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Chapter 1: Q1E (page 48)


Show that in any base b≥2, the sum of any three single-digit numbers is at most two digits long.

Short Answer

Expert verified

Sum of any three single digit number is not as long as of a two-digit number.

Step by step solution

01

Find equation of sum of digits

Let’s assume p,q,r be the single digit numbers in base b.

Let sum of these number besumgiven as:

sum=p+q+r

Given that:

Value of p,q,rlies in range b≥2

Here, p,q,ris at most b−1. For example: in base 10 maximum single digit is 9.

02

Calculate the value of sum

sum=p+q+r=(b−1)+(b−1)+(b−1)=3(b−1)

For b=2,

sum=3(2−1)=3×1=3

Here, sum is a single digit number.

For b=3,

sum=3(3−1)=3×2=6

Here, sum is a single digit number.

For b=9,

sum=3(9−1)=3×8=24

Here, sum is a two-digit number.

Thus, the sum is less than b2 and b2 in any base, sum is less than b2 that is represented in two digits.

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Most popular questions from this chapter

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].

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.)

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 ?

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.

Suppose that instead of using a compositeN=pqin the RSA cryptosystem (Figure 1.9), we simply use a prime modulus p . As in RSA, we would have an encryption exponent e, and the encryption of a message mmodpwould be memodp.Prove that this new cryptosystem is not secure, by giving an efficient algorithm to decrypt: that is, an algorithm that given and p,e,andmemodp as input, computes . Justify the correctness and analyze the running time of your decryption algorithm.
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