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Some of the proofs of Fermat's little theorem given below depend on two simplifications. The first is that we may assume that a is in the range 0 ≤ a ≤ p − 1 . This is a simple consequence of the laws of modular arithmetic ; we are simply saying that we may first reduce a modulo p .
The Miller–Rabin primality test uses the following extension of Fermat's little theorem: [14] If p is an odd prime and p − 1 = 2 s d with s > 0 and d odd > 0, then for every a coprime to p , either a d ≡ 1 (mod p ) or there exists r such that 0 ≤ r < s and a 2 r d ≡ −1 (mod p ) .
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation a n + b n = c n for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b.
The first part of the book concludes with chapter 4, on the history of prime numbers and primality testing, including the prime number theorem (in a weakened form), applications of prime numbers in cryptography, and the widely used Miller–Rabin primality test, which runs in randomized polynomial time.
Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on Euler's theorem. We want to show that m ed ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)).
Gerhard Frey (German:; born 1 June 1944) is a German mathematician, known for his work in number theory.Following an original idea of Hellegouarch, [1] he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to Wiles's proof of Fermat's Last Theorem.
The encryption of the message M is given by C ≡ M e (mod N) and the decryption of cipher text C is given by C d ≡ (M e) d ≡ M ed ≡ M (mod N) (using Fermat's little theorem). Using the Euclidean algorithm, one can efficiently recover the secret key d if one knows the factorization of N. By having the secret key d, one can efficiently ...
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]