By A. Campillo

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**Example text**

4 Primality Testing 37 There are composite numbers n (called Carmichael numbers) with the amazing property that an−1 ≡ 1 (mod n) for all a with gcd(a, n) = 1. The first Carmichael number is 561, and it is a theorem that there are infinitely many such numbers ([AGP94]). 2. Is p = 323 prime? We compute 2322 (mod 323). Making a table as above, we have i 0 1 2 3 4 5 6 7 8 m 322 161 80 40 20 10 5 2 1 εi 0 1 0 0 0 0 1 0 1 i 22 mod 323 2 4 16 256 290 120 188 137 35 Thus 2322 ≡ 4 · 188 · 35 ≡ 157 (mod 323), so 323 is not prime, though this computation gives no information about how 323 factors as a product of primes.

The set of units in Z/nZ is a group (Z/nZ)∗ = {a ∈ Z/nZ : gcd(a, n) = 1} that has order ϕ(n). The theorem then asserts that the order of an element of (Z/nZ)∗ divides the order ϕ(n) of (Z/nZ)∗ . 1 Congruences Modulo n 27 the more general fact (Lagrange’s Theorem) that if G is a finite group and g ∈ G, then the order of g divides the cardinality of G. We now give an elementary proof of the theorem. Let P = {a : 1 ≤ a ≤ n and gcd(a, n) = 1}. 12, we see that the reductions modulo n of the elements of xP are the same as the reductions of the elements of P .

Until we find an exponent n such that bn = a. For example, suppose a = 18, b = 5, and p = 23. Working modulo 23, we have b1 = 5, b2 = 2, b3 = 10, . . , b12 = 18, so n = 12. When p is large, computing the discrete log this way soon becomes impractical, because increasing the number of digits of the modulus makes the computation take vastly longer. 3. Perhaps part of the reason that computing discrete logarithms is difficult, is that the logarithm in the real numbers is continuous, but the (minimum) logarithm of a number mod n bounces around at random.