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Chapter 1: Q10E (page 48)

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

Short Answer

Expert verified

It is proved that if a=b(modN)andM dividesN then a=b(modM).

Step by step solution

01

Introduction

For,

a=b(modN)

AndM divides N.

To prove, a=b(modN).

02

To prove a≡b(modM) 

Assume that,

a=b(modN)

Then,

For some k∈Z,

By using division algorithm,

a and bcan be defined as,

a=nx+yb=nx'+y'

For0≤y andy'<n.

Then,

a=b+nk=(nx'+y')+nk=n(x'+k)+y

Comparing the above equations,

a=nx+y      0≤y<na=n(x'+k)+y  0≤y'<n

After division, a and bleaves the same remainder when it is divided by nthen,

a=nx+yb=nx'+y'

Subtracting these two equations,

a−b=n(x−x')

Thus,

a=b(modn)

Therefore, ifa=b(modN) andM divides Nthen a=b(modM).

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Most popular questions from this chapter

A d-ary tree is a rooted tree in which each node has at most dchildren. Show that any d-ary tree with n nodes must have a depth ofΩ(lognlogd) .Can you give a precise formula for the minimum depth it could possibly have?

Show that any binary integer is at most four times as long as the corresponding decimal integer. For very large numbers, what is the ratio of these two lengths, approximately?

Let[m]denote the set{0,1,…,m−1}. For each of the following families of hash functions, say whether or not it is universal, and determine how many random bits are needed to choose a function from the family.

(a) H={ha1,a2:a1,a2∈[m]}, wheremis a fixed prime and

ha1·ha1,a2(x1,x2)=a1x1+a2x2modm

Notice that each of these functions has signatureha1,a2:[m]2→[m]that is, it maps a pair of integers in[m]to a single integer in[m].

(b) His as before, except that nowm=2kis some fixed power of.2

(c) His the set of all functionsf:[m]→[m−1].

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?

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−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.
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