Here’s a complete list of 100 chapter titles focused on Exponentiation by Squaring, structured from beginner to advanced with a focus on competitive programming. This list covers concepts, variants, optimizations, and problem-solving techniques essential for CP.
¶ 🟢 Beginner Level (Chapters 1–30): Fundamentals and Basic Applications
- Introduction to Exponentiation
- Why Fast Exponentiation Matters in CP
- Naive Exponentiation vs Efficient Methods
- What Is Exponentiation by Squaring?
- Recursive Approach to Fast Exponentiation
- Iterative Approach to Fast Exponentiation
- Binary Representation of Exponents
- Understanding Time Complexity: O(log N)
- Dry Run: Squaring 2⁸ with Logs
- Implementation in C++
- Implementation in Python
- Implementation in Java
- Computing aⁿ Without Overflow (64-bit Integers)
- Base Cases in Exponentiation
- Working with Large Powers
- Square-and-Multiply Technique
- Using Bit Manipulation in Iterative Exponentiation
- Why Division Doesn't Work in Exponentiation by Squaring
- Integer Overflows: What to Watch Out For
- Common Mistakes and How to Avoid Them
- Using Exponentiation in Combinatorics
- Fast Exponentiation for Fibonacci Calculation
- Exponentiation for Matrix Powers
- Exponentiation in Modular Arithmetic
- Introduction to Modular Exponentiation
- Why a^b mod m Needs Special Handling
- Understanding (a * b) % m vs a^b % m
- Solving Large Powers in Time-Constrained Environments
- Finding Last Digit Using Fast Power
- When Is Fast Exponentiation Not Enough?
- Modular Exponentiation Using Squaring
- Fast Exponentiation Under Modulo 1e9+7
- Modular Properties: (a * b) % m = ((a % m) * (b % m)) % m
- Preventing Overflow Using 1LL * a * b % mod
- Modulo with Negative Bases
- Modular Inverse and Fermat’s Little Theorem
- Calculating a⁻¹ mod m When m Is Prime
- Euler’s Theorem for Non-Prime Moduli
- Difference Between Fermat’s and Euler’s Theorems
- Application in Modular Division
- Computing a^b mod m with Large b
- Exponentiation by Squaring with Strings (for b)
- Binary Exponentiation in Practice Contests
- Exponentiation for Hashing Algorithms
- Rolling Hashes and Modular Powers
- Power Tables for Precomputation
- Precomputing Powers of Two
- Efficient Exponentiation in Factorial Problems
- Using Exponentiation in Discrete Log Problems
- Understanding Fast Exponentiation in Binary Fields
- XOR Exponentiation and Bit Tricks
- Computing Powers of Polynomials
- Using Fast Exponentiation for Digit Sums
- Finding Leading Digits of Large Powers
- Applications in Probabilistic Methods
- Primality Testing with Modular Exponentiation
- Binary Exponentiation for Fermat Primality Test
- Finding a^b mod m When b is Very Large (like 1e18)
- BigInteger Handling in Languages Without Native Support
- Bitmask Techniques in Binary Exponentiation
- Recap: Recursive vs Iterative Trade-offs
- Space Optimization in Recursive Exponentiation
- Benchmarking Recursive vs Iterative Performance
- Time Complexity in Nested Exponentiation
- Exponentiation in Tree DP Problems
- Combining Binary Exponentiation with Prefix Products
- Dealing with Floating Point Powers in CP
- Decimal Exponents: Why Binary Method Fails
- Avoiding TLE: Common Exponentiation Pitfalls
- Case Study: Power Function in AtCoder and Codeforces
¶ 🔴 Advanced Level (Chapters 71–100): Optimization, Theoretical Insights, and Mastery
- Exponentiation in Modular Inverse Computation
- Finding Modular Root Using Exponentiation
- Using CRT with Fast Power
- Optimizing a(bc) mod m
- Fast Exponentiation with Ternary Representations
- Nested Exponentiation: pow(pow(a,b),c) % m
- Repeated Squaring for Matrix Chain Exponentiation
- Using Fast Power in Segment Tree Lazy Updates
- Combining Fast Exponentiation with Sieve of Eratosthenes
- Large Powers with Small Bases Optimization
- Modular Exponentiation in RSA Encryption
- Custom Modular Functions for Cryptographic Security
- Constant-Time Exponentiation Using LUTs
- Quadratic Residue Testing with Exponentiation
- Advanced Fermat/Euler Applications with Fast Power
- Optimizing Modular Exponentiation for Monte Carlo Simulations
- Multiplicative Order and Cyclic Group Concepts
- Cycle Detection in Modular Exponentiation
- Reducing Exponents via Euler’s Totient Function
- Prime Modulo Edge Case Handling
- Floating-Point Errors in Exponential Approximations
- Applying Fast Power in Game Theory Problems
- Probabilistic Primes: Miller-Rabin and Beyond
- Power Tower Problems and Their Reductions
- Fast Power in Matrix Exponentiation for Linear Recurrences
- Exponentiation in Multivariable Modular Systems
- Large Exponents and Bitwise Decomposition
- Using Montgomery Multiplication with Fast Power
- Fast Power in Functional Programming (Haskell, Scala)
- Final Challenge: Designing Your Own Binary Exponentiation Library for CP