By Raphael Pass, Wei-lung Tseng

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The first few composites are: 4,6,8,9,10,12,14,15. How can we determine if a number n is prime or not? This is called a primality test. Given the above observation, we can try to divide n by every √ positive integer ≤ n; this is not very efficient, considering that in today’s cryptographic applications, we use 1024 or 2048-bit primes. A deterministic polynomial-time algorithm to for primality tests was not known until Agarwal, Saxena and Kayal constructed the first such algorithm in 2002; even after several improvements, the algorithm is still not fast enough to practically feasible.

We show only item 1; the other proofs are similar (HW). By definition, a|b ⇒ there exist k1 ∈ Z such that b = k1 a a|c ⇒ there exist k2 ∈ Z such that c = k2 a Therefore b + c = k1 a + k2 a = (k1 + k2 )a, so a|(b + c). 4. Let a, b, c ∈ Z. If a|b and a|c, then a|mb+nc for any m, n ∈ Z. We learn in elementary school that even when integers don’t divide evenly, we can compute the quotient and the remainder. 5 (Division Algorithm). t. a = dq + r and 0 ≤ r < d. q is called the quotient and denoted by q = a div d.

We omit the proof here and give an example instead. 14. Suppose we want to find s, t ∈ Z such that s(252) + t(198) = gcd(252, 198) = 18. Run Euclid’s algorithm, but write out the equation a = bq + r for each recursive call of EuclidAlg. 1) = 4(252) − 5(198) We conclude that gcd(252, 198) = 18 = 4(252) − 5(198). 2 Modular Arithmetic Modular arithmetic, as the name implies, is arithmetic on the remainders of integers, with respect to a fixed divisor. A central idea to modular arithmetic is congruences: two integers are considered “the same” if they have the same remainder with respect to the fixed divisor.