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Chapter 1: Q31E (page 50)

Consider the problem of computing N!=1·2·3···N.

(a) If Nis an role="math" localid="1658397956489" n-bit number, how many bits long is N!, approximately ( inΘ(·)form)?

(b) Give an algorithm to compute N!and analyze its running time.

Short Answer

Expert verified

The running time complexity of the given algorithm can be analyzed as it runs for n iteration and that is multiplied and the final time complexity O(NlogN).

Step by step solution

01

Introduction

The factorial can be defined as the product of all the numbers between 1 and the given number including the given number. It can be represented as an integer with an exclamation mark.

02

Calculation of bits long with N!

We have,

N!=1·2·3···N.

It is known that,

Log2(N!)=Θ(Nlog2N)

So, N-factorial can be written as,

N!=Θ(Nlog2N)

It can be written as,Θ(n2n)

Given that, N is a n-bit number, then

N!isΘ(n2n)bitslong.

03

Algorithm to compute N!

Normal algorithm flow is shown below,

Factorial(N)If(N==0)Return1ElseReturnNfactorial(N-1)

The running time complexity of the given algorithm can be analyzing as it runs for n iteration and that are multiplied and the final time complexity O(NlogN).

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Most popular questions from this chapter

Find the inverse of:.20mod79,3mod62,21mod91,5mod23

1.38. To see if a number, say 562437487, is divisible by 3, you just add up the digits of its decimalrepresentation, and see if the result is divisible by role="math" localid="1658402816137" 3.

( 5+6+2+4+3+7+4+8+7=46, so it is not divisible by 3).

To see if the same number is divisible by 11, you can do this: subdivide the number into pairs ofdigits, from the right-hand end(87,74,43,62,5) , add these numbers and see if the sum is divisible by11 (if it's too big, repeat).

How about 37? To see if the number is divisible by 37, subdivide it into triples from the end(487,437,562) add these up, and see if the sum is divisible by37 .


This is true for any prime pother than2 and 5. That is, for any prime p≠2,5, there is an integer rsuch that in order to see ifp divides a decimal number n, we breakn into r-tuples of decimal digits (starting from the right-hand end), add up these r-tuples, and check if the sum is divisible by p.

(a) What is the smallest rsuch for p=13? Forp=13 ?

(b) Show thatr is a divisor of p-1.

Justify the correctness of the recursive division algorithm given in page 25, and show that it takes time O(n2)onn- bit inputs.

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

In the RSA cryptosystem, Alice’s public key (N,e)is available to everyone. Suppose that her private key d is compromised and becomes known to Eve. Show that e=3if (a common choice) then Eve can efficiently factor N.
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