Modular Exponentiation (Fast Power with Modulo)
Efficiently compute (base^exponent) mod m without overflow, crucial for cryptographic operations. Computing large powers like 2^100 mod 1000000007 is impossible with naive exponentiation due to integer overflow, but modular exponentiation achieves this in O(log exponent) time using the binary representation of the exponent. This algorithm is the foundation of RSA encryption, Diffie-Hellman key exchange, and many cryptographic protocols.
Visualization
Interactive visualization for Modular Exponentiation (Fast Power with Modulo)
Interactive visualization with step-by-step execution
Implementation
1function modularExponentiation(base: number, exp: number, mod: number): number {
2 let result = 1;
3 base = base % mod;
4
5 while (exp > 0) {
6 if (exp % 2 === 1) {
7 result = (result * base) % mod;
8 }
9 exp = Math.floor(exp / 2);
10 base = (base * base) % mod;
11 }
12
13 return result;
14}Deep Dive
Theoretical Foundation
Modular exponentiation leverages two key properties: (1) Modular multiplication: (a × b) mod m = ((a mod m) × (b mod m)) mod m, and (2) Binary exponentiation: we can express any exponent as a sum of powers of 2 (binary representation). For example, 13 = 8 + 4 + 1 = 2³ + 2² + 2⁰, so x¹³ = x⁸ × x⁴ × x¹. We build these powers by repeatedly squaring, taking modulo at each step to prevent overflow.
Complexity
Time
O(1)
O(log exp)
O(log exp)
Space
O(1)
Applications
Industry Use
RSA encryption/decryption (computing m^e mod n)
Diffie-Hellman key exchange
Digital signatures (DSA, ECDSA)
Primality testing (Fermat, Miller-Rabin)
Discrete logarithm problems
Cryptographic hash functions
Modular multiplicative inverse
Chinese Remainder Theorem applications
ElGamal encryption
Zero-knowledge proofs
Use Cases
Related Algorithms
GCD (Euclidean Algorithm)
Compute the Greatest Common Divisor (GCD) of two integers using the Euclidean algorithm. Dating back to around 300 BC and appearing in Euclid's Elements, it's one of the oldest algorithms still in common use. The algorithm is based on the principle that GCD(a,b) = GCD(b, a mod b) and is remarkably efficient with O(log min(a,b)) time complexity. The GCD is the largest positive integer that divides both numbers without remainder.
LCM (Least Common Multiple)
Calculate the Least Common Multiple (LCM) of two integers - the smallest positive integer that is divisible by both numbers. The LCM is intimately related to the GCD through the formula: LCM(a,b) = |a×b| / GCD(a,b). This relationship allows us to compute LCM efficiently using the Euclidean algorithm for GCD, achieving O(log min(a,b)) time complexity instead of naive factorization methods.
Sieve of Eratosthenes
Ancient and highly efficient algorithm to find all prime numbers up to a given limit n. Invented by Greek mathematician Eratosthenes of Cyrene (276-194 BC), this sieve method systematically eliminates multiples of primes, leaving only primes in the array. With O(n log log n) time complexity, it remains one of the most practical algorithms for generating large lists of primes, vastly superior to trial division which runs in O(n² / log n) time.
Prime Factorization
Decompose a positive integer into its unique prime factor representation. Every integer greater than 1 can be expressed as a product of prime numbers in exactly one way (Fundamental Theorem of Arithmetic). This algorithm uses trial division optimized to check only up to √n, as any composite number must have a prime factor ≤ √n. Returns a map of prime factors to their powers, e.g., 360 = 2³ × 3² × 5¹.