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Chapter 1: Q40E (page 52)

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.)

Short Answer

Expert verified

It can be proved by the proof by contradiction method.

Step by step solution

01

Explain composite numbers

The multiple of two small positive integers that has atleast one divisor other than one.

02

Prove the given problem

By proof by contradiction method, Consider Ifx2≡1modp,x≠±1modp and is a prime number. Then,

(xmodp)2-1≡0modp(xmodp+1)(xmodp-1)≡0modp

Since,x≠±1modp , it is known that1≤xmodp-1<xmodp+1<p, ,and P is a prime number. Knowing thatxmodp±1 is relatively prime to p . Then by(xmodp+1)(xmodp-1)≡0modp, according to the division theorems,

xmodp±1≡0modp, This is a contradiction.

So, p is a composite.

Therefore, it has been proved.

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Most popular questions from this chapter

Consider the problem of computing x y for given integers x and y: we want the whole answer, not modulo a third integer. We know two algorithms for doing this: the iterative algorithm which performs y − 1 multiplications by x; and the recursive algorithm based on the binary expansion of y. Compare the time requirements of these two algorithms, assuming that the time to multiply an n-bit number by an m-bit number is O(mn).

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].

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 ?

Show that if a has a multiplicative inverse modulo N, then this inverse is unique (modulo N).

Quadratic residues. Fix a positive integer N. We say that a is a quadratic residue modulo N ifthere exists a such that a≡x2modN.
(a) Let N be an odd prime and be a non-zero quadratic residue modulo N. Show that there are exactly two values in{0,1,....,N-1} satisfying x2≡amodN.
(b) Show that if N is an odd prime, there are exactly(N+1)2 quadratic residues in {0,1,...,N-1}.
(c) Give an example of positive integers a and N such thatx2≡amodNhas more than two solutions in {0,1,...,N-1}.
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