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Introduction to the Theory of Computation

Sipser Michael

Chapter 7: Chapter 7

Problem 13

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Let $M O D E X P=\\{\langle a, b, c, p\rangle \mid a, b, c$, and $p$ are positive binary integers such that $\left.a^{b} \equiv c(\bmod p)\right\\}$. Show that $M O D E X P \in P$. (Note that the most obvious algorithm doesn't run in polynomial time. Hint: Try it first where $b$ is a power of 2 .)

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The problem $MODEXP$, defined as $MODEXP =\\{\langle a, b, c, p\rangle \mid a, b, c, and p$ are binary integers such that $a^{b} \equiv c(\bmod p)\right\\}$, can be solved using an algorithm running in polynomial time. In the case where $b$ is a power of 2, the result can be efficiently computed by successively squaring the base mod $p$. This concept can be extended to arbitrary exponents $b$ with a divide-and-conquer approach, breaking down the exponentiation into smaller components and utilizing the squaring method. This approach, known as "exponentiation by squaring", evaluates even and odd exponents differently: for even exponents, we compute $(a^{(b/2)} \cdot a^{(b/2)}) \pmod{p}$, and for odd exponents, we compute $(a^{(b-1)} \cdot a) \pmod{p}$. This algorithm shows that $MODEXP$ is indeed a class $P$ problem, thus, the problem $MODEXP \in P$, as required.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Understanding the Modular Exponentiation Problem

Begin by observing that we're trying to calculate $a^b \equiv c \pmod{p}$, which means that we need to find the remainder when $a^b$ is divided by $p$. Our task is to prove that an algorithm for this problem exists that can operate in polynomial time with respect to the input length. Step 2:

Step 2: Employing the Squaring Technique

For the case where $b$ is a power of 2, we can efficiently find the result using the squaring technique. Essentially, this method computes the modular exponentiation by successively squaring the base modulo $p$. Here's how the algorithm would look: 1. Initialize the result as $a$ and the exponent counter as $k=1$. 2. While $k < b$: a. Set the result as the square of the current result, modulo $p$. b. Double the exponent counter: $k = 2k$. At the end of this process, the result will be equal to $a^b \pmod{p}$. Step 3:

Step 3: Extending to Arbitrary Exponents

For general exponents $b$, we need to adjust the algorithm to accommodate any non-power-of-two values. We can do this by employing a divide-and-conquer strategy, breaking down the exponentiation into smaller components and reusing the squaring method. The adapted algorithm, known as "exponentiation by squaring," can be outlined as follows: 1. If $b = 0$, return 1 (because $a^0 = 1$). 2. If $b = 1$, return $a$ (since $a^1 = a$). 3. If $b$ is even, compute $a^{b/2}$, then square the result and take the modulo $p$: $(a^{(b/2)} \cdot a^{(b/2)}) \pmod{p}$. 4. If $b$ is odd, compute $a^{(b-1)}$, then multiply it by $a$ modulo $p$: $(a^{(b-1)} \cdot a) \pmod{p}$$. Applying the algorithm recursively, we can evaluate the modular exponentiation in polynomial time with respect to the input length. Step 4:

Step 4: Showing that $MODEXP$ is a Class P Problem

Since we've successfully designed a polynomial-time algorithm for evaluating the modular exponentiation operation using the "exponentiation by squaring" technique, it demonstrates that $MODEXP$ is indeed a class P problem. Now we've shown that $MODEXP \in P$, as required by the problem.

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Chapter 7: Chapter 7

Short Answer

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Step 1: Understanding the Modular Exponentiation Problem

Step 2: Employing the Squaring Technique

Step 3: Extending to Arbitrary Exponents

Step 4: Showing that \(MODEXP\) is a Class P Problem

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More chapters from the book ‘Introduction to the Theory of Computation’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 8

Chapter 9

Chapter 10

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