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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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Most popular questions from this chapter

Give a polynomial-time algorithm for computing,abcmodp given a,b,c, and prime p.

The grade-school algorithm for multiplying two n-bit binary numbers x and y consist of addingtogethern copies of r, each appropriately left-shifted. Each copy, when shifted, is at most 2n bits long.
In this problem, we will examine a scheme for adding n binary numbers, each m bits long, using a circuit or a parallel architecture. The main parameter of interest in this question is therefore the depth of the circuit or the longest path from the input to the output of the circuit. This determines the total time taken for computing the function.
To add two m-bit binary numbers naively, we must wait for the carry bit from position i-1before we can figure out the ith bit of the answer. This leads to a circuit of depthΟ(m). However, carry-lookahead circuits (seewikipedia.comif you want to know more about this) can add inΟ(logn)depth.

  1. Assuming you have carry-lookahead circuits for addition, show how to add n numbers eachm bits long using a circuit of depth Ο(lognlogm).
  2. When adding three m-bit binary numbers x+y+z, there is a trick we can use to parallelize the process. Instead of carrying out the addition completely, we can re-express the result as the sum of just two binary numbersr+s, such that the ith bits of r and s can be computedindependently of the other bits. Show how this can be done. (Hint: One of the numbers represents carry bits.)
  3. Show how to use the trick from the previous part to design a circuit of depthΟ(logn)for multiplying two n-bit numbers.

Suppose you want to compute the nth Fibonacci number Fn , modulo an integer p. Can you find an efficient way to do this?

On page 38, we claimed that since about a 1nfraction of n-bit numbers are prime, on average it is sufficient to draw O(n)random n -bit numbers before hitting a prime. We now justify this rigorously. Suppose a particular coin has a probability p of coming up heads. How many times must you toss it, on average, before it comes up heads? (Hint: Method 1: start by showing that the correct expression is∑i=1∞i(1-p)i-1p . Method 2: if E is the average number of coin tosses, show that E=1+(1-p)E).

Show that if xis a nontrivial square root of 1 modulo N , that is if x2≡1modNbut x≠±1modN, thenN must be composite. (For instance,42≡1mod15but4≢±1mod15; thus 4 is a nontrivial square root of 1 modulo 15.)
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