Question

Write a recursive \(\Theta(n \lg n)\) algorithm whose parameters are three integers \(x, n,\) and \(p,\) and which computes the remainder when \(x^{n}\) is divided by \(p\) For simplicity, you may assume that \(n\) is a power of \(2-\) that is, that \(n=2^{k}\) for some positive integer \(k\)

Short answer

The required recursive algorithm relies on the method of exponentiation by squaring. By halving \(n\) and squaring the result recursively, and applying modulo \(p\) after each multiplication, it achieves \(\Theta(n \lg n)\) time complexity and computes the remainder when \(x^{n}\) is divided by \(p\).

Step-by-step solution

Define the base case for recursion

If \(n\) equals to 1, the remainder when \(x\) is divided by \(p\) is returned.

Define the recursive case

Otherwise, recursively calculate the remainder when \(x^{\frac{n}{2}}\) is divided by \(p\). Multiply the result by itself, and if \(n\) is odd, multiply the result by \(x\). This helps in reducing the number of multiplication operations needed and achieves better performance.

Apply modulo operation

After each multiplication, apply modulo \(p\) to keep the result within the range. This is due to the property: \((a*b)\%p = ((a\%p)*b)\%p\), which will ensure that we will not overflow for large \(x\) and \(n\).

Return the final result

After all these steps we will get the remainder when \(x^{n}\) is divided by \(p\). The algorithm will calculate the power in \(\Theta(n \lg n)\) time complexity.

Key concepts

Modular Exponentiation

Modular exponentiation is a type of exponentiation performed over a modulus. It is frequently used in the field of cryptography. The goal is to compute the remainder of a large exponent expression under a given modulus in an efficient way. This can be expressed as finding the value of \( x^n \mod p \), where \( x \), \( n \), and \( p \) are integers and \( p \) is the modulus. The direct approach of first calculating \( x^n \) and then taking the modulus is often computationally infeasible due to the large size of \( x^n \).

The recursive algorithm presented in the exercise breaks down the problem using the property that \( x^{n} \mod p = (x^{\frac{n}{2}} \times x^{\frac{n}{2}}) \mod p \). This property is leveraged to divide the larger problem into smaller subproblems, which can be solved recursively, eventually reaching the base case when \( n = 1 \). It is a classic example of using mathematical properties to simplify computations and enhance efficiency.

Time Complexity

Time complexity is a measure of the amount of time an algorithm takes to complete as a function of the length of the input. It provides a high-level understanding of the algorithm's efficiency and helps predict how it will scale with the size of the data. In the provided example, the recursive algorithm has a time complexity of \( \Theta(n \lg n) \), suggesting it is more efficient than a naive approach, which might have a complexity of \( O(n^2) \) for computing \( x^{n} \).

The term \( \Theta \) represents a tight bound, meaning the time complexity is both the upper and lower bound of the algorithm's running time. Thus, a \( \Theta(n \lg n) \) complexity indicates that the running time will increase logarithmically as the size of \( n \) increases. This is significantly faster than polynomial or exponential growth, making the algorithm practical for large values of \( n \).

Divide and Conquer

The divide and conquer strategy is a foundational algorithm design paradigm that works by breaking a problem into smaller subproblems, solving each subproblem independently, and combining their solutions to solve the original problem. This is particularly useful when subproblems can be solved more efficiently than tackling the entire problem in one go.

Our modular exponentiation algorithm uses divide and conquer by dividing the exponent \( n \) into halves until it reaches the base case. After each division, it conquers the problem by recombining the solutions. This method's efficiency comes from reducing the number of multiplicative operations required to reach the solution, optimizing both time and computational resources.

Algorithm Optimization

Algorithm optimization involves refining an algorithm to improve its performance and resource utilization. The exercise's algorithm optimizes the modular exponentiation process by using recursion and the modulo operation's mathematical properties to prevent unnecessary calculations and data overflow.

Step 2 and Step 3 of the solution are crucial for optimization. Multiplying the results from recursive calls and using the modulo operation afterwards ensures that the intermediate values remain within a manageable range. This prevents the algorithm from performing computations with excessively large numbers, which would increase the running time and risk arithmetic overflow errors. By optimizing the algorithm, it achieves better performance and becomes capable of handling larger inputs more effectively.