By Victor Shoup
Quantity thought and algebra play an more and more major function in computing and communications, as evidenced through the amazing purposes of those topics to such fields as cryptography and coding thought.
This introductory ebook emphasises algorithms and purposes, corresponding to cryptography and blunder correcting codes, and is out there to a large viewers. The mathematical must haves are minimum: not anything past fabric in a regular undergraduate direction in calculus is presumed, except a few adventure in doing proofs - every thing else is built from scratch.
Thus the ebook can serve a number of reasons. it may be used as a reference and for self-study through readers who are looking to examine the mathematical foundations of recent cryptography. it's also perfect as a textbook for introductory classes in quantity idea and algebra, specifically these geared in the direction of desktop technological know-how scholars.
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Extra resources for A Computational Introduction to Number Theory and Algebra
6), we obtain β∈Z∗n β . 6) β∈Z∗n β ∈ Z∗n from the left- and right-hand αφ(n) = n . That proves the ﬁrst statement of the theorem. The second follows from the observation made above that αi = n if and only if the multiplicative order of α divides i. 16 (Fermat’s little theorem). For any prime p, and any integer a ≡ 0 (mod p), we have ap−1 ≡ 1 (mod p). Moreover, for any integer a, we have ap ≡ a (mod p). Proof. 15, and the fact that φ(p) = p − 1. The second statement is clearly true if a ≡ 0 (mod p), and if a ≡ 0 (mod p), we simply multiply both sides of the congruence ap−1 ≡ 1 (mod p) by a.
Therefore, it suﬃces to consider the problem of determining the solutions z to congruences of the form az ≡ b (mod n), for given integers a, b, n. 6. Let a, b, n ∈ Z with n > 0. If a is relatively prime to n, then the congruence az ≡ b (mod n) has a solution z; moreover, any integer z is a solution if and only if z ≡ z (mod n). Proof. The integer z := ba , where a is a multiplicative inverse of a modulo n, is clearly a solution. 5 holds if and only if z ≡ z (mod n). ✷ Suppose that a, b, n ∈ Z with n > 0, a = 0, and gcd(a, n) = 1.
Show that a0 + a1 x + · · · + ak xk ≡ a0 + a1 y + · · · + ak y k (mod n). 2. Let a, b, n, n ∈ Z with n > 0 and n | n. Show that if a ≡ b (mod n), then a ≡ b (mod n ). 3. Let a, b, n, n ∈ Z with n > 0, n > 0, and gcd(n, n ) = 1. Show that if a ≡ b (mod n) and a ≡ b (mod n ), then a ≡ b (mod nn ). 4. Let a, b, n ∈ Z such that n > 0 and a ≡ b (mod n). Show that gcd(a, n) = gcd(b, n). 5. Prove that for any prime p and integer x, if x2 ≡ 1 (mod p) then x ≡ 1 (mod p) or x ≡ −1 (mod p). 6. Let a be a positive integer whose base-10 representation is a = (ak−1 · · · a1 a0 )10 .