1.3.2 also attached 1.24. Let N, g, and A be positive integers (note that N need not be prime).Prove that the following algorithm, which is a low-storage variant of the square-and-multiply algorithm described in Section 1.3.2, returns the value gª (mod N).(In Step 4 we use the notation [x] to denote the greatest integer function, i.e.,round x down to the nearest integer.)1.2.Input. Positive integers N, g, and A.Set a g and b = 1.=Loop while A > 0.3. If A 1 (mod 2), set b = b. a (mod N).a² (mod N) and A = [A/2].4.5. If A0, continue with loop at Step 2.6. Return the number 6, which equals g (mod N).Set a= 1.3.2 The fast powering algorithmIn some cryptosystems that we will study, for example the RSA and Diffie-Hellman cryptosystems, Alice and Bob are required to compute large powersof a number g modulo another number N, where N may have hundreds ofdigits. The naive way to compute gª is by repeated multiplication by g. Thus91 = g (mod N),92 = 9.91 (mod N), 93 9.92 (mod N),94 = 9.93 (mod N), 959.94 (mod N),....It is clear that gA = gª (mod N), but if A is large, this algorithm is completelyimpractical. For example, if A≈ 21000, then the naive algorithm would takelonger than the estimated age of the universe! Clearly if it is to be useful, weneed to find a better way to compute gª (mod N).The idea is to use the binary expansion of the exponent A to convertthe calculation of g4 into a succession of squarings and multiplications. Anexample will make the idea clear, after which we give a formal description ofthe method.

Answered: Image /qna-images/answer/f774c1de-5c44-4fed-86d7-704dbb69c708.jpg

1.24. Let N, g, and A be positive integers (note that N need not be prime). Prove that the following algorithm, which is a low-storage variant of the square- and-multiply algorithm described in Section 1.3.2, returns the value gª (mod N). (In Step 4 we use the notation [x] to denote the greatest integer function, i.e., round x down to the nearest integer.) 1. 2. Input. Positive integers N, g, and A. Set a = g and b = 1. Loop while A > 0. 3. If A = 1 (mod 2), set b = b. a (mod N). 4. Set a a² (mod N) and A = = [A/2]. 5. If A > 0, continue with loop at Step 2. 6. Return the number b, which equals gª (mod N).

1.24. Let N, g, and A be positive integers (note that N need not be prime). Prove that the following algorithm, which is a low-storage variant of the square- and-multiply algorithm described in Section 1.3.2, returns the value gª (mod N). (In Step 4 we use the notation [x] to denote the greatest integer function, i.e., round x down to the nearest integer.) 1. 2. Input. Positive integers N, g, and A. Set a = g and b = 1. Loop while A > 0. 3. If A = 1 (mod 2), set b = b. a (mod N). 4. Set a a² (mod N) and A = = [A/2]. 5. If A > 0, continue with loop at Step 2. 6. Return the number b, which equals gª (mod N).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.7: Introduction To Coding Theory (optional)

Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...

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