|
Integer Factorization Algorithms
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] |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
AKS Primality Test
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". The algorithm was the first one which is able to determine in polynomial time, whether a given number is prime or composite without relying on mathematical conjectures such as the generalized Riemann hypothesis. The proof is also notable for not relying on the field of analysis. In 2006 the authors received both the Gödel Prize and Fulkerson Prize for their work. Importance AKS is the first primality-proving algorithm to be simultaneously ''general'', ''polynomial-time'', ''deterministic'', and ''unconditionally correct''. Previous algorithms had been developed for centuries and achieved three of these properties at most, but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Polynomial Time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is '' trial division'': given an input number, n, check whether it is divisible by any prime number between 2 and \sqrt n (i.e., whether the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For any divisor p \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Empty Product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of addition, adding no numbers—is by convention 0, zero, or the additive identity. When numbers are implied, the empty product becomes 1, one. The term ''empty product'' is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing Set theory, set-theoretic intersections, categorical products, and products in computer programming. Nullary arithmetic product Definition Let ''a''1, ''a''2, ''a''3, ... be a sequence of numbers, and let :P_m = \prod_^m a_i = a_1 \cdots a_m be the product of the first ''m'' elements of the sequence. Then :P_m = P_ a_m for all ''m'' = 1, 2, ... provided that we use the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Prime Factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Fundamental Theorem Of Arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is prime or can be represented uniquely as a product of prime numbers, up to the order of the factors. For example, : 1200 = 2^4 \cdot 3^1 \cdot 5^2 = (2 \cdot 2 \cdot 2 \cdot 2) \cdot 3 \cdot (5 \cdot 5) = 5 \cdot 2 \cdot 5 \cdot 2 \cdot 3 \cdot 2 \cdot 2 = \ldots The theorem says two things about this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, 12 = 2 \cdot 6 = 3 \cdot 4). This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Public-key
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security. There are many kinds of public-key cryptosystems, with different security goals, including digital signature, Diffie–Hellman key exchange, public-key key encapsulation, and public-key encryption. Public key algorithms are fundamental security primitives in modern cryptosystems, including applications and protocols that offer assurance of the confidentiality and authenticity of electronic communications and data storage. They underpin numerous Internet standards, such as Transport Layer Security (TLS), SSH, S/MIME, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
RSA (algorithm)
The RSA (Rivest–Shamir–Adleman) cryptosystem is a public-key cryptosystem, one of the oldest widely used for secure data transmission. The initialism "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ), the British signals intelligence agency, by the English mathematician Clifford Cocks. That system was declassified in 1997. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone via the public key, but can only be decrypted by someone who knows the private key. The security of RSA relies on the practical difficulty of factoring the product of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
RSA Problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a ''message'' to an '' exponent'', modulo a composite number ''N'' whose factors are not known. Thus, the task can be neatly described as finding the ''e''th roots of an arbitrary number, modulo N. For large RSA key sizes (in excess of 1024 bits), no efficient method for solving this problem is known; if an efficient method is ever developed, it would threaten the current or eventual security of RSA-based cryptosystems—both for public-key encryption and digital signatures. More specifically, the RSA problem is to efficiently compute ''P'' given an RSA public key (''N'', ''e'') and a ciphertext ''C'' ≡ ''P'' ''e'' (mod ''N''). The structure of the RSA public key requires that ''N'' be a large semiprime (i.e., a product of two large prime numbers), that 2 < ''e'' < ''N'', that ''e'' be |