By William Stein
ISBN-10: 1441927522
ISBN-13: 9781441927521
This can be a textbook approximately classical straight forward quantity thought and elliptic curves. the 1st half discusses undemanding themes equivalent to primes, factorization, endured fractions, and quadratic kinds, within the context of cryptography, computation, and deep open learn difficulties. the second one half is set elliptic curves, their purposes to algorithmic difficulties, and their connections with difficulties in quantity thought reminiscent of Fermat’s final Theorem, the Congruent quantity challenge, and the Conjecture of Birch and Swinnerton-Dyer. The meant viewers of this ebook is an undergraduate with a few familiarity with simple summary algebra, e.g. earrings, fields, and finite abelian teams.
Read Online or Download Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach PDF
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Extra resources for Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach
Example text
12 (Number of primitive roots). If there is a primitive root modulo n, then there are exactly ϕ(ϕ(n)) primitive roots modulo n. Proof. The primitive roots modulo n are the generators of (Z/nZ)∗ , which by assumption is cyclic of order ϕ(n). Thus they are in bijection with the generators of any cyclic group of order ϕ(n). In particular, the number of primitive roots modulo n is the same as the number of elements of Z/ϕ(n)Z with additive order ϕ(n). An element of Z/ϕ(n)Z has additive order ϕ(n) if and only if it is coprime to ϕ(n).
Then each element of (Z/pZ)∗ has order one of 1, 2, (p − 1)/2, or p − 1. Proof. Since p is prime, the group (Z/pZ)∗ is of order p−1. By assumption, the prime factorization of p − 1 is 2 · ((p − 1)/2). Let a ∈ (Z/pZ)∗ . 20, ap−1 = 1, so the order of a is a divisor of p − 1, which proves the proposition. 2 The Diffie-Hellman Key Exchange 55 Given a prime p with (p − 1)/2 prime, find an element of order p − 1 as follows. If 2 has order p − 1, we are done. If not, 2 has order (p − 1)/2 since 2 does not have order either 1 or 2.
7 is a lift of 1 mod 3, since 7 + 3Z = 1 + 3Z. 8). 9. A number n ∈ Z is divisible by 3 if and only if the sum of the digits of n is divisible by 3. Proof. Write n = a + 10b + 100c + · · · , where the digits of n are a, b, c, etc. Since 10 ≡ 1 (mod 3), n = a + 10b + 100c + · · · ≡ a + b + c + · · · (mod 3), from which the proposition follows. 1 Linear Equations Modulo n In this section, we are concerned with how to decide whether or not a linear equation of the form ax ≡ b (mod n) has a solution modulo n.
Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach by William Stein
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