Algebraic Circuits

This page contains notes on Algebraic Circuits by Antonio Lloris Ruiz, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos

Summary

Chapter Summaries

• Chapter 1: Number Systems: Devoted to various number systems, providing a complete revision of different numeral representations for integer numbers, including signed, unsigned, and redundant systems. It covers positional notation, base conversion, binary arithmetic, modular arithmetic, and fixed-point representation for fractional numbers. The main procedures for basic arithmetic operations like addition, subtraction, multiplication, division, and square root are also introduced. This chapter lays the foundational understanding of how numbers are represented and manipulated, which is essential for designing arithmetic circuits.

• Chapter 2: Basic Arithmetic Circuits: This chapter presents the design of fundamental arithmetic circuits used for operating with integer numbers, which serve as building blocks for more complex algebraic circuits. It details circuits for addition, subtraction, multiplication, and division, with special attention given to modular reduction techniques. Additionally, the chapter describes comparator circuits for comparing numerical values and shifter circuits for bitwise shifting operations.

• Chapter 3: Residue Number Systems: This chapter deals with Residue Number Systems (RNS), which are numerical representation systems with interesting applications, particularly due to their carry-free properties for certain arithmetic operations. It introduces the concept of representing integers using their residues with respect to a set of moduli and explores arithmetic operations within RNS. The chapter also introduces Galois fields ${\text{GF}}(p)$, as modular operations with a prime modulus $p$ are implemented in ${\text{GF}}(p)$.

• Chapter 4: Basic Algebraic Circuits: This chapter defines and explores basic algebraic circuits, focusing primarily on Linear Feedback Shift Registers (LFSRs) and Cellular Automata (CAs) . It introduces classic LFSRs (LFSR mod 2) and generalizes them to LFSRmod$2^{m}$, LFSRmod$p$, and LFSRmod$p^{m}$, discussing their association with polynomials. The chapter studies mainly one-dimensional linear CAs but also defines two-dimensional CAs, outlining their structure and rules.

• Chapter 5: Galois Fields ${\text{GF}}(2^{m})$: This chapter is dedicated to the study of Galois Fields ${\text{GF}}(2^{m})$, which are fundamental in many applications like coding theory and cryptography. It presents circuits for implementing key arithmetic operations such as addition, multiplication, squaring, square root, exponentiation, inversion, and division of polynomials over these fields. These operations are explored using various representations including power representation, standard basis, normal basis, and dual basis, and also covers operations in composite Galois fields ${\text{GF}}((2^{n})^{m})$.

• Chapter 6: Galois Fields ${\text{GF}}(p^{m})$: This chapter parallels Chapter 5 but focuses on Galois Fields of the form ${\text{GF}}(p^{m})$ . It covers the arithmetic operations (addition, subtraction, multiplication, exponentiation, inversion, division) within these fields. The chapter details implementations using different polynomial representations, including power representation and standard, normal, and dual bases, extending the concepts from ${\text{GF}}(2^{m})$ to fields of prime characteristic $p>2$. It also includes operations in GF(p).

• Chapter 7: Two Galois Fields Cryptographic Applications: This chapter illustrates the practical use of Galois fields by describing two cryptographic applications. The first application is based on discrete logarithms, using the Galois field ${\text{GF}}(2^{233})$ as a real example. The second application is devoted to elliptic curve cryptography, exemplified with the Galois field ${\text{GF}}(2^{192}-2^{64}-1)$.

• Appendices: The appendices provide essential mathematical background for the topics discussed in the book.
  • Appendix A: Finite or Galois Fields: Outlines the postulates and theorems concerning Galois fields.
  • Appendix B: Polynomial Algebra: Focuses on the algebra of polynomials, paying particular attention to their different forms of representation.
  • Appendix C: Elliptic Curves: Includes all matters relating to elliptic curves that are used in the application examples of Galois fields developed in Chapter 7.

Details

Chapter 1 - Key Conceptual Operations and Ideas

Summary: Chapter 1 of Algebraic Circuits lays the crucial groundwork for understanding all subsequent topics in the book. It covers various number systems, their representations, and fundamental arithmetic operations. Below is a list of key conceptual operations and ideas from Chapter 1, along with direct connections and examples of their relevance to later material in the book.

• Section 1.1: Introduction
  • Key Ideas:
    • Distinction between additional notation (like Roman numerals) and positional notation (e.g., decimal, binary).
    • Concept of a base or radix in positional systems.
  • Relevance to Later Material:
    • Foundation: Positional notation is fundamental to representing numbers in digital systems and is implicitly used in all circuits discussed later for arithmetic (Chapter 2), Residue Number Systems (Chapter 3), and Galois Field elements (Chapters 3, 5, 6, 7).
    • Example: The representation of polynomial coefficients in ${\text{GF}}(p^{m})$ (Chapters 5, 6, Appendix B) relies on positional notation for the coefficients themselves if $p$ or $m$ is greater than 2.

• Section 1.2: Positional Notation Using One Base
  • 1.2.1 Most Efficient Radix
    • Key Idea: Theoretical derivation showing that base $e$ (Euler's number) is most efficient for representation, followed by base 3, and then base 2. This justifies the use of binary (base 2) in digital computers due to implementation reliability.
    • Relevance: Explains the prevalence of binary systems (${\text{GF}}(2)$, ${\text{GF}}(2^{m})$) which are central to Chapters 4, 5, and many circuits in Chapter 2.
  • 1.2.2 Base Conversion
    • Key Idea: Algorithms for converting numbers between different bases (e.g., decimal to binary, binary to base $b$).
    • Relevance:
      • Chapter 2 (Basic Arithmetic Circuits): Digital systems often interface with decimal inputs/outputs, requiring conversion to/from internal binary representations for arithmetic circuits.
      • Chapter 3 (Residue Number Systems): Converting numbers to their residue representations for different moduli ($m_{i}$) is a form of base conversion (Section 3.7).
      • Example: Converting a decimal number to its binary form before processing by binary adders or multipliers (Chapter 2).
  • 1.2.3 Bases Power of Two (Binary, Octal, Hexadecimal)
    • Key Idea: Focus on binary representation, octal, and hexadecimal as compact forms. Introduction to binary arithmetic operations (addition, subtraction, multiplication, division examples).
    • Relevance:
      • Chapter 2 (Basic Arithmetic Circuits): Directly underpins the design of all binary arithmetic circuits (adders, subtractors, multipliers, dividers in Sections 2.2-2.6). The binary arithmetic rules shown in Table 1.1 (p. 9) are implemented by these circuits.
      • Chapter 5 (Galois Fields ${\text{GF}}(2^{m})$): Elements of ${\text{GF}}(2^{m})$ are represented as polynomials with binary coefficients, and operations involve binary arithmetic.
  • 1.2.4 Modular Arithmetic
    • Key Idea: Defines $N{\pmod {M}}$, properties of modular addition and multiplication, and the concept of modular reduction, including multiplicative modular reduction.
    • Relevance: This is one of the most critical foundational sections.
      • Chapter 2 (Basic Arithmetic Circuits): Section 2.6.4 discusses modular reduction circuits, and Section 2.6.7 revisits it.
      • Chapter 3 (Residue Number Systems): The entire chapter is built upon modular arithmetic. RNS represents numbers by their residues modulo a set of bases (Section 3.1, 3.2). Modular circuits for addition, subtraction, multiplication, etc., are detailed in Section 3.8.
      • Chapter 4 (Basic Algebraic Circuits): LFSRmod$p$ (Section 4.2) and Cellular Automata mod$p$ (Section 4.5.1) operate using arithmetic modulo $p$.
      • Chapter 6 (Galois Fields ${\text{GF}}(p^{n})$): Operations in ${\text{GF}}(p)$ (Section 6.1) are modular arithmetic operations. Operations in ${\text{GF}}(p^{m})$ involve polynomial arithmetic where coefficients are handled using ${\text{GF}}(p)$ arithmetic.
      • Chapter 7 (Cryptographic Applications): Cryptography heavily relies on operations in finite fields, which are based on modular arithmetic (e.g., discrete logarithm problems, elliptic curve operations over finite fields).
      • Example: The expression for $N\operatorname {mod} M=\{a_{n}(b^{n}\operatorname {mod} M)+\dots +a_{0}(b^{0}\operatorname {mod} M)\}\operatorname {mod} M$ is used in Chapter 3 (e.g., Example 3.1, p. 137, for reducing modulo 251 for RNS related operations) and informs circuit design for modular reduction.
  • 1.2.5 Fractional Numbers: Fixed Point Representation
    • Key Idea: Representing fractional numbers with a fixed number of fractional digits (unit in the last position - ulp).
    • Relevance: While the book primarily focuses on integer arithmetic for algebraic circuits and Galois fields, this concept is fundamental in digital signal processing where such circuits might be applied. The core arithmetic operations (Chapter 2) can be adapted for fixed-point numbers.

• Section 1.3: Multiple Radix Representations
  • 1.3.1 Double Radix
    • Key Idea: Representing an integer $N$ as $\sum d_{ij}2^{i}3^{j}$.
    • Relevance: This is a specialized representation. While not central to the main themes of LFSRs or standard Galois Field arithmetic in the book, it shows alternative algebraic structures.
  • 1.3.2 Mixed Radix System
    • Key Idea: Representing an integer $X$ as $a_{n}p_{n}+\dots +a_{1}p_{1}$ where weights $p_{i}$ are products of different radices $b_{i}$.
    • Relevance:
      • Chapter 3 (Residue Number Systems): This is explicitly stated as being of "special interest for the Residue Number System, as will be outlined in Chap. 3". The Mixed Radix System is used for converting numbers from RNS back to a positional system (an alternative to or component of applying the Chinese Remainder Theorem).
      • Example: Example 1.1 (p. 17) shows converting a number to a mixed radix system, a process analogous to parts of RNS conversion.

• Section 1.4: Negative Integer Numbers
  • Key Ideas: Representations for signed integers: Sign-Magnitude (SM), Complement (base complement like Two's Complement, and base-1 complement like One's Complement), and Biased Representation. Includes how addition/subtraction work for each.
  • Relevance:
    • Chapter 2 (Basic Arithmetic Circuits): Crucial for designing circuits that handle signed numbers.
      • SM (Section 1.4.1): Informs SM adders/subtractors, multiplication where sign is handled separately (Section 2.4.1).
      • Complement representations (Section 1.4.2): Lead to efficient adder/subtractor designs (e.g., two's complement adder/subtractor in Fig. 2.6e, p. 79). Overflows in complement arithmetic (Section 1.4.2.5) are detected in circuits.
      • Biased Representation (Section 1.4.3): Relevant for comparators (Section 2.8) as it preserves order, and potentially for exponent representation in more advanced arithmetic units. Binary adders can be adapted for biased numbers (Section 2.2.1, p. 77).
    • Example: The rules for addition/subtraction in two's complement (Section 1.4.2.6, p. 29-31) directly translate to the design of two's complement arithmetic circuits in Chapter 2.

• Section 1.5: Binary Numbers Multiplication
  • Key Ideas: Multiplication algorithms for binary numbers in SM and complement representations.
  • Relevance:
    • Chapter 2 (Basic Arithmetic Circuits): Forms the basis for combinational and sequential multiplier designs (Section 2.4).
      • SM multiplication (Section 1.5.1) is often simpler to conceptualize for magnitude and sign separately, influencing some multiplier architectures.
      • Complement multiplication (Section 1.5.2) often requires corrections or conversion, which must be handled by the circuits if numbers are kept in complement form (e.g., Booth algorithm, which relates to signed digit concepts from Section 1.8, is used for efficient complement multiplication).
    • Example: The method of generating partial products and summing them (implied in Section 1.2.3 and built upon in 1.5) is implemented in array multipliers in Chapter 2 (e.g., Fig 2.8, p. 82).

• Section 1.6: Division and Square Root of Binary Integer Numbers
  • Key Ideas: Algorithms for integer division (obtaining quotient and remainder) and square root, typically based on shifting and subtraction.
  • Relevance:
    • Chapter 2 (Basic Arithmetic Circuits): These algorithms are implemented as hardware circuits in Section 2.6 (Divisors and Square Root circuits, p. 98-101, 111-112).
    • Example: The division algorithm described in Section 1.6.1 (Table 1.6, p. 42) is realized by the sequential divisor in Fig 2.14e (p. 100).
    • While division and square root in Galois Fields (Chapters 5, 6) often use different algebraic techniques (like finding multiplicative inverses), the basic understanding of these integer operations is foundational.

• Section 1.7: Decimal Numbers
  • Key Ideas: BCD (Binary-Coded Decimal) representation, other decimal codes (Excess-3, Gray), BCD arithmetic (sum, negative numbers), and Packed BCD (CHC) for efficiency.
  • Relevance:
    • Chapter 2 (Basic Arithmetic Circuits): Leads to the BCD Adder/Subtracter designs in Section 2.7 (p. 112-113).
    • Example: The BCD sum correction rule (adding 6 when sum > 9 or carry out, Section 1.7.1, p. 45-46) is implemented in BCD adder circuits (Fig 2.21, p. 113).
    • This is less central to the later chapters on Galois fields based on ${\text{GF}}(2)$ or ${\text{GF}}(p)$ where $p$ is a small prime, but important for systems needing direct decimal arithmetic.

• Section 1.8: Signed Digits
  • Key Ideas: Introducing redundancy by allowing digits to be negative (e.g., $\{-1,0,1\}$ for BSD). Covers conversion, Binary Signed Digits (BSD), Canonical Codification (Non-Adjacent Form - NAF), and Booth codification.
  • Relevance:
    • Speeding up Arithmetic (Chapter 2): The primary motivation is to manage or eliminate long carry-propagation chains in addition (explained further in Section 1.9).
    • Exponentiation (Chapters 2, 5, 6, 7): Canonical codification (NAF, Section 1.8.3.2, p. 60-61) is highly relevant for efficient exponentiation algorithms (used in Chapters 2.5, 5.7, 6.7.3, and crucial for cryptographic applications in Chapter 7). NAF minimizes non-zero terms in an exponent, reducing the number of multiplications needed. The exponentiation circuit in Fig 2.13c (p. 94) explicitly mentions using canonical development.
    • Multiplication (Chapter 2): Booth encoding (Section 1.8.3.3, related to Table 1.13, p. 59) is a common technique to improve multipliers, especially for signed numbers, by reducing the number of partial products.
    • Example: The algorithm for converting to NAF (Table 1.14, p. 61) helps in optimizing $N^{P}$ where $P$ is the exponent, by representing $P$ in NAF.

• Section 1.9: Redundant Number Systems
  • Key Ideas: How using redundant number systems (like those with signed digits) can limit or eliminate long carry propagation in addition, making addition constant-time or faster.
  • Relevance:
    • Chapter 2 (Basic Arithmetic Circuits): Provides a theoretical basis for designing high-speed adders.
    • Chapter 3 (Residue Number Systems): RNS achieves carry-free addition (and multiplication and subtraction) between the different moduli channels by using a redundant representation across the system, directly relating to the core idea of avoiding full-length carry propagation.
    • Example: The binary signed digit addition in Table 1.18 and 1.19 (p. 69) shows how carry is limited to adjacent positions, a principle that makes redundant adders faster.

In essence, Chapter 1 provides the entire numerical and basic algorithmic toolkit that is then translated into hardware (Chapter 2) and applied within more complex algebraic structures like Residue Number Systems (Chapter 3) and Galois Fields (Chapters 4-7) for building advanced algebraic circuits.

Notes

Applications of Algebraic Circuits to Cellular Automata

The Connection:

The book defines algebraic circuits in two main ways:

1. Digital circuits whose behavior can be linked to an algebraic structure, often a polynomial, where the circuit's evolution mirrors the polynomial's algebraic properties. Cellular automata fall into this category.
2. Digital circuits designed to implement operations within specific algebraic structures, such as Galois fields (finite fields).

Conway's Game of Life, with its grid of cells evolving based on simple rules concerning their neighbors, is a prime example of a cellular automaton. Although its rules are non-linear, leading to complex emergent behavior, its operation can be conceptualized in terms of circuitry and analyzed using algebraic methods.

Applications and Interpretations:

• Computational Universality and Circuit Implementation: A key aspect of Conway's Game of Life is its Turing completeness, meaning it possesses the computational power of a universal Turing machine. This implies that any algorithmic computation, including the operations of digital logic circuits, can theoretically be performed within the Game of Life. Researchers have demonstrated how to construct logic gates (like NOT and AND gates) using specific patterns of live and dead cells, such as "gliders" and "glider guns," which interact in predictable ways to perform logical operations. These constructed logic gates are, in essence, a form of algebraic circuit implemented within the cellular automaton framework.

• State Transition and Algebraic Representation: The evolution of each cell in the Game of Life is determined by the state of its eight neighbors (the Moore neighborhood). This transition rule can be expressed as a logical function, which in turn can be implemented by a digital logic circuit. Furthermore, algebraic approaches, such as using state transition matrices, can be employed to represent and analyze the dynamics of the Game of Life, even for its complex, non-linear rules. While the provided book discusses characteristic polynomials primarily for linear CAs, the general concept of using algebraic structures to describe CA behavior is relevant.

• Modeling and Analysis: The study of cellular automata like the Game of Life often involves analyzing their emergent patterns, stability, and growth. Algebraic properties can aid in this analysis. For instance, understanding the algebraic structure underlying simpler CAs can provide insights into methodologies for analyzing more complex ones. The book "Algebraic Circuits" details how characteristic polynomials describe the temporal evolution of linear CAs, and similar algebraic thinking can be applied, albeit with more complexity, to non-linear systems.

• Hardware Implementation of CAs: The book "Algebraic Circuits" describes how cellular automata are structures composed of cells (like flip-flops) with local interconnections, making them suitable for hardware implementation. While Conway's Game of Life is often simulated in software, its rules could be directly translated into a hardware algebraic circuit consisting of an array of logic units, each representing a cell and connected to its neighbors.

Conway's Game of Life

Translating Conway's Game of Life to an Algebraic Circuit

Breaking down how a cellular automaton like Conway's Game of Life, with its specific neighborhood and rules, can be translated into an algebraic circuit representation:

The foundation for this translation lies in representing the state of each cell and the rules governing its transition to the next state using algebraic (specifically Boolean, in this case) expressions. These expressions then directly map to logic circuits.

The Algebraic Circuits book discusses how the behavior of Cellular Automata (CAs) can be described using algebraic structures like polynomials, particularly for linear CAs.

However, Conway's Game of Life is non-linear.

The principle of defining the local update rule algebraically still applies, though, so we can go through the step-by-step of constructing an algebraic circuit for a single cell in Conway's Game of Life.

Here's a step-by-step construction of the algebraic circuit representation for a single cell in Conway's Game of Life:

1. Define States and Neighborhood

• Cell States: Conway's Game of Life has two states for each cell:
  • Dead: Represented by the value 0.
  • Alive: Represented by the value 1.
  • These states naturally align with elements in the Galois Field GF(2)

• Neighborhood: The game uses a 2D grid. Each cell's next state is determined by its current state and the states of its eight immediate neighbors (the Moore neighborhood).
• Let c be the current state of the cell in question ($c\in \{0,1\}$).
• Let $n_{1},n_{2},\dots ,n_{8}$ be the states of the 8 neighbors, each neighbor being 0 or 1.

2. Define Conway's Game of Life Rules Algebraically:

• Rules of cell transition to next state:
  • Survival: Live cell c = 1 stay alive if there are 2 or 3 live neighbors.
  • Birth: Dead cell c = 0 becomes alive if there are 3 live neighbors.
  • Death: in all other cases, live cells will die (due to overpopulation or underpopulation)

• Let S be sum of states of 8 neighbors: $S=\sum _{i=1}^{8}n_{i}$. This means S is an integer from 0 to 8

3. Construct Boolean Polynomial for Next State

Define a boolean indicator variable P2, P3 such that P2 = 1 if the cell has 2 neighbors (S = 2), and is 0 otherwise, and P3 = 1 if the cell has 3 neighbors, and is 0 otherwise

The conditions that must be met for the cell to be alive at the next generation is:

• it is currently alive, c = 1, and P2 = 1 or P3 = 1
• or,
• it is currently dead, c = 0, and P3 = 1

Translating into a boolean expression for the new cell state, c prime:

$c^{\prime }=\left(c\land \left(P_{2}\lor P_{3}\right)\right)\lor \left(\neg c\land P_{3}\right)$

We mentioned above the binary nature of the Game of Life lends itself to representation on the Galois Field GF(2), so this gives us a way to write and evaluate polynomials on GF(2).

On GF(2), the AND operation is multiplication, the NOT c operation is $1\oplus c$, and the OR operation is $A\oplus B\oplus AB$ (or just $A\oplus B$ if independent)

Now there are two conditions where c will be alive at the end of the round:

• If c survives (is alive at the beginning), and meets those survival criteria
• If c is born (is dead at the beginning), and meets those birth criteria

So when we translate the logical expression for c prime above into a binary operation on GF(2), we can start by XORing those two expressions:

$c^{\prime }={\mbox{SurvivalTerm}}\oplus {\mbox{BirthTerm}}$

Now handle each term one at a time:

Survival term:

${\mbox{SurvivalTerm}}=c\cdot \left(P_{2}\lor P_{3}\right)=c\cdot \left(P_{2}\oplus P_{3}\right)$

Birth term:

${\mbox{BirthTerm}}=\left(1\oplus c\right)\cdot P_{3}$

So now we have our algebraic expression for the next state:

$c^{\prime }=\left[c\cdot \left(P_{2}\oplus P_{3}\right)\right]\oplus \left[\left(1\cdot c\right)\cdot P_{3}\right]$

(note that the multiplication operator would involve AND gates, and the bitwise oplus operator would involve XOR gates)

The terms $P_{2}(n_{1},\dots ,n_{8}),P_{3}(n_{1},\dots ,n_{8})$ are symmetric boolean functions of the neighbor states.

4. Algebraic Circuit Representation

• Boolean polynomial derived above defines the algebraic circuit for a single cell

• Circuit components:
  • Neighbor summation circuit - takes the state of the 8 neighbors as input, calculates sum S; can be implemented using a network of full adders and half adders; outputs a 4-bit binary word (there are 9 possible states, requiring slightly more than 3 bits), representing the count S
  • Sum comparison logic - indicator circuits for P2 and P3, so that we know when we meet the P2 or P3 condition; P2 circuit outputs 1 if $S=(0010)_{2}$, 0 otherwise, and P3 circuit outputs 1 if $S=(0011)_{2}$, 0 otherwise

• Applying the logic:
  • Use logic gates and/xor/not, or GF(2) equivalents like multiplication/addition, to get the next state

Now, we'e assembled an algebraic circuit for a single cell, and it can be connected to 8 other algebraic circuits for the neighbor cells, and if all circuits were driven by a common clock, they could all update synchronously.

Connecting to Turing Completeness and Life in Life

This concept of treating each cell as a circuit, and treating the entire grid as a set of interconnected circuits, highlights a connection to the mind-blowing fact that you can use the Game of Life to implement circuits that perform logical operations like AND and XOR, which allow you to implement the circuits that run the Game of Life, in the Game of Life.

YouTube video: "Life in life" by user mrphlip (Philip Bradbury): https://www.youtube.com/watch?v=xP5-iIeKXE8

...

Video shows a large grid of Game of Life patterns where each "metapixel" (a large, stable or oscillating collection of Game of Life cells) itself represents a single cell of a higher-level Game of Life grid. The interactions between these metapixels are engineered to follow the rules of Conway's Game of Life, simulating the game at a macroscopic level using the game's own microscopic rules.

The Golly software (often mentioned in the descriptions of these videos, e.g.,) is a crucial tool for exploring such complex patterns and for running scripts that can generate and manage these "Life in Life" constructions like the OTCA Metapixel.

Emergent behavior: a set of simple rules can lead to layers upon layers of emergent complexity and computation.

Complications

As mentioned above, Conway's Game of Life is non-linear, so the rule to update an entire grid to find its next state is not simply an operation that can be applied to a bitvector or done in one go.

To express this a different way,

$c^{\prime }=f(c,n_{1},\dots ,n_{8})$

is a nonlinear function. But any boolaen function can be uniquely expressed as a polynomial in GF(2) (Reed-Muller expansion). The polynomial that represents the local update rule is the algebraic core of the circuit of a single cell.

Galois Fields and Cellular Automata

GF(2)

We saw above that when a cellular automaton has a binary state (alive/dead, 0 or 1), it can be mapped to the Galois Field GF(2). That essentially means we can think about it, represent it, and operate on it as a bitvector (because it is).

However, I was curious about the Galois Field connection and wanted to dig deeper into examples of automata with more complex internal states, and what those algebraic circuits might look like.

GF(3) and Beyond

The provided book, "Algebraic Circuits," lays some groundwork for thinking about this case, although it mainly focuses on GF(2) (binary automata).

The extensions the book does cover are for $GF(2^{n})$ or $GF(p)$, powers of 2 or prime numbers. In this case, we can think about the cellular automaton's state being updated by carrying out the update rules/operations over $GF(2^{n})$ or $GF(p)$.

Role of Galois Fields and the Number of Cell States

The number of distinct states a cell in a CA can take directly determines the type of Galois Field (GF) that provides the natural algebraic framework for describing its behavior:

• If a cell has $k$ states, and $k$ is a prime number $p$: The states can be mapped to the elements of the Galois Field ${\text{GF}}(p)=\{0,1,\dots ,p-1\}$. All arithmetic operations (addition, multiplication) used in defining the CA's rules are then performed modulo $p$.
  • Example:** If a CA has 3 states (e.g., "Alive," "Decaying," "Dead"), these can be mapped to the elements 0, 1, 2 of ${\text{GF}}(3)$.
• If a cell has $k$ states, and $k=p^{m}$ for some prime $p$ and integer $m\geq 1$: The states can be mapped to the elements of the Galois Field ${\text{GF}}(p^{m})$. Operations are then defined by the arithmetic of this extension field.
  • Example:** If a CA has 4 states, these can be mapped to the elements of ${\text{GF}}(2^{2})$. Operations would involve polynomial arithmetic modulo an irreducible polynomial of degree 2 over ${\text{GF}}(2)$.
• Your Example CA: You mentioned a CA where cells might cycle through several "dead" states that affect their ability to be reborn. If these states are distinct, they increase the total number of states per cell. For instance:
  • State 0: Alive
  • State 1: Dead_Recently (can be reborn)
  • State 2: Dead_Longer (cycles to State 3, cannot be reborn)
  • State 3: Dead_VeryLong (cycles to State 1, cannot be reborn from this state)

Here, we have $k=4$ states. We could map these to the elements of ${\text{GF}}(2^{2})$. The transition rules, including the cycling through dead states and the conditions for rebirth (which would depend on the cell being in State 1, for instance), would be defined using the arithmetic of ${\text{GF}}(2^{2})$.

The Galois Field provides the consistent mathematical system (the "algebra") within which the cell state transitions (the "circuit" logic) are defined.

Polynomial Representation of Multi-State CAs

The "polynomial representation" of a CA generally refers to expressing its local update rule (the function that determines a cell's next state based on its current state and its neighbors' states) as a polynomial.

• Let $c_{0}$ be the current state of the cell and $c_{1},\dots ,c_{N}$ be the states of its $N$ neighbors. All these states are elements of ${\text{GF}}(k)$.
• The next state of the cell, $c'_{0}$, is given by a function $f:({\text{GF}}(k))^{N+1}\rightarrow {\text{GF}}(k)$.

$c'_{0}=f(c_{0},c_{1},\dots ,c_{N})$

• It's a fundamental result in algebra that any function $f$ mapping from a finite set of inputs from ${\text{GF}}(k)$ to an output in ${\text{GF}}(k)$ can be represented as a polynomial in $N+1$ variables with coefficients in ${\text{GF}}(k)$.
  • Example: If we have 3 states (${\text{GF}}(3)$) and one neighbor ($c_{1}$), the rule $c'_{0}=(c_{0}+2c_{1}^{2}){\pmod {3}}$ is a polynomial over ${\text{GF}}(3)$.
• This polynomial is the algebraic description of the local rule. The "algebraic circuit" for one cell is then the logical implementation of this polynomial using adders, multipliers, and constant multipliers defined in ${\text{GF}}(k)$.

Impact of Dimensions and Linearity

• Linearity:
  • Linear CAs: If the local update rule $f$ is a linear combination of the states in the neighborhood (i.e., $c'_{0}=a_{0}c_{0}+a_{1}c_{1}+\dots +a_{N}c_{N}$, where coefficients $a_{i}$ and states $c_{i}$ are in ${\text{GF}}(k)$, and operations are in ${\text{GF}}(k)$), the CA is linear.
    • For linear CAs, a global transition matrix $T$ (with elements from ${\text{GF}}(k)$) can be defined for the entire grid (if finite).
    • A characteristic polynomial (e.g., $\det(T-\lambda I)$, as described in Source 1, p. 203 for ${\text{GF}}(2)$ CAs and extended to ${\text{GF}}(p)$ on p. 214) can be associated with this matrix. The roots and properties of this characteristic polynomial can reveal much about the CA's global behavior, such as cycle lengths and patterns. These CAs are often considered "easily computable" in an analytical sense due to the power of linear algebra.
  • Non-Linear CAs: (e.g., Conway's Game of Life, and likely your example with conditional rebirth based on specific dead states). The local update rule $f$ is a non-linear polynomial over ${\text{GF}}(k)$.
    • While the local rule still has a polynomial representation, the tools of linear algebra and characteristic polynomials for the global system do not apply as directly or simply. Superposition doesn't hold.
    • The global behavior is often complex and emergent, typically studied through simulation. However, the local polynomial representation is crucial for defining the CA's logic and for designing the circuit for each cell.

• Dimensions (1D vs. 2D, etc.):
  • The dimensionality primarily affects the size and structure of the neighborhood. A 1D CA might have 2 neighbors (left and right), while a 2D CA like Conway's Game of Life has 8 (Moore neighborhood).
  • This means the local update function $f$ (and its polynomial representation) will have more input variables for higher-dimensional CAs or CAs with larger neighborhoods.
  • The choice of the Galois Field ${\text{GF}}(k)$ for the cell states is independent of the CA's dimensionality.
  • If a linear CA is defined on a higher-dimensional grid, its global transition matrix $T$ will be larger and more complex, but the principle of using a characteristic polynomial (if the CA is linear over ${\text{GF}}(k)$) still holds. The book "Algebraic Circuits" mentions this extension for mod$p$ CAs in 2D
  • Conway's Game of Life is 2D and non-linear. Its local rule (next state of a cell based on its state and 8 neighbors) can be expressed as a complex Boolean polynomial (a polynomial over ${\text{GF}}(2)$). "Rule 30" (an example of a 1D elementary CA) has the rule $c'_{i}=c_{i-1}\oplus (c_{i}\lor c_{i+1})$. This is non-linear over ${\text{GF}}(2)$, and its polynomial representation is $c'_{i}=c_{i-1}\oplus c_{i}\oplus c_{i+1}\oplus c_{i}c_{i+1}$. Some elementary 1D rules are linear (e.g., Rule 90: $c'_{i}=c_{i-1}\oplus c_{i+1}$), and for these, the characteristic polynomial analysis is very effective.

Multi-State CAs with Cycling Dead States

1. Determine the total number of states, $k$. (e.g., Alive, Dead_1, Dead_2, Dead_3 $\rightarrow k=4$).
2. Choose the appropriate Galois Field. If $k=4$, use ${\text{GF}}(2^{2})$. Assign each state to an element of ${\text{GF}}(2^{2})$ (e.g., using a polynomial basis like $ax+b$ over ${\text{GF}}(2)$, the states could be $(0,0),(0,1),(1,0),(1,1)$).
3. Write down the transition rules explicitly. For each possible state of the central cell and each possible configuration of its neighbors (or a summary like "sum of 'Alive' neighbors"), define the next state of the central cell. This defines the function $f$.
   • The "cycling through dead states" means if $c={\text{Dead2}}$, then $c'={\text{Dead3}}$ (perhaps regardless of neighbors, or with some conditions).
   • The "cannot be reborn" condition means: if $c\in \{{\text{Dead2, Dead3}}\}$ AND (birth condition based on neighbors is met), then $c'$ is not "Alive", but perhaps stays in its cycle or goes to Dead_1.
4. Convert this set of rules into a polynomial over ${\text{GF}}(k)$ for $f$. This can be complex for many states and non-linear rules but is always theoretically possible. For programmatic implementation, this often translates to lookup tables or nested conditional statements which are themselves implementations of this polynomial.
5. The algebraic circuit for one cell is the implementation of this polynomial using adders, multipliers, and constants from ${\text{GF}}(k)$.

In essence, the Galois Field provides the number system and arithmetic for the cell states. The polynomial representation captures the logic of the local update rule, which forms the basis of the algebraic circuit for each cell in the automaton, regardless of its linearity or dimensionality. Linear CAs over ${\text{GF}}(k)$ just happen to have very elegant global algebraic descriptions via their characteristic polynomials.