Question

Assume \(2_{R} \in R^{*}\). Suppose that we are given two polynomials \(g, h \in R[X]\) of length at most \(\ell,\) along with a primitive \(2^{n}\) th root of unity \(\omega \in R,\) where \(2 \ell \leq 2^{n}<4 \ell .\) Let us "pad" \(g\) and \(h,\) writing \(g=\sum_{i=0}^{2^{n}-1} a_{i} X^{i}\) and \(h=\sum_{i=0}^{2^{n}-1} b_{i} X^{i},\) where \(a_{i}\) and \(b_{i}\) are zero for \(i \geq \ell\). Show that the following algorithm correctly computes the product of \(g\) and \(h\) using \(O(\ell \operatorname{len}(\ell))\) operations

Short answer

Short Answer: The algorithm pads the polynomials to a power of two, evaluates them using FFT, takes their element-wise product, and performs the inverse FFT to obtain the coefficients of the product polynomial. We prove its correctness by showing that after evaluating using FFT, the element-wise product corresponds to multiplying coefficients in the Fourier basis, thus reconstructing the coefficients of the product polynomial. Finally, since the FFT and inverse FFT have time complexity of O(n log n) and n = 2^x, where 2l ≤ n < 4l, the algorithm has a complexity of O(l log l) as required.

Step-by-step solution

Padding the polynomials

First, we are given two polynomials \(g\) and \(h\), and we need to "pad" them. This means representing these polynomials with coefficients up to \(2^n-1\) by adding zero coefficients to each polynomial, i.e., for \(i \geq \ell\), set \(a_i = 0\) for \(g\) and \(b_i = 0\) for \(h\). This ensures that the polynomials have degree \(2^n - 1\).

Understand the algorithm

We do not have the algorithm directly specified, but we can use the given information to extract it: 1. Pad the polynomials \(g\) and \(h\) as described in Step 1, writing \(g=\sum_{i=0}^{2^{n}-1} a_{i} X^{i}\), \(h=\sum_{i=0}^{2^{n}-1} b_{i} X^{i}\). 2. Evaluate \(g\) and \(h\) at the \(2^n\) powers of \(\omega\) (i.e., \(g(\omega^i)\) and \(h(\omega^i)\) for \(0 \leq i < 2^n\)) using the Cooley-Tukey FFT algorithm in \(O(\ell \operatorname{len}(\ell))\) time. 3. Take the product of the evaluated values (i.e., \(c_i = g(\omega^i) \cdot h(\omega^i)\)) for all \(i\). 4. Perform the inverse FFT on the resulting values \(c_i\) to get the coefficients of the product \(gh\).

Prove the correctness of the algorithm

Since the algorithm evaluates \(g\) and \(h\) using Fourier-transform-based techniques, we need to prove that this algorithm computes the desired polynomial product. First, notice that evaluating \(g\) and \(h\) at the \(2^n\) powers of \(\omega\) essentially gives us the values of \(g\) and \(h\) in the Fourier basis. When we take their element-wise products (i.e., compute \(c_i = g(\omega^i) \cdot h(\omega^i)\)), we are multiplying the coefficients of the Fourier representation of \(g\) and \(h\) pointwise.\ Finally, taking the inverse FFT of the obtained values \(c_i\) reconstructs the coefficients of the product polynomial \(gh\). Thus, the algorithm correctly computes the product of polynomials \(g\) and \(h\).

Analyzing the algorithm's complexity

Since the Cooley-Tukey FFT and its inverse both have \(O(n\log n)\) time complexity for \(n\)-point FFTs, and since we've padded \(g\) and \(h\) to have \(2^n\) points (where \(2\ell \leq 2^n < 4\ell\)), we can write the complexity as\ \(O((2^n)\log(2^n)) = O(2^n\cdot n)\).\ Now, we know that \(2^n = O(\ell)\) based on the problem constraints, so we can substitute that in:\ \(O((2^n) \cdot n) = O(\ell \cdot \log{\ell})\).\ Therefore, the algorithm has a complexity of \(O(\ell \operatorname{len}(\ell))\) as required.

Key concepts

Computational Complexity

The concept of computational complexity is a cornerstone of computer science and mathematics, particularly in algorithm analysis. It refers to the amount of resources required for an algorithm to solve a problem, given the size of the input, known as the 'problem size' or 'input size'. These resources can be computational time or memory space.

Complexity is often expressed using Big O notation, which provides an upper bound on the growth rate of the resource requirements. For instance, an algorithm with a complexity of O()) takes approximately n steps to complete, growing linearly with the input size. In the case of polynomial multiplication, we focus on time complexity—the number of operations needed to compute the product.

When examining the complexity of polynomial multiplication, specifically through the use of the Fast Fourier Transform (FFT), we aim to transform a naively quadratic problem—one with complexity of O(n^2)—into a more efficient one. The Cooley-Tukey FFT algorithm, as used in this textbook exercise, achieves an impressive O(nlogn) time complexity for multiplying polynomials padded to length n. This efficiency gain is a game-changer for computations in numerous fields, such as signal processing, data analysis, and even cryptography.

Cooley-Tukey FFT Algorithm

The Cooley-Tukey Fast Fourier Transform (FFT) algorithm revolutionized the way we compute the discrete Fourier Transform (DFT) of a sequence. It is a divide-and-conquer algorithm that recursively breaks down the DFT of a sequence into smaller DFTs, significantly reducing the number of calculations required.

For a sequence of length n, a naive DFT computation would require O(n^2) operations; however, the Cooley-Tukey FFT reduces this to O(nlogn), which is a dramatic improvement, particularly for large n. This reduction is achieved by exploiting the periodic and symmetrical properties of the complex roots of unity, which are the foundation of the algorithm's operation.

In the context of polynomial multiplication, the key step involves evaluating the polynomials at the points corresponding to the nth roots of unity, which then allows the coefficients of the product polynomial to be determined efficiently. This algorithm is the backbone of many digital technologies we use today, due to its ability to handle large-scale computations swiftly.

Primitive Root of Unity

A primitive root of unity is a concept taken from the field of complex numbers and plays a fundamental role in the FFT algorithm and by extension, in polynomial multiplication. Specifically, an nth primitive root of unity is a complex number omega) that satisfies the equation omega^n = 1), while omega^k eq 1) for any smaller positive integer k.

The powers of a primitive root of unity are particularly useful in FFT because they form a cycle that covers all nth distinct complex roots of uni.. These are exactly the points where the polynomial is evaluated in the FFT algorithm. The use of a primitive root of unity ensures that the transform has the necessary properties to permit the efficient combination of the results from the divided subproblems.

In the given polynomial multiplication problem, the mention of a primitive 2^n th root of unity is crucial since it guarantees the right set of evaluation points for the polynomials before applying the FFT algorithm. This harmonious link between algebra and computation underlies many of the optimizations that make modern data processing possible.

Polynomial Padding

Polynomial padding, as encountered in the following textbook exercise, is a technique used to prepare polynomials for efficient multiplication using the FFT algorithm. The essential idea is to add zero-coefficients to the polynomials until they reach a desired length, which is usually a power of two.

Padding serves multiple purposes. First, it standardizes the length of the polynomials, simplifying the application of the FFT algorithm. Second, it prevents the wraparound effect, where the coefficients of the product of two polynomials could interfere with each other due to the cyclical nature of the FFT. Finally, padding to a power of two is particularly important in the Cooley-Tukey FFT algorithm because it enables the algorithm to split the polynomial into halves repeatedly during the recursive computation.

Through padding, we can ensure that these polynomials are accurately represented in a way that facilitates their multiplication using the sophisticated mechanisms of FFT. This step, though seemingly simple, is an integral part of the polynomial multiplication process that relies on the FFT algorithm.