By William Stein

This can be a booklet approximately best numbers, congruences, mystery messages, and elliptic curves so that you can learn disguise to hide. It grew out of undergr- uate classes that the writer taught at Harvard, UC San Diego, and the collage of Washington. The systematic learn of quantity concept used to be initiated round 300B. C. while Euclid proved that there are in?nitely many top numbers, and likewise cleverly deduced the basic theorem of mathematics, which asserts that each optimistic integer components uniquely as a fabricated from primes. Over one thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent quantity challenge that asks for how to come to a decision even if a given optimistic integer n is the realm of a correct triangle, all 3 of whose aspects are rational numbers. Then one other thousand years later (in 1976), Di?e and Hellman brought the ?rst ever public-key cryptosystem, which enabled humans to speak secretely over a public communications channel without predetermined mystery; this invention and those that it revolutionized the area of electronic communique. within the Eighties and Nineteen Nineties, elliptic curves revolutionized quantity conception, supplying amazing new insights into the congruent quantity challenge, primality trying out, publ- key cryptography, assaults on public-key structures, and taking part in a primary position in Andrew Wiles’ answer of Fermat’s final Theorem.

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**Example text**

2. [Finished] Output x. 34 2. The Ring of Integers Modulo n Proof. Reduce ax+ny = 1 modulo n to see that x satisfies ax ≡ 1 (mod n). 9. Solve 17x ≡ 1 (mod 61). 7 to find x, y such that 17x + 61y = 1: 61 = 3 · 17 + 10 17 = 1 · 10 + 7 10 = 1 · 7 + 3 3=2·3+1 10 = 61 − 3 · 17 7 = −61 + 4 · 17 3 = 2 · 61 − 7 · 17 1 = −5 · 61 + 18 · 17 Thus 17 · 18 + 61 · (−5) = 1 so x = 18 is a solution to 17x ≡ 1 (mod 61). 10. Sage implements the above algorithm for quickly computing inverses modulo n. 2 How to Compute am (mod n) Let a and n be integers, and m a nonnegative integer.

Prove that if an | bn , then a | b. (b) Suppose p is a prime and a and k are positive integers. Prove that if p | ak , then pk | ak . 13 (a) Prove that if a positive integer n is a perfect square, then n cannot be written in the form 4k + 3 for k an integer. ) (b) Prove that no integer in the sequence 11, 111, 1111, 11111, 111111, . . is a perfect square. 14 Prove that a positive integer n√is prime if and only if n is not divisible by any prime p with 1 < p ≤ n. 2 The Ring of Integers Modulo n A startling fact about numbers is that it takes less than a second to decide with near certainty whether or not any given 1,000 digit number n is a prime, without actually factoring n.

All three of these algorithms are of fundamental importance to the cryptography algorithms of Chapter 3. 1 2. The Ring of Integers Modulo n How to Solve ax ≡ 1 (mod n) Suppose a, n ∈ N with gcd(a, n) = 1. 13 the equation ax ≡ 1 (mod n) has a unique solution. How can we find it? 1 (Extended Euclidean Representation). Suppose a, b ∈ Z and let g = gcd(a, b). Then there exists x, y ∈ Z such that ax + by = g. 2. If e = cg is a multiple of g, then cax + cby = cg = e, so e = (cx)a + (cy)b can also be written in terms of a and b.