point numbers. It is a fundamental building block of many types of computing circuits, including the central processing unit (CPU) of computers, FPUs, and graphics processing units (GPUs). The inputs to an ALU are the data to be operated on, called operands, and a code indicating the operation to be performed (opcode); the ALU's output is the result of the performed operation. In many designs, the ALU also has status inputs or outputs, or both, which convey information about a previous operation or the current operation, respectively, between the ALU and external status registers. Signals An ALU has a variety of input and output nets, which are the electrical conductors used to convey digital signals between the ALU and external circuitry. When an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Montgomery (mathematician)
Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research. He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in applications of elliptic curves to integer factorization and other problems, and the Montgomery ladder, which is used to protect against side-channel attacks in elliptic curve cryptography. Education and career Montgomery began his undergraduate career at the University of California, Riverside, in 1965 and transferred to Berkeley in 1967, earning a BA in mathematics in 1969 and an MA in mathematics in 1971, He joined the System Development Corporation (SDC) in 1972, where he worked for many years as a programmer implementing algorithms for the CDC 7600 and PDP series of computers, inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
