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In number theory, integer factorization is the decomposition, when possible, of a positive integer into a product of smaller integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime.
A prime number 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, 1 × 5 or 5 × 1, 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.
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem, one of the oldest, that is widely used for secure data transmission. The acronym "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) by the English mathematician Clifford Cocks. That system was declassified in 1997.
Fermat's little theorem states that if p is a prime number, then for any integer a, the number is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let p be a prime number of the form 8k + 7 and to let n = p – 1. In this case, is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if p is an odd prime and 2p + 1 is also prime, then p must divide x, y, or z. Otherwise, . This case where p does not divide x, y, or z is called the first case. Sophie Germain’s work was the most progress achieved on Fermat’s last theorem at that time. Later work by Kummer and others always divided the problem into first and second cases. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare.
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. 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.
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test.
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA cryptosystem. The Solovay–Strassen test is essentially an Euler–Jacobi probable prime test.
The security of cryptographic systems depends on some secret data that is known to authorized persons but unknown and unpredictable to others. To achieve this unpredictability, some randomization is typically employed. Modern cryptographic protocols often require frequent generation of random quantities. Cryptographic attacks that subvert or exploit weaknesses in this process are known as random number generator attacks.
In cryptography, a verifiable random function (VRF) is a public-key pseudorandom function that provides proofs that its outputs were calculated correctly. The owner of the secret key can compute the function value as well as an associated proof for any input value. Everyone else, using the proof and the associated public key, can check that this value was indeed calculated correctly, yet this information cannot be used to find the secret key.
In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications, for example hashing, public-key cryptography, and search of prime factors in large numbers.
Robert Martin Solovay is an American mathematician specializing in set theory.
The Baillie–PSW primality test is a probabilistic or possibly deterministic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff.
In mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. "Succinct" usually means that the proof should be at most polynomially larger than the number of digits in the number itself.
Industrial-grade primes are integers for which primality has not been certified, but they have undergone probable prime tests such as the Miller–Rabin primality test, which has a positive, but negligible, failure rate, or the Baillie–PSW primality test, which no composites are known to pass.
A physical unclonable function, or PUF, is a physical object that for a given input and conditions (challenge), provides a physically defined "digital fingerprint" output (response) that serves as a unique identifier, most often for a semiconductor device such as a microprocessor. PUFs are often based on unique physical variations occurring naturally during semiconductor manufacturing. A PUF is a physical entity embodied in a physical structure. PUFs are implemented in integrated circuits, including FPGAs, and can be used in applications with high-security requirements, more specifically cryptography, Internet of Things (IOT) devices and privacy protection.
In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and François Morain, in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing followed quickly.
A pseudoprime is a probable prime that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.
The ROCA vulnerability is a cryptographic weakness that allows the private key of a key pair to be recovered from the public key in keys generated by devices with the vulnerability. "ROCA" is an acronym for "Return of Coppersmith's attack". The vulnerability has been given the identifier CVE-2017-15361.
References
- ↑ C. Couvreur and J. J. Quisquater (1982), An Introduction to Fast Generation of Large Prime Numbers, Philips Journal of Research, vol. 37, pp. 231–264
- ↑ Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective. Springer. pp. 174–178. ISBN 978-0387-25282-7.
- ↑ Mollin, Richard A. (2002), RSA and Public-Key Cryptography, Discrete Mathematics and Its Applications, CRC Press, pp. 124–125, ISBN 9781420035247 .
- ↑ Christophe, Clavier. "Generating Provable Primes Efficiently on Embedded Devices" (PDF). The International Association for Cryptologic Research.
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