By Victor Shoup

Quantity concept and algebra play an more and more major position in computing and communications, as evidenced by way of the amazing purposes of those topics to such fields as cryptography and coding thought. This introductory e-book emphasises algorithms and purposes, corresponding to cryptography and blunder correcting codes, and is available to a wide 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 little thing else is constructed from scratch. hence 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 study the mathematical foundations of contemporary cryptography. it's also perfect as a textbook for introductory classes in quantity idea and algebra, in particular these geared in the direction of laptop technology scholars.

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Now consider the two equations: r = sn + ty ri = si n + ti y. 7) 52 Chapter 4. 7), we obtain n(sti − si t) = 0, and hence sti − si t = 0. 5, we know that gcd(si , ti ) = 1, we must have ti | t. So t = ti α for some α, and we must have α = 0 since t = 0. 8) yields r = ri α and s = si α. That proves the second statement of the theorem. 1 Application: Chinese Remaindering with Errors One interpretation of the Chinese Remainder Theorem is that if we “encode” an integer z, with 0 ≤ z < n, as the sequence (a1 , .

46 Chapter 4. Euclid’s Algorithm If we define M0 to be the identity matrix, then it is easy to see that Mi = si ti si+1 ti+1 , for 0 ≤ i ≤ . 5 is immediate, and part (ii) follows from the fact that Mi is the product of i matrices, each of determinant −1, and the determinant of Mi is evidently si ti+1 − ti si+1 . 4 that efficiently computes d = gcd(a, b), along with integers s and t such that as + bt = d, and which uses only addition, subtraction, and “shift” operations. 3 Computing Modular Inverses and Chinese Remaindering One application of the extended Euclidean algorithm is to the problem of computing multiplicative inverses in Zn , where n > 1.

We also have x x s x 1 = + < + ≤ n y y y2 y y x y + y −1 y + 1 . y Thus, we have x/y < x /y + 1, and hence, x/y ≤ x /y . 3. Basic Integer Arithmetic 31 From this theorem, one sees that the following algorithm correctly computes the quotient and remainder in time O(k): carry ← 0 for i ← k − 1 down to 0 do tmp ← carry · B + ai qi ← tmp/b0 carry ← tmp rem b0 output the quotient q = (qk−1 · · · q0 )B and the remainder carry Note that in every loop iteration, the value of carry lies between 0 and b0 ≤ B − 1, and the value of tmp lies between 0 and B · b0 + (B − 1) ≤ B 2 − 1.

### A computational introduction to number theory and algebra by Victor Shoup

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