A computational introduction to number theory and algebra by Victor Shoup

By Victor Shoup

Quantity conception and algebra play an more and more major position in computing and communications, as evidenced through the outstanding purposes of those topics to such fields as cryptography and coding conception. This introductory e-book emphasises algorithms and functions, similar to cryptography and mistake correcting codes, and is offered to a wide viewers. The mathematical must haves are minimum: not anything past fabric in a standard undergraduate direction in calculus is presumed, except a few event in doing proofs - every little thing else is constructed from scratch. therefore the e-book 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 excellent as a textbook for introductory classes in quantity thought and algebra, in particular these geared in the direction of laptop technological know-how scholars.

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For any integer n > 1, and for all 1 ≤ r, t ≤ , we have crt ≡ ars bst (mod n). 4) then we must have crt = crt . , that n divides (crt − crt ). 4), we obtain |crt − crt | ≤ |crt | + |crt | ≤ M + n/2 < n/2 + n/2 = n. So we see that the quantity (crt − crt ) is a multiple of n, while at the same time this quantity is strictly less than n in absolute value; hence, this quantity must be zero. 5). 4. Speeding up Algorithms via Modular Computation 49 So from the above discussion, to compute C, it suffices to compute the entries of C modulo n, where we have to make sure that we compute “balanced” remainders in the interval [−n/2, n/2), rather than the more usual “least nonnegative” remainders.

4) then we must have crt = crt . , that n divides (crt − crt ). 4), we obtain |crt − crt | ≤ |crt | + |crt | ≤ M + n/2 < n/2 + n/2 = n. So we see that the quantity (crt − crt ) is a multiple of n, while at the same time this quantity is strictly less than n in absolute value; hence, this quantity must be zero. 5). 4. Speeding up Algorithms via Modular Computation 49 So from the above discussion, to compute C, it suffices to compute the entries of C modulo n, where we have to make sure that we compute “balanced” remainders in the interval [−n/2, n/2), rather than the more usual “least nonnegative” remainders.

2r∗ From the equalities ri = si n + ti y and r = sn + ty, we have the two congruences: r ≡ ty (mod n), ri ≡ ti y (mod n). Subtracting ti times the first from t times the second, we obtain rti ≡ ri t (mod n). This says that n divides rti − ri t; however, using the bounds |r| ≤ r ∗ , |ti | < n/(2r ∗ ), |ri | ≤ 2r ∗ , |t| ≤ t∗ , and 4r ∗ t∗ ≤ n, we obtain (verify) |rti − ri t| ≤ |rti | + |ri t| < n. Since n divides rti − ri t and |rti − ri t| < n, the only possibility is that rti − ri t = 0. Now consider the two equations: r = sn + ty ri = si n + ti y.

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