Enhancements To NTL-extra Polynomial Algebra For Cryptanalysis

Hey guys! I've been diving deep into algebraic cryptanalysis, and I've cooked up some potentially handy functions that could be a sweet addition to PML. I wanted to share what I've been working on and get your thoughts on what might be a good fit for a pull request. Let's dive in!

Factorial Store: Turbocharging Factorial Computations

Currently, the zz_pX_shift_large_characteristic function handles the recomputation of factorials and their inverses. But, these factorials are super useful in many different algorithms, and the current class structure makes it tricky to share them effectively. To solve this, I've built a factorial store. Think of it like the zz_p::init – a global context that manages an auto-resizable array of factorials and their inverses. This way, we can avoid redundant calculations and speed things up across the board.

Why a Factorial Store?

The core idea behind a factorial store is to centralize the management of factorial computations and their inverses. In various algorithms used in cryptanalysis, factorials pop up frequently. Recomputing them every single time is not only inefficient but also a waste of computational resources. By storing these precomputed values, we can significantly reduce the overhead associated with these calculations. The factorial store is designed to be:

• Globally Accessible: Much like the initialization context of prime fields (zz_p::init), the factorial store is designed as a global context. This makes it accessible from anywhere in the code, ensuring that factorial values can be retrieved quickly and easily.
• Auto-Resizable: The store uses an auto-resizable array, which means it can dynamically adjust its size as needed. This is crucial because the maximum factorial value required might not be known in advance. The array grows automatically, accommodating larger factorials without requiring manual resizing or pre-allocation of memory.
• Efficient Storage and Retrieval: The store maintains both factorial values and their inverses. This is particularly useful in cryptographic algorithms where both values are often required. By storing both, we avoid the need to compute inverses on the fly, which can be computationally expensive.

How It Works

The factorial store operates on a simple principle: precompute and store. When a factorial value (or its inverse) is requested, the store first checks if it has already been computed and stored. If it has, the value is returned immediately. If not, the store computes the value, stores it in the array, and then returns it.

This process involves several key steps:

1. Initialization: The store is initialized when the program starts, creating an empty array for factorial values and their inverses.
2. Request Handling: When a factorial or its inverse is requested, the store checks if the value is within the currently computed range.
3. Computation (If Needed): If the value is not in the store, the store calculates it. This involves iteratively computing factorials and their inverses up to the requested value.
4. Storage: Once computed, the factorial and its inverse are stored in the array.
5. Retrieval: The stored value is then returned to the caller.

Benefits in Cryptanalysis

The factorial store can significantly benefit several areas of cryptanalysis:

• Improved Performance: By avoiding redundant computations, the factorial store accelerates algorithms that heavily rely on factorial values. This is particularly noticeable in computations involving binomial coefficients, permutations, and combinations.
• Resource Efficiency: Precomputing and storing factorials reduces the computational load, freeing up resources for other tasks. This is crucial in resource-constrained environments or when dealing with large-scale cryptanalytic problems.
• Code Simplification: The centralized nature of the store simplifies code by providing a single point of access for factorial values. This reduces code duplication and makes the codebase easier to maintain.

In the context of the zz_pX_shift_large_characteristic function, integrating the factorial store would mean that factorials and their inverses are computed once and then reused, rather than being recomputed each time the function is called. This can lead to significant performance gains, especially in scenarios where this function is used repeatedly.

Bivariate Polynomials (zz_pXY): Leveling Up the Polynomial Game

I've been playing around with extending bivariate polynomials (zz_pXY) with some cool new features. Here's what I've added:

• Overloaded operator>>: Makes it super easy to load polynomials.
• power function: Follows NTL's convention for consistency.
• 2D Taylor Shift: This is the big one! It computes f(x + a, y + b) in a single bivariate multiplication. There might even be a way to optimize this further by using middle_product for the top part of the multiplication, similar to how the 1D Taylor Shift is implemented. This could seriously boost performance.
• Representation switch of X and Y: Might not be super useful for everyone, but it's there if you need it.

Digging Deeper into Bivariate Polynomials

Bivariate polynomials, which involve two variables (typically denoted as x and y), are fundamental in numerous areas of mathematics and computer science, including cryptography. Extending the functionalities for handling these polynomials can open up new avenues for cryptanalysis and other applications. My extensions to the zz_pXY class aim to provide a more robust and efficient toolkit for working with these polynomials.

Overloading operator>> for Ease of Use

One of the primary goals in software development is to make tools as user-friendly as possible. The overloaded operator>> for zz_pXY facilitates the easy loading of polynomials from input streams. This means that you can read polynomial coefficients directly from a file or any other input source without having to write custom parsing code. The new operator>> simplifies the process of initializing and manipulating bivariate polynomials, reducing the potential for errors and saving valuable development time.

The Power of the power Function

Exponentiation is a common operation in polynomial algebra. The power function, which I've added to the zz_pXY class, efficiently computes polynomial powers. By following NTL's established conventions, the function ensures consistency and ease of use for those already familiar with NTL's polynomial arithmetic. This function is crucial for cryptographic applications that involve polynomial exponentiation, such as key exchange protocols and signature schemes.

2D Taylor Shift: A Game Changer

The 2D Taylor shift is perhaps the most significant addition to the zz_pXY class. This operation computes f(x + a, y + b) for a bivariate polynomial f(x, y) and constants a and b. The traditional method of computing this shift involves multiple polynomial multiplications and additions, which can be computationally intensive. My implementation performs this computation in a single bivariate multiplication, leading to a substantial improvement in performance.

Potential Optimization with middle_product

Further optimizing the 2D Taylor shift is an exciting prospect. The middle_product technique, which is used in the 1D Taylor shift implementation, could potentially be adapted for the 2D case. The idea is to focus on the essential parts of the multiplication to reduce computational overhead. By utilizing middle_product, we might be able to achieve even faster Taylor shift computations, making complex cryptanalytic tasks more feasible.

Representation Switch of X and Y: Niche but Useful

While not universally applicable, the ability to switch the representation of x and y in a polynomial can be valuable in certain situations. This feature allows you to view the polynomial from a different perspective, which might simplify specific computations or analyses. Although its use cases may be limited, having this option available adds flexibility to the zz_pXY class.

Tri-variate Polynomials (zz_pXYZ): Extending to Three Dimensions

I've also been experimenting with tri-variate polynomials (zz_pXYZ). These polynomials have three variables, making them even more powerful for representing complex relationships. I've tried to keep the same conventions and overloads as zz_pXY to make the transition smooth. Think of it as taking everything we love about bivariate polynomials and adding another dimension!

The Power of Three Variables

Tri-variate polynomials, involving three variables, represent an extension of bivariate polynomials into a higher dimension. This added complexity allows for the modeling of more intricate relationships and systems, making them valuable in various fields, including cryptography, coding theory, and algebraic geometry. The zz_pXYZ class aims to provide a comprehensive set of tools for manipulating these polynomials efficiently.

Consistency with zz_pXY

To ensure a seamless transition for users already familiar with the zz_pXY class, I've maintained similar conventions and overloads in the zz_pXYZ implementation. This consistency simplifies the learning curve and allows users to apply their existing knowledge to the new class. The goal is to make the zz_pXYZ class feel like a natural extension of the bivariate polynomial tools.

Key Features and Overloads

The zz_pXYZ class includes several key features and overloads designed to facilitate the manipulation of tri-variate polynomials:

• Basic Arithmetic Operations: Standard arithmetic operations such as addition, subtraction, and multiplication are overloaded, allowing for intuitive polynomial manipulation.
• Input/Output Overloads: Overloads for input and output streams enable easy reading and writing of polynomials from and to files or other data sources.
• Evaluation Functions: Functions for evaluating the polynomial at specific points, which are crucial for various applications in cryptography and algebraic cryptanalysis.
• Differentiation and Integration (Potential): While not yet fully implemented, future extensions could include differentiation and integration functions, further expanding the utility of the class.

Applications in Cryptography

Tri-variate polynomials can be particularly useful in cryptographic applications where complex relationships between multiple variables need to be modeled. For example, they can be used in:

• Multivariate Cryptography: Cryptosystems based on multivariate polynomials often use tri-variate or higher-order polynomials to achieve security. The zz_pXYZ class provides a tool for analyzing and potentially breaking these systems.
• Secret Sharing Schemes: Tri-variate polynomials can be used to construct secret sharing schemes where a secret is divided among multiple parties, and a certain number of parties are required to reconstruct the secret. The class can aid in the design and analysis of such schemes.
• Algebraic Cryptanalysis: Techniques that involve solving systems of polynomial equations can benefit from the zz_pXYZ class, as it provides efficient tools for manipulating and analyzing these equations.

N-variate Polynomials (zz_pXi): Experimental Territory!

Now, things get really interesting! I've been playing with N-variate polynomials (zz_pXi) using a recursive template. WARNING: This is still experimental! The representation is super fast, but it can be a bit tricky to use because of the templates. But, the potential here is huge – imagine polynomials with any number of variables!

Exploring the Realm of N-variate Polynomials

N-variate polynomials represent a significant generalization of polynomial algebra, allowing for an arbitrary number of variables. This capability is essential for modeling highly complex systems and relationships in various fields, including cryptography, machine learning, and computational mathematics. The zz_pXi<N> class, although experimental, aims to provide a powerful tool for handling these polynomials.

Recursive Template Approach

The implementation of zz_pXi<N> uses a recursive template, a technique that allows for the creation of a class that can adapt to different numbers of variables at compile time. This approach offers several advantages:

• Flexibility: The class can handle any number of variables, making it highly versatile for various applications.
• Efficiency: The recursive structure can lead to efficient computations, as the compiler can optimize the code for specific numbers of variables.
• Code Reusability: The template-based approach reduces code duplication, as the same code structure is used for different numbers of variables.

However, the use of templates also introduces some challenges:

• Complexity: Template code can be more complex to write and debug than non-template code.
• Usability: The need to specify the number of variables at compile time can make the class less user-friendly in some scenarios.
• Compile Times: Excessive use of templates can lead to longer compile times.

Fast Representation

One of the key goals in designing zz_pXi<N> was to achieve a fast representation of N-variate polynomials. The recursive template structure allows for a compact and efficient storage of coefficients, which is crucial for performance. This fast representation is particularly beneficial in cryptographic applications where large polynomials with many variables are often used.

Potential Applications

N-variate polynomials have a wide range of potential applications:

• Multivariate Cryptography: Cryptosystems based on multivariate polynomials can leverage zz_pXi<N> to represent and manipulate polynomials with a large number of variables. This can lead to more secure and efficient cryptographic schemes.
• Machine Learning: Polynomial models are used in various machine learning algorithms. The ability to handle N-variate polynomials efficiently can improve the performance of these algorithms.
• Coding Theory: N-variate polynomials can be used in the construction and analysis of error-correcting codes. The zz_pXi<N> class can aid in the development of more powerful codes.

Pretty Print (pp): Making Polynomials Look Pretty

Finally, I've overloaded a function called pp (for