WORK IN PROGRESS
Since the influential works of Carl Friedrich Gauss, we have learned that a polynomial (in multiple variables) over a field can be broken down into a single and unique product of irreducible polynomials. The question of how this decomposition can be done efficiently is the foundation of one of the most prominent and successful areas of computer algebra.
The extended Euclidean algorithm for univariate polynomials over a field is an incredibly important tool, enabling us to solve a variety of complex mathematical problems. Its applications are vast and varied, including the Chinese remainder theorem, interpolation, inversion in extension rings, linear recurrence sequences, linear algebra for sparse matrices using the Krylov method, and more. Perhaps most famously, it is used in the Berlekamp-Massey algorithm for the decoding of BCH-codes. Its importance in modern mathematics cannot be understated, and researchers are constantly finding new and innovative ways of leveraging this powerful tool. Its real world applications range from quantum cryptography, satellite communications, and storage device technology particularly in error correction.
In cryptography, factorization algorithms are used to solve the discrete logarithm problem (DLP), a computational puzzle that is central to many cryptographic protocols. The extended Euclidean algorithm for univariate polynomials can be used with elliptic curves and other algebraic structures to facilitate faster computations in solving DLP problems. This makes them an invaluable tool in modern cryptography, allowing us to create more secure systems by making it difficult for adversaries to break our codes.
Another application of the extended Euclidean algorithm for univariate polynomials is in zero-knowledge succinct non interactive arguments of knowledge (zkSNARKs). These allow one party (the prover) to prove its possession of certain information without revealing any details about what exactly this information is or how they obtained it. zkSNARKs rely on the effectiveness of the Extended Euclidean Algorithm as part of their security protocol; thus, if we want stronger security guarantees from these cryptographic proofs then we need efficient implementations of this algorithm.
Finally, multi-party computation has become increasingly important as organizations seek ways to securely exchange data between parties while maintaining privacy and integrity at all times. Factorization algorithms such as those based on Berlekamp’s Method or Cantor–Zassenhaus’ Algorithm have proven particularly useful here since they allow multiple parties involved in a transaction process different parts simultaneously but still arrive at the same result through a trusted third-party mediator who verifies correctness before finalizing transactions.
The importance and applications of factorization algorithms over univariate polynomials should not be underestimated – researchers today are constantly finding new uses for these tools which further enhance our ability both mathematically and cryptographically speaking! In short, these techniques enable us greater levels accuracy when performing complex calculations related error correction, cryptography, ZKSnarks, and Multi Party Computation.
To summarize some key algorithms and topics within each domain of modern computing and communications:
- Error Correction for data storage and communications The Berlekamp Massey Algorithm enables decoding BCH Codes using Univariate Polynomial Factorizations
- Cryptography Elliptic Curve Cryptography relies upon Univariate Polynomial Factorizations via EEA
- zkSNARKS A combination between Boolean Satisfiability Problem & Univariate Polynomial Factoring helps generate Zero Knowledge Proofs
- Multi Party Computation Berlekamp’s Method & Cantor–Zassenhaus’ Algorithm as it relates to root finding over Galois fields.
After stochastic calculus and combinatronics, polynomial factorization is one of my favorite topics for a long time but this has been much more significant to me for the last 10 years due to my interest in cryptography, blockchains, and secure multi-party computation.
I’ve found myself revisiting concepts that we all learned in high school and college like computing greatest common divisors (gcds), and squarefree factorization. I’ve looked at things I learned during my primary education with a different perspective and interest. Who would have thought I would spend 30 minutes learning more about partial fractions, something I learned in the early 90s in high school and how to decompose them just so I can wrap my head around Taylor series expressions.
Fundamental Theorem of Algebra and finding the roots
A given polynomial of degree n with complex coefficients has exactly n roots (multiplicities counted).”
I was very particular about mentioning Gauss (Gauß) in my first paragraph because he was the first to have created the most convincing proof of this. The quote above was from Alber De Girard in 1629, nearly 170 years before this statement was formally proven in a way that is considered to this day as the most complete. This illustrates how this small subset of algebra is one that has baffled even the greatest mathematical minds of our time and the reason why we are not all cryptoanalysts working for the NSA.
The statement above has profound implications in the world of computers, science, and finance but without a basic grasp of the terminologies it’s difficult for anyone to appreciate it.
A given polynomial of degree \(n\)
Degree refers to the largest exponent in a given polynomial such as the following form:
\[\displaylines{ \begin{equation} p(x) = ax^{n} + bx^{n - 1} + \cdots + cx^{1}\\ \end{equation}\\ \text{where a, b, and c are constants} }\]For my example I will be using a 3rd degree polynomial:
\[p(x) = ax^3 + bx^2 + cx + d\]with complex coefficients Refers to an number with a real and imaginary part. Such as:
\[ax^3 + bx + c\]Here \(a\), \(b\), and \(c\) are real numbers and \(x\) is imaginary. So the complex coefficients for our given polynomial are \(ax^3\) and \(bx\)
has exactly \(n\) roots
\[p(x) = ax^3 + bx^2 + cx + d\]Here roots refer to the values at \(x\) that would make the right hand expression evaluate to \(0\). An example would be a polynomial for a parabola which is a 2nd degree polynomial that intersects the X axis at \(0\) 2 times.
Example
Given a polynomial: \(p(x) = 3x^3 + 4x - 2\) which can be written as \(p(x) = 3x^3 + 0x^2 + 4x^1 - 2\)
Polynomial Factorization
There are many ways of factoring a polynomial but I’m particularly interested in techniques related to computing as they tend to be intuitiely connected to them due to their use in my profession.
The squarefree part \(f_1 \cdots f_r\) of a polynomial \(f = f{_1}^{e_1} \cdots f{_r}^{e_r} \in F[x]\) where \(F\) is a field and \(f_1,\cdots,f_r\) is irreducible and pairwise non-associated is computed by dividing \(f\) by its greatest common divisor (gcd) with its derivative \(f^{'}\), i.e. \(f / gcd(f,f^{'})\).
In fields of characteristic \(p > 0\), one additionally takes some \(p\)-th roots to obtain the squarefree decomposition \(f = lc(f)g_{1}^{1}\cdot g_{2}^{2}\cdots g_{n}^{n}\) where \(g_{i} \in F[X]\) are monic, squarefree, and pairwise relatively prime.
This squarefree decomposition is used in the integration of rational functions via partial fraction decomposition.
Berlekamp [Berlekamp 1967, 1970] devised a deterministic algorithm for factorization of univariate polynomials over finite fields with running time (meaning number of operations in \(\mathbb{F}_{q}\ \ O\sim(n^3 + nq)\) (\(O\sim\) denotes that \(log\ n\) factors are not accounted for).
Berelkamp then devised a faster probabilistic algorithm with a polynomial running time \(O\sim(n^3 + n\ log \ q)\).
Cantor and Zassenhaus [Cantor and Zassenhaus 1981] advanced this further with an algorithm requiring only \(O\sim(n^2\log\ q)\) operations where \(n\) is the degree and length \(\log\ q\) of the representation of a field element.