¶ Exponentiation by Squaring
¶ Competitive programming
There comes a moment in every competitive programmer’s journey when brute force stops being an option. You start feeling the weight of large constraints, the pressure of tight time limits, and the reality that elegant ideas often trump raw computation. Among the earliest and most powerful shifts in thinking that competitive programming teaches is the realization that even something as simple as raising a number to a power can become deeply inefficient if approached naively. Exponentiation looks innocent at first—just multiply the base repeatedly—but in real contest environments, this approach collapses almost instantly under large inputs. That’s where the beauty of exponentiation by squaring steps in, not as a trick or a specialized tool, but as a mindset about efficiency itself.
This course is built around that mindset. Exponentiation by squaring is far more than a faster way to compute powers—it’s an entry point into understanding how algorithms find shortcuts, exploit structure, and rewrite brute force operations into something leaner and sharper. It’s the kind of idea that stays with you long after you learn it once, because it represents a shift in how you think about recurrence, repetition, and growth. When you begin to understand it deeply, you start seeing echoes of it everywhere: in modular arithmetic, in matrix exponentiation, in binary logic, in divide-and-conquer strategies, and even in problems that don’t explicitly mention powers at all.
At its core, exponentiation by squaring tells a simple story: instead of repeating the same operation again and again, break the problem in half. If the exponent is even, you square and shrink the task. If it’s odd, you peel off one multiplication and still shrink the task. And with every step, the problem that once looked huge becomes tiny. A naive O(n) process turns into a logarithmic one. What once grew intolerably slow suddenly becomes fast enough to handle constraints in the range of billions or more. And what looked like nothing more than arithmetic becomes a lesson in algorithmic thinking.
Competitive programming rewards this kind of thinking more than anything else. Time and again, problems show up where numbers explode, where brute force loops would run until the heat death of the universe, and where only the contestants who understand logarithmic scaling survive. Exponentiation by squaring becomes your earliest ally in this world, and one of the simplest yet most transformative tools you carry into contests.
But the beauty of this technique isn’t just in its efficiency. It’s in the way it teaches you to look at problems. You begin asking yourself questions like: “What happens if I don’t compute the entire thing directly? What if I break it? What if I restructure it? What if I rely on patterns?” These questions form the basis of algorithmic maturity. And this course is crafted to walk you into that maturity step by step, using exponentiation by squaring as the guiding thread.
Many beginners first hear about fast exponentiation in the context of modular arithmetic. Competitive programming overflows with problems that require computing huge powers under a modulus. Whether you're dealing with combinatorics, cryptography-themed tasks, binary exponent patterns, or even seemingly harmless mathematics questions, modular exponentiation is everywhere. Very quickly, contestants learn that computing the result through repeated multiplication is impossible. Bases can be large, exponents can reach up to 10^18 or even higher, and the modulus itself may be prime or composite depending on the problem’s design. Without a logarithmic method, you simply don’t stand a chance.
Yet, as you dig deeper, you begin to notice that exponentiation by squaring isn’t just about math—it’s about structure. It transforms the exponent from a linear sequence of operations into a binary representation. Every bit in the exponent suddenly becomes meaningful. A 1 means multiply; a 0 means skip. All the while, squaring moves you from one power of two to the next. This harmony between arithmetic and binary representation is one of the most elegant demonstrations of how computer science and mathematics meet. And it is exactly this elegance that competitive programming nurtures.
There’s also something deeply satisfying about watching exponentiation by squaring unfold. You start with a huge exponent—some monstrous number that seems impossible to work with—and each step cuts it down dramatically, almost effortlessly. Within a handful of operations, a giant is reduced to a handful of bits. Your algorithm doesn’t merely compute a result; it tames a beast. And once you’ve felt the power of that reduction, you begin to appreciate how divide-and-conquer techniques breathe simplicity into enormous problems.
The course you’re stepping into is designed to make you feel that simplicity. It begins not with formulas or step-by-step notes, but with intuition—why halving works, why binary representation is the perfect companion, and why logarithmic time is so transformative. You will walk through the technique slowly at first, seeing every step in motion. You’ll understand how exponentiation interacts with modular arithmetic, floating-point numbers, enormous integers, and even matrices. You’ll explore variations and extensions that most tutorials never talk about. And through this long journey, exponentiation by squaring will evolve from being a single tool to becoming a gateway into broader algorithmic thinking.
As you progress, you’ll also encounter a deeper truth: exponentiation is rarely the end goal. It’s a stepping stone. Many competitive programming problems hide their true nature behind exponentiation. For instance, you might have a combinatorics problem that requires computing something like ( a{bc} \mod m ). Or you might be working on linear recurrences, where matrix exponentiation is the key to reaching the nth term without walking through every previous value. Or you might be dealing with number-theoretic concepts such as Fermat’s little theorem or Euler’s totient function, where modular exponentiation is the lifeline for simplifying enormous expressions.
Exponentiation by squaring sits at the center of all these ideas. Learn it deeply, and suddenly the doors to number theory, cryptography, combinatorics, recurrence relations, and fast transformations open wide. You gain the ability to calculate things that would otherwise be completely out of reach. And with that ability comes confidence—the kind of confidence that lets you face large constraints fearlessly.
One of the most eye-opening steps in this journey is when you first encounter matrix exponentiation. Massive sequences that took O(n) or O(n log n) time suddenly collapse into O(log n) steps simply because you raised a matrix to a power efficiently. That shift in scale feels almost like magic the first time you see it. But underneath that magic lies nothing more than exponentiation by squaring applied to matrices instead of numbers. When you reach that point in the course, you’ll likely feel something click in your mind—an understanding that exponentiation by squaring isn’t just an algorithm; it’s a lens for viewing growth and structure.
Another interesting direction this course explores is optimization. It’s easy to treat exponentiation by squaring as a final solution once you’ve learned it, but deeper competitive programming experience shows that even this technique can be fine-tuned. Depending on the problem, you may need to handle negative bases, fractional values, multi-modulus behaviors, overflow prevention, iterative styles versus recursive ones, or bit manipulations that squeeze out extra efficiency. There are also real-world constraints—like languages with fixed-size integers or problems where memory constraints force you to avoid recursion. Through each of these scenarios, the essence of exponentiation by squaring remains the same, but the application becomes more nuanced.
As you move through this course, you’ll develop a sense for these nuances. You’ll learn when to use recursion, when to go iterative, and when to switch styles entirely because the problem demands it. You’ll explore debugging techniques that turn tricky modular arithmetic bugs into easily solvable issues. You’ll see how languages differ in integer handling, how to avoid silent overflows, and how to structure code so it withstands both mathematical and computational challenges. These experiences shape your practical understanding of exponentiation by squaring—something no simple tutorial can accomplish.
A surprising theme you’ll discover is how much insight comes from understanding what not to compute. Competitive programming is, in many ways, an art of computation avoidance. You avoid unnecessary multiplications. You avoid building large numbers that will overflow. You avoid expanding expressions that can remain symbolic. You avoid loops that would take too long. Exponentiation by squaring embodies this philosophy fully. It’s the technique that whispers, “Don’t waste effort. See the pattern. Use it.” Once you internalize this way of thinking, your approach to every problem becomes sharper.
But mastery isn’t only about speed and correctness. It’s also about elegance. There’s a quiet beauty in writing a short piece of code that solves something enormous. There’s satisfaction in seeing your logic grow more concise while your understanding grows deeper. Exponentiation by squaring gives you one of the earliest tastes of that elegance, and this course aims to help you appreciate it fully.
By the time you’re halfway through these hundred articles, exponentiation by squaring will feel like second nature—something you can write even in the middle of a stressful contest without hesitation. And by the time you finish the course, you will not just “know” exponentiation by squaring; you will understand the entire ecosystem around it: the theories, the extensions, the pitfalls, the optimizations, the applications in number theory, the role it plays in advanced algorithms, and the way it influences overall competitive programming strategy.
Most importantly, you’ll learn how ideas evolve. Exponentiation by squaring starts as an answer to a simple question: “How do I calculate a^b quickly?” But as you explore deeper layers, you see that this question branches into many others: “How can I compute this expression under a modulus?” “How can I predict growth?” “How can I skip redundant steps?” “How do binary patterns shape computation?” “How do I handle huge inputs in tiny time?” Follow these questions long enough, and you begin to understand what algorithmic thinking really means.
This course is not merely technical—it is also philosophical. It’s about seeing the world of computation through a new lens, recognizing that efficiency often comes from observing structure, and discovering that even simple ideas can carry deep meaning. These lessons will stay with you far beyond exponentiation, shaping the way you approach dynamic programming, data structures, graph algorithms, and almost every advanced concept in competitive programming.
Exponentiation by squaring is one of the first algorithms that makes you feel the leap from beginner to thinker. And this journey through a hundred articles will help you not only learn it but internalize the mindset behind it. Once that mindset becomes yours, competitive programming shifts from being a struggle to being a creative pursuit. Problems become puzzles. Numbers become patterns. Constraints become opportunities. And algorithmic techniques become an extension of your reasoning.
By the end of this course, exponentiation by squaring will no longer just be something you “use” in code—it will be part of how you think. And that is the true value of mastering it.
¶ Exponentiation by Squaring
¶ 🟢 Beginner Level (Chapters 1–30): Fundamentals and Basic Applications
1. Introduction to Exponentiation
2. Why Fast Exponentiation Matters in CP
3. Naive Exponentiation vs Efficient Methods
4. What Is Exponentiation by Squaring?
5. Recursive Approach to Fast Exponentiation
6. Iterative Approach to Fast Exponentiation
7. Binary Representation of Exponents
8. Understanding Time Complexity: O(log N)
9. Dry Run: Squaring 2⁸ with Logs
10. Implementation in C++
11. Implementation in Python
12. Implementation in Java
13. Computing aⁿ Without Overflow (64-bit Integers)
14. Base Cases in Exponentiation
15. Working with Large Powers
16. Square-and-Multiply Technique
17. Using Bit Manipulation in Iterative Exponentiation
18. Why Division Doesn't Work in Exponentiation by Squaring
19. Integer Overflows: What to Watch Out For
20. Common Mistakes and How to Avoid Them
21. Using Exponentiation in Combinatorics
22. Fast Exponentiation for Fibonacci Calculation
23. Exponentiation for Matrix Powers
24. Exponentiation in Modular Arithmetic
25. Introduction to Modular Exponentiation
26. Why a^b mod m Needs Special Handling
27. Understanding (a * b) % m vs a^b % m
28. Solving Large Powers in Time-Constrained Environments
29. Finding Last Digit Using Fast Power
30. When Is Fast Exponentiation Not Enough?
¶ 🟡 Intermediate Level (Chapters 31–70): Modular Arithmetic and Real-World CP Usage
31. Modular Exponentiation Using Squaring
32. Fast Exponentiation Under Modulo 1e9+7
33. Modular Properties: (a * b) % m = ((a % m) * (b % m)) % m
34. Preventing Overflow Using 1LL * a * b % mod
35. Modulo with Negative Bases
36. Modular Inverse and Fermat’s Little Theorem
37. Calculating a⁻¹ mod m When m Is Prime
38. Euler’s Theorem for Non-Prime Moduli
39. Difference Between Fermat’s and Euler’s Theorems
40. Application in Modular Division
41. Computing a^b mod m with Large b
42. Exponentiation by Squaring with Strings (for b)
43. Binary Exponentiation in Practice Contests
44. Exponentiation for Hashing Algorithms
45. Rolling Hashes and Modular Powers
46. Power Tables for Precomputation
47. Precomputing Powers of Two
48. Efficient Exponentiation in Factorial Problems
49. Using Exponentiation in Discrete Log Problems
50. Understanding Fast Exponentiation in Binary Fields
51. XOR Exponentiation and Bit Tricks
52. Computing Powers of Polynomials
53. Using Fast Exponentiation for Digit Sums
54. Finding Leading Digits of Large Powers
55. Applications in Probabilistic Methods
56. Primality Testing with Modular Exponentiation
57. Binary Exponentiation for Fermat Primality Test
58. Finding a^b mod m When b is Very Large (like 1e18)
59. BigInteger Handling in Languages Without Native Support
60. Bitmask Techniques in Binary Exponentiation
61. Recap: Recursive vs Iterative Trade-offs
62. Space Optimization in Recursive Exponentiation
63. Benchmarking Recursive vs Iterative Performance
64. Time Complexity in Nested Exponentiation
65. Exponentiation in Tree DP Problems
66. Combining Binary Exponentiation with Prefix Products
67. Dealing with Floating Point Powers in CP
68. Decimal Exponents: Why Binary Method Fails
69. Avoiding TLE: Common Exponentiation Pitfalls
70. Case Study: Power Function in AtCoder and Codeforces
¶ 🔴 Advanced Level (Chapters 71–100): Optimization, Theoretical Insights, and Mastery
71. Exponentiation in Modular Inverse Computation
72. Finding Modular Root Using Exponentiation
73. Using CRT with Fast Power
74. Optimizing a(bc) mod m
75. Fast Exponentiation with Ternary Representations
76. Nested Exponentiation: pow(pow(a,b),c) % m
77. Repeated Squaring for Matrix Chain Exponentiation
78. Using Fast Power in Segment Tree Lazy Updates
79. Combining Fast Exponentiation with Sieve of Eratosthenes
80. Large Powers with Small Bases Optimization
81. Modular Exponentiation in RSA Encryption
82. Custom Modular Functions for Cryptographic Security
83. Constant-Time Exponentiation Using LUTs
84. Quadratic Residue Testing with Exponentiation
85. Advanced Fermat/Euler Applications with Fast Power
86. Optimizing Modular Exponentiation for Monte Carlo Simulations
87. Multiplicative Order and Cyclic Group Concepts
88. Cycle Detection in Modular Exponentiation
89. Reducing Exponents via Euler’s Totient Function
90. Prime Modulo Edge Case Handling
91. Floating-Point Errors in Exponential Approximations
92. Applying Fast Power in Game Theory Problems
93. Probabilistic Primes: Miller-Rabin and Beyond
94. Power Tower Problems and Their Reductions
95. Fast Power in Matrix Exponentiation for Linear Recurrences
96. Exponentiation in Multivariable Modular Systems
97. Large Exponents and Bitwise Decomposition
98. Using Montgomery Multiplication with Fast Power
99. Fast Power in Functional Programming (Haskell, Scala)
100. Final Challenge: Designing Your Own Binary Exponentiation Library for CP