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Chapter 1: Q23E (page 49)

Show that if a has a multiplicative inverse modulo N, then this inverse is unique (modulo N).

Short Answer

Expert verified

It is proved that the inverse multiplicative modulo is N a distinct modulo.

Step by step solution

01

First we will find the expression of x

Initially, take the two multiplicative inverses modulo N of x as y1andy2 .

Then,

xy1鈮�1(modN)xy2鈮�1(modN)

Subtract both equations as follows:

role="math" localid="1658916438802" xy1-xy2鈮�1-1(modN)鈬�xy1-xy2鈮�0(modN)鈬�xy1-y2鈮�0(modN)

02

Proving modulo N as distinct modulo

Takey1 as the multiplicative inverse. Then,

y1路xy1-y2鈮�x-1.0modN鈬�y1-y2鈮�0modN鈬�y1鈮�y2modN

From this, it is derived that the inverse multiplicative modulo N is a distinct modulo N .

Therefore, the inverse multiplicative modulo N is a distinct modulo.

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Most popular questions from this chapter

Consider the problem of computing x y for given integers x and y: we want the whole answer, not modulo a third integer. We know two algorithms for doing this: the iterative algorithm which performs y 鈭� 1 multiplications by x; and the recursive algorithm based on the binary expansion of y. Compare the time requirements of these two algorithms, assuming that the time to multiply an n-bit number by an m-bit number is O(mn).

The Fibonacci numbers F0,F1,...are given by the recurrenceFn+1=Fn+Fn-1鈥�,F0=0,F1=1. Show that for anyn鈮�1,gcd(Fn+1,Fn)=1.

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

Show that if a鈮�b(modN)and if M Divides Nthena鈮�b(modM)

Consider an RSA key set with p = 17 , q = 23, N = 23 and e = 3 (as in Figure 1.9). What value of d should be used for the secret key? What is the encryption of the message M = 41 ?
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