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[Utility & Helpers](/categories/utility) 4. / 5. guillaumetissier/galois-fields ActiveLibrary[Utility & Helpers](/categories/utility) guillaumetissier/galois-fields ============================== A PHP library for working with Galois fields (finite fields) v1.3.1(3mo ago)07MITPHPPHP ^8.1 Since Feb 2Pushed 3mo agoCompare [ Source](https://github.com/guillaumetissier/galois-fields)[ Packagist](https://packagist.org/packages/guillaumetissier/galois-fields)[ RSS](/packages/guillaumetissier-galois-fields/feed)WikiDiscussions main Synced 1mo ago READMEChangelogDependencies (4)Versions (6)Used By (0) Galois Fields (Finite Fields) Library for PHP ============================================= [](#galois-fields-finite-fields-library-for-php) A comprehensive PHP library for working with Galois fields (finite fields), supporting both prime fields GF(p) and binary extension fields GF(2^n). What are Galois Fields? ----------------------- [](#what-are-galois-fields) Galois fields (also called finite fields) are mathematical structures used in cryptography, error correction codes, and computer science. Every finite field has order p^n where p is prime and n ≥ 1. **Key properties:** - **GF(p)**: Prime fields with p elements (e.g., GF(7), GF(11)) - **GF(2^n)**: Binary extension fields (e.g., GF(256) used in QR codes and AES) - All fields of the same order are isomorphic (essentially identical) Installation ------------ [](#installation) ``` composer require guillaumetissier/galois-fields ``` Features -------- [](#features) - ✅ Prime fields GF(p) for any prime p - ✅ Binary extension fields GF(2^n) for n = 2 to 16 - ✅ All basic field operations: add, subtract, multiply, divide, power, inverse - ✅ Efficient logarithm table implementation for GF(2^n) - ✅ Full test coverage - ✅ Type-safe with PHP 8.1+ strict types - ⚠️ Extension fields GF(p^n) where p > 2 (planned) Usage ----- [](#usage) ### Basic Example [](#basic-example) ``` use Guillaumetissier\GaloisFields\GaloisField; // Create GF(256) - used in QR codes $gf256 = new GaloisField(256); // Basic operations $sum = $gf256->add(123, 45); // Addition (XOR for binary fields) $product = $gf256->multiply(53, 78); // Multiplication $quotient = $gf256->divide(200, 17); // Division $inverse = $gf256->inverse(123); // Multiplicative inverse // Check field properties echo $gf256->getOrder(); // 256 echo $gf256->getCharacteristic(); // 2 echo $gf256->getDegree(); // 8 ``` ### Prime Fields [](#prime-fields) ``` // Create GF(7) $gf7 = new GaloisField(7); // All operations work modulo 7 echo $gf7->add(5, 3); // 1 (8 mod 7) echo $gf7->multiply(4, 2); // 1 (8 mod 7) echo $gf7->power(3, 2); // 2 (9 mod 7) // Every non-zero element has an inverse echo $gf7->multiply(3, $gf7->inverse(3)); // 1 ``` ### QR Code Error Correction (GF(256)) [](#qr-code-error-correction-gf256) ``` $gf = new GaloisField(256); // Reed-Solomon error correction uses GF(256) // Polynomial evaluation in GF(256) function evaluatePolynomial(GaloisField $gf, array $coefficients, int $x): int { $result = 0; foreach ($coefficients as $coeff) { $result = $gf->add($gf->multiply($result, $x), $coeff); } return $result; } $polynomial = [1, 2, 3]; // 1*x^2 + 2*x + 3 $value = evaluatePolynomial($gf, $polynomial, 5); ``` ### Different Field Sizes [](#different-field-sizes) ``` // Small fields $gf4 = new GaloisField(4); // GF(2^2) $gf8 = new GaloisField(8); // GF(2^3) $gf16 = new GaloisField(16); // GF(2^4) // Medium fields $gf256 = new GaloisField(256); // GF(2^8) - QR codes $gf65536 = new GaloisField(65536); // GF(2^16) - requires implementation // Prime fields $gf2 = new GaloisField(2); // GF(2) - simplest field $gf11 = new GaloisField(11); // GF(11) $gf31 = new GaloisField(31); // GF(31) ``` ### Field Properties [](#field-properties) ``` $gf = new GaloisField(256); // Get detailed information $info = $gf->getInfo(); print_r($info); // Output: // [ // 'order' => 256, // 'characteristic' => 2, // 'degree' => 8, // 'notation' => 'GF(2^8)' // ] // Validate elements $gf->isValidElement(123); // true $gf->isValidElement(256); // false (out of range) ``` Mathematical Background ----------------------- [](#mathematical-background) ### Field Operations [](#field-operations) In a Galois field GF(q), the following operations are defined: 1. **Addition**: Closed, associative, commutative with identity 0 2. **Multiplication**: Closed, associative, commutative with identity 1 3. **Inverse**: Every non-zero element has a multiplicative inverse 4. **Distributivity**: a × (b + c) = (a × b) + (a × c) ### Binary Fields GF(2^n) [](#binary-fields-gf2n) - Elements represented as polynomials over GF(2) - Addition is XOR (⊕) - Multiplication uses a primitive polynomial - Efficient implementation using logarithm tables ### Prime Fields GF(p) [](#prime-fields-gfp) - Elements are integers {0, 1, ..., p-1} - Operations are modular arithmetic mod p - Simple and efficient implementation Supported Fields ---------------- [](#supported-fields) ### Currently Supported [](#currently-supported) - **Prime fields**: GF(p) for any prime p - **Binary extension fields**: GF(2^n) for n in \[2, 16\] ### Coming Soon [](#coming-soon) - Extension fields GF(p^n) for p > 2 (e.g., GF(3^5), GF(5^3)) - Polynomial arithmetic over fields - Reed-Solomon encoding/decoding Applications ------------ [](#applications) - **Cryptography**: AES, elliptic curve cryptography - **Error correction**: Reed-Solomon codes in QR codes, CDs, DVDs - **Coding theory**: Linear codes, cyclic codes - **Computer algebra**: Symbolic computation - **Networking**: CRC checksums Testing ------- [](#testing) ``` composer test ``` Performance ----------- [](#performance) The library uses logarithm tables for efficient multiplication in GF(2^n): - Addition: O(1) - Multiplication: O(1) using log tables - Division: O(1) using log tables - Inverse: O(1) using log tables - Power: O(1) using log tables For prime fields GF(p), operations are O(1) modular arithmetic. Contributing ------------ [](#contributing) Contributions are welcome! Please feel free to submit pull requests. License ------- [](#license) MIT License References ---------- [](#references) - [Finite Field Arithmetic](https://en.wikipedia.org/wiki/Finite_field_arithmetic) - [Reed-Solomon Error Correction](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction) - [Primitive Polynomials](https://en.wikipedia.org/wiki/Primitive_polynomial_(field_theory)) ### Health Score 36 — LowBetter than 82% of packages Maintenance82 Actively maintained with recent releases Popularity4 Limited adoption so far Community6 Small or concentrated contributor base Maturity46 Maturing project, gaining track record Bus Factor1 Top contributor holds 100% of commits — single point of failure How is this calculated?**Maintenance (25%)** — Last commit recency, latest release date, and issue-to-star ratio. Uses a 2-year decay window. **Popularity (30%)** — Total and monthly downloads, GitHub stars, and forks. Logarithmic scaling prevents top-heavy scores. **Community (15%)** — Contributors, dependents, forks, watchers, and maintainers. Measures real ecosystem engagement. **Maturity (30%)** — Project age, version count, PHP version support, and release stability. ### Release Activity Cadence Every ~0 days Total 5 Last Release 97d ago ### Community Maintainers ![](https://www.gravatar.com/avatar/295253ce7789fa9279c56ac58b2d42bd808f97bb7f312fa2c78e2e782a31cfaf?d=identicon)[guillaume.tissier](/maintainers/guillaume.tissier) --- Top Contributors [![guillaumetissier](https://avatars.githubusercontent.com/u/8750033?v=4)](https://github.com/guillaumetissier "guillaumetissier (6 commits)") --- Tags cryptographymathematicsalgebragaloisfinite-fields ### Code Quality TestsPHPUnit Static AnalysisPHPStan Code StylePHP\_CodeSniffer Type Coverage Yes ### Embed Badge ![Health badge](/badges/guillaumetissier-galois-fields/health.svg) ``` [![Health](https://phpackages.com/badges/guillaumetissier-galois-fields/health.svg)](https://phpackages.com/packages/guillaumetissier-galois-fields) ``` ### Alternatives [brick/math Arbitrary-precision arithmetic library 2.1k504.0M277](/packages/brick-math)[markrogoyski/math-php Math Library for PHP. 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