By Victor Shoup

ISBN-10: 0511113633

ISBN-13: 9780511113635

Quantity idea and algebra play an more and more major function in computing and communications, as evidenced via the notable purposes of those topics to such fields as cryptography and coding concept.

This introductory e-book emphasises algorithms and functions, equivalent to cryptography and blunder correcting codes, and is out there to a vast viewers. The mathematical necessities are minimum: not anything past fabric in a standard undergraduate path in calculus is presumed, except a few event in doing proofs - every little thing else is constructed from scratch.

Thus the e-book can serve a number of reasons. it may be used as a reference and for self-study via readers who are looking to examine the mathematical foundations of contemporary cryptography. it's also excellent as a textbook for introductory classes in quantity thought and algebra, in particular these geared in the direction of machine technological know-how scholars.

**Read Online or Download A Computational Introduction to Number Theory and Algebra PDF**

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**Extra resources for A Computational Introduction to Number Theory and Algebra**

**Example text**

6. Let n = 7. For each value a = 1, . . , 6, we can compute successive powers of a modulo n to ﬁnd its multiplicative order modulo n. 1i 2i 3i 4i 5i 6i i mod 7 mod 7 mod 7 mod 7 mod 7 mod 7 1 1 2 3 4 5 6 2 1 4 2 2 4 1 3 1 1 6 1 6 6 4 1 2 4 4 2 1 5 1 4 5 2 3 6 6 1 1 1 1 1 1 So we conclude that modulo 7: 1 has order 1; 6 has order 2; 2 and 4 have order 3; and 3 and 5 have order 6. 15 (Euler’s Theorem). For any positive integer n, and any integer a relatively prime to n, we have aφ(n) ≡ 1 (mod n).

6, we can compute successive powers of a modulo n to ﬁnd its multiplicative order modulo n. 1i 2i 3i 4i 5i 6i i mod 7 mod 7 mod 7 mod 7 mod 7 mod 7 1 1 2 3 4 5 6 2 1 4 2 2 4 1 3 1 1 6 1 6 6 4 1 2 4 4 2 1 5 1 4 5 2 3 6 6 1 1 1 1 1 1 So we conclude that modulo 7: 1 has order 1; 6 has order 2; 2 and 4 have order 3; and 3 and 5 have order 6. 15 (Euler’s Theorem). For any positive integer n, and any integer a relatively prime to n, we have aφ(n) ≡ 1 (mod n). In particular, the multiplicative order of a modulo n divides φ(n).

6. Let n = 7. For each value a = 1, . . , 6, we can compute successive powers of a modulo n to ﬁnd its multiplicative order modulo n. 1i 2i 3i 4i 5i 6i i mod 7 mod 7 mod 7 mod 7 mod 7 mod 7 1 1 2 3 4 5 6 2 1 4 2 2 4 1 3 1 1 6 1 6 6 4 1 2 4 4 2 1 5 1 4 5 2 3 6 6 1 1 1 1 1 1 So we conclude that modulo 7: 1 has order 1; 6 has order 2; 2 and 4 have order 3; and 3 and 5 have order 6. 15 (Euler’s Theorem). For any positive integer n, and any integer a relatively prime to n, we have aφ(n) ≡ 1 (mod n).

### A Computational Introduction to Number Theory and Algebra by Victor Shoup

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