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18 Οκτ 2024 · Example 1: Find the remainder when 7100 is divided by 13. Solution: Since 13 is a prime number, we can apply Fermat's Little Theorem, which states: ap−1 ≡ 1 (mod p) where p is a prime number, and a is an integer not divisible by p. Here, a = 7 and p = 13. By Fermat's Little Theorem: 712 ≡ 1 (mod 13) 7100 = 712×8+4 = (712)8 ⋅ 74.
27 Σεπ 2015 · By Fermat’s Little Theorem, we know that ap a (mod p) and aq a (mod q) no matter what integer a is. Combining with what is given, we have that ap a (mod p) =)(ap)q aq a (mod p) =)apq a (mod p) aq a (mod q) =)(aq)p ap a (mod q) =)apq a (mod q) This means that apq = px+ a = qy + a for some integers x and y. However, this then implies
21 Αυγ 2022 · Take an Example How Fermat’s little theorem works. Example 1: P = an integer Prime number . a = an integer which is not multiple of P . Let a = 2 and P = 17 . According to Fermat's little theorem . 2 17 - 1 ≡ 1 mod(17) we got 65536 % 17 ≡ 1 . that mean (65536-1) is an multiple of 17 . Example 2:
The following pages contain solutions to core problems from exams in Cryptography given at the Faculty of Mathematics, Natural Sciences and Information Technologies at the University of Primorska.
Compute gcd(85; 289) using Euclid's extended algorithm. Then compute x and y such that 85x + 289y = gcd(85; 289). We stop when we reach a remainder of 0, that is, when rn+1 = 0. We obtain gcd(a; b) = rn. Fact 1 For all a; b 2 N, if gcd(a; b) = d, then there exists x; y 2 Z such that ax + by = d.
Fermat’s little theorem gave us a way to think about whether a number is prime or not without factoring it. Compute 2N−1(mod N), 3N−1(mod N), 5N−1(mod N), and 7N−1(mod N) for N = 1729. What can you conclude? What happens if you try to factor 1729 using a pocket calculator?
27 Σεπ 2015 · An alternative proof of Fermat’s Little Theorem, in two steps: (a) Show that (x+ 1) p x p + 1 (mod p) for every integer x, by showing that the coe cient of x k is the same on both sides for every k = 0;:::;p.
