In cryptography, key size or key length is the number of bits in a key used by a cryptographic algorithm.
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and distinct from the decryption key which is kept secret (private). In RSA, this asymmetry is based on the practical difficulty of factoring the product of two large prime numbers, the "factoring problem". The acronym RSA is the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who publicly described the algorithm in 1977. Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), had developed an equivalent system in 1973, which was not declassified until 1997.
In the mathematics of the real numbers, the logarithm logba is a number x such that bx = a, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logba is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indra (mod m) for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1.
Daniel Julius Bernstein is a German-American mathematician, cryptologist, and programmer. He is a personal professor in the department of mathematics and computer science at the Eindhoven University of Technology, as well as a Research Professor of Computer Science at the University of Illinois at Chicago.
Articles related to cryptography include:
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 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, but 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 mathematics, the RSA numbers are a set of large semiprimes that are part of the RSA Factoring Challenge. The challenge was to find the prime factors but it was declared inactive in 2007. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers.
Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n, the integer to be factored, can be divided by each number in turn that is less than n. For example, for the integer n = 12, the only numbers that divide it are 1, 2, 3, 4, 6, 12. Selecting only the largest powers of primes in this list gives that 12 = 3 × 4 = 3 × 22.
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm.
TWINKLE is a hypothetical integer factorization device described in 1999 by Adi Shamir and purported to be capable of factoring 512-bit integers. It is also a pun on the twinkling LEDs used in the device. Shamir estimated that the cost of TWINKLE could be as low as $5000 per unit with bulk production. TWINKLE has a successor named TWIRL which is more efficient.
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 eth roots of an arbitrary number, modulo N. For large RSA key sizes, 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.
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.
In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.
Arjen Klaas Lenstra is a Dutch mathematician. He studied mathematics at the University of Amsterdam. He is currently a professor at the EPFL (Lausanne), in the Laboratory for Cryptologic Algorithms, and previously worked for Citibank and Bell Labs.
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography. The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes.
The Texas Instruments signing key controversy refers to the controversy which resulted from Texas Instruments' (TI) response to a project to factorize the 512-bit RSA cryptographic keys needed to write custom firmware to TI devices.
Ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption. Public-key cryptography relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known.
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.
