Project 6: Arbitrary-Precision Arithmetic

Implement big integer arithmetic and core algorithms from TAOCP Vol 2.

Quick Reference

Attribute Value
Difficulty Expert
Time Estimate 3-4 weeks
Language C
Prerequisites Arrays, math toolkit
Key Topics big ints, carry, algorithms

1. Learning Objectives

  1. Implement big integer representation.
  2. Support add, subtract, multiply, divide.
  3. Implement exponentiation and gcd on big ints.
  4. Validate correctness with known identities.

2. Theoretical Foundation

2.1 Core Concepts

  • Radix representation: Arrays of limbs base 2^k or 10^k.
  • Carry propagation: Correctness depends on carry handling.
  • Algorithmic tradeoffs: Schoolbook vs Karatsuba.

2.2 Why This Matters

TAOCP arithmetic chapters assume precise multi-precision reasoning. This library underpins later algorithms.

2.3 Historical Context / Background

Multi-precision arithmetic predates modern crypto and is essential for number theory.

2.4 Common Misconceptions

  • “Use strings”: This is too slow and error-prone.
  • “Division is trivial”: It is the hardest operation.

3. Project Specification

3.1 What You Will Build

A big integer library with core arithmetic, comparison, and conversion routines.

3.2 Functional Requirements

  1. Add/subtract with sign handling.
  2. Multiply (schoolbook) and optional Karatsuba.
  3. Divide with quotient and remainder.
  4. Convert to/from decimal strings.

3.3 Non-Functional Requirements

  • Correctness: All operations exact.
  • Performance: Reasonable for medium sizes.
  • Usability: Clear API and tests.

3.4 Example Usage / Output

big_add("123456789", "987654321") -> "1111111110"

3.5 Real World Outcome

You can compute large factorials or RSA-like values beyond 64-bit.

4. Solution Architecture

4.1 High-Level Design

┌──────────────┐     ┌──────────────┐
│ big int API  │────▶│ limb ops     │
└──────────────┘     └──────────────┘

4.2 Key Components

Component Responsibility Key Decisions
Representation Limbs + sign base 2^32
Arithmetic add/mul/div schoolbook first
Conversion string I/O base 10 conversion

4.3 Data Structures

typedef struct {
    uint32_t *limbs;
    size_t len;
    int sign;
} BigInt;

4.4 Algorithm Overview

Key Algorithm: Schoolbook multiply

  1. For each limb i in a, for each limb j in b, accumulate a[i]*b[j].
  2. Propagate carries.

Complexity Analysis:

  • Time: O(n^2) for multiply.
  • Space: O(n).

5. Implementation Guide

5.1 Development Environment Setup

gcc --version

5.2 Project Structure

project-root/
├── include/bigint.h
├── src/bigint.c
├── tests/
└── Makefile

5.3 The Core Question You’re Answering

“How do you compute correctly when numbers exceed machine word size?”

5.4 Concepts You Must Understand First

  • Limb representation and base choice
  • Carry propagation
  • Long division algorithm

5.5 Questions to Guide Your Design

  1. What base minimizes overflow risk?
  2. How will you normalize numbers after operations?
  3. How will you handle negative values?

5.6 Thinking Exercise

Prove that your carry propagation always terminates.

5.7 The Interview Questions They’ll Ask

  1. How does big integer multiplication work?
  2. Why is division harder than multiplication?
  3. How do you choose a limb size?

5.8 Hints in Layers

Hint 1: Start with unsigned addition. Hint 2: Add subtraction with borrow. Hint 3: Add multiplication and division.

5.9 Books That Will Help

Topic Book Chapter
Multi-precision TAOCP Vol 2 Ch. 4

5.10 Implementation Phases

Phase 1: Foundation (1 week)

  • Representation + add/subtract.

Phase 2: Core Functionality (1-2 weeks)

  • Multiplication + division.

Phase 3: Polish & Edge Cases (1 week)

  • Conversion + tests + optimizations.

5.11 Key Implementation Decisions

Decision Options Recommendation Rationale
Base 10^k vs 2^k 2^32 fast ops

6. Testing Strategy

6.1 Test Categories

Category Purpose Examples
Basic ops correctness add/sub/mul/div
Identities validation (a*b)/b = a
Edge cases zero/negative sign handling

6.2 Critical Test Cases

  1. Carry across many limbs.
  2. Division with remainder.
  3. Negative values and sign.

6.3 Test Data

Large factorials and powers

7. Common Pitfalls & Debugging

7.1 Frequent Mistakes

Pitfall Symptom Solution
Overflow in multiply wrong results use 64-bit accumulators
Incorrect normalization trailing zeros trim limbs
Division errors wrong remainder test with small values

7.2 Debugging Strategies

  • Cross-check with Python big ints.
  • Add asserts on limb ranges.

7.3 Performance Traps

Naive algorithms slow for large n; optional Karatsuba later.

8. Extensions & Challenges

8.1 Beginner Extensions

  • Add modular exponentiation.

8.2 Intermediate Extensions

  • Implement Karatsuba multiply.

8.3 Advanced Extensions

  • Implement FFT-based multiplication.

9. Real-World Connections

  • Cryptography, large integer computation, symbolic math.

10. Resources

  • TAOCP Vol 2 Chapter 4.

11. Self-Assessment Checklist

  • I can explain limb representation.
  • All arithmetic operations pass tests.

12. Submission / Completion Criteria

Minimum: Add/sub/mul implemented. Full: Division + conversion. Excellence: Advanced multiplication.

This guide was generated from LEARN_TAOCP_DEEP_DIVE.md. For the complete learning path, see the parent directory README.