By Edsger W. Dijkstra

He starts off by means of contemplating the questions, «What is an algorithm?» and «What are we doing once we program?» those questions lead him to an attractive digression at the semantics of programming languages, which, in flip, results in essays on programming language constructs, scoping of variables, and array references. Dijkstra then promises, as promised, a suite of lovely algorithms. those algorithms are a ways ranging, masking mathematical computations, different types of sorting difficulties, development matching, convex hulls, and extra. simply because this can be an previous ebook, the algorithms awarded are often not the easiest to be had. notwithstanding, the price in analyzing A self-discipline of Programming is to soak up and comprehend the way in which that Dijkstra thought of those difficulties, which, in many ways, is extra priceless than one thousand algorithms.

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It follows immediately that t = s and that the original factorizations of a were identical up to rearranging the order of the factors. ) Deﬁnition. 20) says that in the factorization of a positive integer a into primes, each prime p appears to a particular power. We denote this power by ordp (a) and call it the order (or exponent) of p in a. ) For example, the factorization of 1728 is 1728 = 26 · 33 , so ord2 (1728) = 6, ord3 (1728) = 3, and ordp (1728) = 0 for all primes p ≥ 5. Using the ordp notation, the factorization of a can be succinctly written as pordp (a) .

42 1. An Introduction to Cryptography Let p be a large prime,17 say 2159 < p < 2160 . Alice and Bob take their key space K, plaintext space M, and ciphertext space C to be the same set, K = M = C = {1, 2, 3, . . , p − 1}. In fancier terminology, K = M = C = F∗p are all taken to be equal to the group of units in the ﬁnite ﬁeld Fp . 9) ek (m) ≡ k · m (mod p). Here we mean that ek (m) is set equal to the unique positive integer between 1 and p that is congruent to k · m modulo p. The corresponding decryption function dk is dk (c) ≡ k · c (mod p), where k is the inverse of k modulo p.

Modular Arithmetic 19 In general, if a and b are relatively prime and if q1 , q2 , . . 2 on page 13, then the box has the form 0 1 1 0 q1 P1 Q1 q2 P2 Q2 ... ... qt−1 Pt−1 Qt−1 qt a b The entries in the box are calculated using the initial values P1 = q 1 , Q1 = 1, P2 = q2 · P1 + 1, Q2 = q2 · Q1 , and then, for i ≥ 3, using the formulas Pi = qi · Pi−1 + Pi−2 and Qi = qi · Qi−1 + Qi−2 . The final four entries in the box satisfy a · Qt−1 − b · Pt−1 = (−1)t . Multiplying both sides by (−1)t gives the solution u = (−1)t Qt−1 and v = (−1)t+1 Pt−1 to the equation au + bv = 1.

### A Discipline of Programming by Edsger W. Dijkstra

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