General Number Field Sieve (GNFS)

The General Number Field Sieve (GNFS) is currently the most efficient algorithm for factoring integers larger than 100 digits. It is a generalization of the Special Number Field Sieve and represents the state of the art in integer factorization. The algorithm combines techniques from algebraic number theory, linear algebra, and sieving methods.

Algorithm Overview

The GNFS works by finding pairs of integers that are smooth over two different number fields. These relations are then used to construct a system of linear equations, whose solution leads to a non-trivial factorization of the target number.

Mathematical Foundation

The algorithm is based on the following principle: if we can find integers x and y such that:

and , then and are non-trivial factors of n. The GNFS achieves this by working in number fields to find these relations.

Time and Space Complexity

The time complexity of the GNFS is:

The space complexity is for storing the factor base and relations.

JavaScript Implementation

code.js
1function generalNumberFieldSieve(n) {
2    // Polynomial selection
3    const polynomial = selectPolynomial(n);
4    
5    // Factor base generation
6    const rationalFactorBase = generateRationalFactorBase(n);
7    const algebraicFactorBase = generateAlgebraicFactorBase(polynomial);
8    
9    // Sieving phase
10    const relations = [];
11    const sieveArray = new Array(sieveSize).fill(0);
12    
13    // Rational sieving
14    for (const p of rationalFactorBase) {
15        const roots = findRoots(n, p);
16        for (const root of roots) {
17            for (let j = root; j < sieveSize; j += p) {
18                sieveArray[j] += Math.log(p);
19            }
20        }
21    }
22    
23    // Algebraic sieving
24    for (const p of algebraicFactorBase) {
25        const roots = findAlgebraicRoots(polynomial, p);
26        for (const root of roots) {
27            for (let j = root; j < sieveSize; j += p) {
28                sieveArray[j] += Math.log(p);
29            }
30        }
31    }
32    
33    // Finding smooth numbers
34    for (let i = 0; i < sieveSize; i++) {
35        if (isSmooth(sieveArray[i], threshold)) {
36            const relation = createRelation(i, polynomial);
37            if (relation) {
38                relations.push(relation);
39            }
40        }
41    }
42    
43    // Linear algebra phase
44    const matrix = buildMatrix(relations);
45    const solution = solveLinearSystem(matrix);
46    
47    // Factorization phase
48    return findFactors(n, relations, solution);
49}
50
51// Helper functions
52function selectPolynomial(n) {
53    // Select an appropriate polynomial for the number field
54}
55
56function generateRationalFactorBase(n) {
57    // Generate the rational factor base
58}
59
60function generateAlgebraicFactorBase(polynomial) {
61    // Generate the algebraic factor base
62}
63
64function findRoots(n, p) {
65    // Find roots modulo p
66}
67
68function findAlgebraicRoots(polynomial, p) {
69    // Find roots of the polynomial modulo p
70}
71
72function isSmooth(value, threshold) {
73    // Check if a number is smooth
74}
75
76function createRelation(x, polynomial) {
77    // Create a relation from a smooth number
78}
79
80function buildMatrix(relations) {
81    // Build the matrix for linear algebra phase
82}
83
84function solveLinearSystem(matrix) {
85    // Solve the system of linear equations
86}
87
88function findFactors(n, relations, solution) {
89    // Use the solution to find factors
90}

Example Usage

The GNFS is typically used for very large numbers:

code.js
1const n = 1234567890123456789012345678901234567890123456789012345678901234567890n;
2const factors = generalNumberFieldSieve(n);
3console.log(factors);

Optimization Techniques

Several optimizations can improve the performance of the GNFS:

1. Using multiple polynomials to increase the yield of smooth numbers
2. Implementing block sieving to improve cache efficiency
3. Using parallel processing for the sieving phase
4. Optimizing the linear algebra phase with sparse matrix techniques
5. Using lattice sieving for better memory efficiency

Limitations

The main limitations of the GNFS are:

• Extremely high memory requirements
• Complex implementation and parameter tuning
• Not suitable for numbers with special forms
• Requires significant computational resources

Applications

The GNFS is particularly useful in:

• Breaking RSA keys
• Cryptanalysis of cryptographic systems
• Research in number theory
• Integer factorization competitions
• Security analysis of cryptographic protocols