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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≡1(modN)xy2≡1(modN)

Subtract both equations as follows:

role="math" localid="1658916438802" xy1-xy2≡1-1(modN)⇒xy1-xy2≡0(modN)⇒xy1-y2≡0(modN)

02

Proving modulo N as distinct modulo

Takey1 as the multiplicative inverse. Then,

y1·xy1-y2≡x-1.0modN⇒y1-y2≡0modN⇒y1≡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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