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Chapter 1: Q12E (page 48)

What is 222006(mod3)?

Short Answer

Expert verified

The solution is 222006(mod3)=1.

Step by step solution

01

Introduction

A function that returns a number or variable's absolute value is known as a modulus function. The magnitude of the number of variables is produced. Another name for it is an absolute value function. No matter what input was provided to this function, a positive result is always the outcome.

02

Calculating 222006(mod3).

The value starts from n=0. i.e.,

For n=1,  21=2(mod3)For n=2,  22=1(mod3)For n=3,  23=2(mod3)For n=4,  24=1(mod3)

The even value ofn remainder is1 and for odd it is 2. i.e.,

22n=1(mod3)22n+1=2(mod3)

Therefore, the solution is222006(mod3)=1 .

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Most popular questions from this chapter

The algorithm for computing abmodCby repeated squaring does not necessarily lead to the minimum number of multiplications. Give an example of b>10where the exponentiation can be performed using fewer multiplications, by some other method.

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.

Wilson's theorem says that a numberis prime if and only if
(N-1)!=-1(modN).

(a) If is prime, then we know every number1≤x<p is invertible modulo . Which of thesenumbers is their own inverse?
(b) By pairing up multiplicative inverses, show thatrole="math" localid="1658725109805" (p-1)!=-1(modp) for prime p.
(c) Show that if N is not prime, then(N-1)!≢(modN) .(Hint: Considerd=gcd(N,(N-1)!.)
(d) Unlike Fermat's Little Theorem, Wilson's theorem is an if-and-only-if condition for primality. Why can't we immediately base a primality test on this rule?

Show that log(n!)=θ(nlogn)

(Hint: To show an upper bound, compare (n!) with nn. To show a lower bound, compare it with (n2)n2).

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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