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Chapter 1: Q24E (page 49)

If p is prime, how many elements of{0,1,...pn-1} have an inverse modulopn ?

Short Answer

Expert verified

The total number of inverses isPn鈥�1(P-1). .

Step by step solution

01

Introduction

The modular inverse of AmodCis the B value that makesA*BmodC=1 . The modular multiplicative inverse can be defined as theGCD(a,b) must be equal to 1 and鈥�a鈥�and鈥�b鈥� are relatively prime.

02

Inverse modulo of pn

Given elements are, 0,1,...pn-1.

If is a prime, then for all given elements i.e. 0,1,...pn-1, it is not the multiple of p and it contains the inverse of pn.

We have given that the range of elements is0,1,...pn-1

Now,pn-1 is the only element that get divisible bypn .

So, total number of inverses are,

Pn鈥�Pn鈥�1=Pn鈥�1(P-1)

Therefore, total number of inverse isPn鈥�1(P-1) .

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Most popular questions from this chapter

Digital signatures, continued.Consider the signature scheme of Exercise 1.45.

(a) Signing involves decryption, and is therefore risky. Show that if Bob agrees to sign anything he is asked to, Eve can take advantage of this and decrypt any message sent by Alice to Bob.

(b) Suppose that Bob is more careful, and refuses to sign messages if their signatures look suspiciously like text. (We assume that a randomly chosen message鈭�that is, a random number in the range{1,...,N-1} is very unlikely to look like text.) Describe a way in which Eve can nevertheless still decrypt messages from Alice to Bob, by getting Bob to sign messages whose signatures look random.

In an RSA cryptosystem, p = 7and q = 11(as in Figure 1.9). Find appropriate exponents and .

Prove or disprove: If a has an inverse modulo b, then b has an inverse modulo a.

Give an efficient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by bothx and y. What is the running time of your algorithm as a function of n?

Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.
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