# VICTOR SHOUP's A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA PDF

By VICTOR SHOUP

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Extra resources for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)

Example text

14. Work out the details of algorithms for arithmetic on signed integers, using the above algorithms for unsigned integers as subroutines. 3 Basic integer arithmetic 45 division with remainder of arbitrary signed integers (for division with remainder, your algorithm should compute a/b and a mod b). Make sure your algorithm correctly computes the sign bit of the result, and also strips leading zero digits from the result. 15. Work out the details of an algorithm that compares two signed integers a and b, determining which of a < b, a = b, or a > b holds.

Suppose that a ≡ a (mod n) and b ≡ b (mod n). This means that there exist integers c and d such that a = a + cn and b = b + dn. Therefore, a + b = a + b + (c + d)n, which proves the ﬁrst congruence of the theorem, and a b = (a + cn)(b + dn) = ab + (ad + bc + cdn)n, which proves the second congruence. 3 allow one to work with congruence relations modulo n much as one would with ordinary equalities: one can add to, subtract from, or multiply both sides of a congruence modulo n by the same integer; also, if x is congruent to y modulo n, one may substitute y for x in any simple arithmetic expression (more precisely, any polynomial in x with integer coeﬃcients) appearing in a congruence modulo n.

Also, on a real machine, the behavior of the cache may signiﬁcantly aﬀect the time it takes to load or store the operands of an instruction. ” However, despite all of these problems, it still turns out that measuring the running time on a RAM as we propose here is nevertheless a good “ﬁrst order” predictor of performance on real machines in many cases. ” If we have an algorithm for solving a certain type of problem, we expect that “larger” instances of the problem will require more time to solve than “smaller” instances.