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Chapter 14: Problem 5

Let \(g\) be a primitive root of \(Z_{n}^{*}\). Recall that \(Z_{n}^{*}\) has \(\phi(n)\) elements. Show that \(g^{a}\) is a primitive root of \(Z_{n}^{*}\) if and only if \(\operatorname{gcd}(a, \phi(n))=1\). Note: It follows that, for every \(n\), there are either 0 or \(\phi(\phi(n))\) primitive roots mod \(n\).

Short Answer

Expert verified
To be a primitive root, \(\operatorname{gcd}(a, \phi(n)) = 1\). The number of primitive roots is \(\phi(\phi(n))\).

Step by step solution

01

Definition of a primitive root

Recall that a number g is a primitive root modulo n if the powers of g generate all the elements of the group \(Z_n^*\). That means if \(g^a\) is a primitive root, the smallest positive integer k such that \(g^a \equiv 1\, (\text{mod n})\) is k = \(\phi(n)\).
02

Determine when \(g^a\) generates all elements

We want to show that \(g^a\) is a primitive root if \(\operatorname{gcd}(a, \phi(n)) = 1\). If \(g^a\) is a primitive root, it should generate all elements of \(Z_n^*\), which means we should have \((g^a)^k \equiv 1\, (\text{mod n})\) only if k is a multiple of \(\phi(n)\).
03

Analyze gcd(a, \phi(n)) = 1

If \(\operatorname{gcd}(a, \phi(n)) = 1\), then there exist integers x and y such that \(ax + \phi(n)y = 1\). This implies a has an inverse modulo \(\phi(n)\). Hence, if \(k\) is the smallest positive integer such that \((g^a)^k \equiv 1\, (\text{mod n})\), it follows that k must be \(\phi(n)\).
04

Confirm the inverse

Since \(ax + \phi(n)y = 1\), we have \(g^{ax} \equiv g^{\text{gcd}(a, \phi(n))} \equiv 1\, (\text{mod n})\) because \(g^{\phi(n)} \equiv 1\, (\text{mod n})\). Therefore, the order of \(g^a\) must indeed be \(\phi(n)\) which shows \(g^a\) is a primitive root.
05

Converse direction

For the converse, assume \(\operatorname{gcd}(a, \phi(n)) e 1\). If \(d = \operatorname{gcd}(a, \phi(n)) > 1\), then there is no integer x such that \(ax + \phi(n)y = 1\). Thus, \(g^a\) can't generate all elements of \(Z_n^*\), which means \(g^a\) is not a primitive root.
06

Primitive roots count

It follows that the number of primitive roots modulo n is count of choices for a where \(\operatorname{gcd}(a, \phi(n)) = 1\). This count is exactly \(\phi(\phi(n))\), the Euler's totient function of \(\phi(n)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Primitive Root
In group theory, a primitive root of a number modulo n is an integer g such that every integer coprime to n is congruent to a power of g. That means for a given n, if you take g and keep raising it to increasing powers, you'll cycle through all integers that are coprime to n before repeating the sequence. For example, if g is a primitive root modulo n, then the smallest positive integer k for which g^k is congruent to 1 modulo n is exactly equal to Euler's Totient Function of n, denoted as \(\phi(n)\). This property is very useful in cryptography and number theory.
Euler's Totient Function
Euler's Totient Function, \(\phi(n)\), counts the number of integers from 1 to n that are coprime to n. Two numbers are coprime if their greatest common divisor (GCD) is 1. For instance, if n is a prime number, then every integer from 1 to n-1 is coprime with n, so \(\phi(n)=n-1\). If n is a product of two distinct prime numbers p and q, then \(\phi(n) = (p-1) \cdot (q-1)\). Calculating \(\phi(n)\) is essential for understanding the structure of multiplicative groups modulo n.
Greatest Common Divisor (GCD)
The Greatest Common Divisor of two integers is the largest positive integer that divides both of them without leaving a remainder. For example, the GCD of 8 and 12 is 4. In the context of primitive roots, we are often interested in knowing \(\operatorname{gcd}(a, \phi(n))\). If this GCD is 1, it implies that the number a and \(\phi(n)\) are coprime. This property is crucial for proving whether a power of a primitive root remains a primitive root in a multiplicative group modulo n.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers where numbers 'wrap around' upon reaching a certain value, called the modulus. In this system, two numbers are said to be congruent modulo n if they have the same remainder when divided by n. For instance, 17 and 5 are congruent modulo 12, written as \(\17 \equiv 5 \pmod{12}\). Modular arithmetic is fundamental in various fields such as cryptography, computer science, and number theory. It helps simplify problems involving large numbers by reducing them within a smaller range.
Group Theory
Group Theory is a mathematical discipline that studies the algebraic structures known as groups. A group consists of a set equipped with an operation that combines any two elements to form a third element while satisfying four key properties: closure, associativity, identity, and invertibility. When working with primitive roots, we often consider the multiplicative group of integers modulo n, denoted as \(\mathbb{Z}_n^*\). This group consists of integers from 1 to n-1 that are coprime to n, providing a rich structure for exploring properties like primitive roots and the distribution of numbers.

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Most popular questions from this chapter

Explain what is wrong in the following argument: In Diffie-Hellman key agreement, Alice sends \(A=g^{a}\) and Bob sends \(B=g^{b} .\) Their shared key is \(g^{a b} .\) To break the scheme, the eavesdropper can simply compute \(A \cdot B=\left(g^{a}\right)\left(g^{b}\right)=g^{a b}\)

Let \(\langle g\rangle\) be a cyclic group with \(n\) elements and generator \(g .\) Show that for all integers \(a\), it is true that \(g^{a}=g^{a \% n}\) Note: As a result, \(\langle g\rangle\) is isomorphic to the additive group \(Z_{n}\)

Let \(G\) be a cyclic group with \(n\) elements and generator \(g\). Consider the following algorithm: \(\frac{\operatorname{RAND}(A, B, C):}{r, s, t \leftarrow Z_{n}}\) \(A^{\prime}:=A^{t} g^{r}\) \(B^{\prime}:=B g^{s}\) \(C^{\prime}:=C^{t} B^{r} A^{s t} g^{r s}\) return \(\left(A^{\prime}, B^{\prime}, C^{\prime}\right)\) Let \(D H=\left\\{\left(g^{a}, g^{b}, g^{a b}\right) \in G^{3} \mid a, b, \in Z_{n}\right\\}\). (a) Suppose \((A, B, C) \in D H .\) Show that the output distribution of \(\operatorname{RAND}(A, B, C)\) is the uniform distribution over \(D H\) (b) Suppose \((A, B, C) \notin D H\). Show that the output distribution of \(\operatorname{RAND}(A, B, C)\) is the uniform distribution over \(G^{3}\). (c) Consider the problem of determining whether a given triple \((A, B, C)\) is in the set \(D H\). Suppose you have an algorithm \(\mathcal{A}\) that solves this problem on average slightly better than chance. That is: \(\operatorname{Pr}[\mathcal{A}(A, B, C)=1]>0.51\) when \((A, B, C)\) chosen uniformly in \(D H\) \(\operatorname{Pr}[\mathcal{A}(A, B, C)=0]>0.51\) when \((A, B, C)\) chosen uniformly in \(\mathrm{G}^{3}\) The algorithm \(\mathcal{A}\) does not seem very useful if you have a particular triple \((A, B, C)\) and you really want to know whether it is in \(D H\). You might have one of the triples for which \(\mathcal{A}\) gives the wrong answer, and there's no real way to know. Show how to construct a randomized algorithm \(\mathcal{A}^{\prime}\) such that: for every \((A, B, C) \in \mathbb{G}^{3}\) : $$ \operatorname{Pr}\left[\mathcal{A}^{\prime}(A, B, C)=[(A, B, C) \stackrel{?}{\in} D H]\right]>0.99 $$ Here the input \(A, B, C\) is fixed and the probability is over the internal randomness in A'. So on every possible input, \(\mathcal{A}^{\prime}\) gives a very reliable answer.

Let \(p\) be an odd prime, as usual. Recall that \(Q \mathbb{R}_{p}^{*}\) is the set of quadratic residues mod \(p\) \- that is, \(Q R_{p}^{*}=\left\\{x \in \mathbb{Z}_{p}^{*} \mid \exists y: x \equiv_{p} y^{2}\right\\} .\) Show that if \(g\) is a primitive root of \(Z_{p}^{*}\) then \(\left\langle g^{2}\right\rangle=Q \mathbb{R}_{p^{*}}^{*}\) Note: This means that \(g^{a} \in Q R_{p}^{*}\) if and only if \(a\) is even \(-\) and in particular, the choice of generator \(g\) doesn't matter.

Suppose you are given \(X \in\langle g\rangle .\) You are allowed to choose any \(X^{\prime} \neq X\) and learn the discrete log of \(X^{\prime}\) (with respect to base \(g\) ). Show that you can use this ability to learn the discrete log of \(X\).
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