binary exponentiation (also known as fast modular exponentiation).
I want the code in cpp
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Binary exponentiation, also known as fast modular exponentiation, is a powerful algorithm for computing large powers efficiently. Instead of multiplying the base number repeatedly, it uses the binary representation of the exponent to reduce computation time from linear to logarithmic complexity.
The binary exponentiation algorithm follows a systematic approach. We start by initializing the result to one and reducing the base modulo our modulus. Then we examine each bit of the exponent. When a bit is set, we multiply our current result by the corresponding power of the base. We continuously square the base to get the next power, and divide the exponent by two to move to the next bit.
Let's work through a concrete example: computing 2 to the power of 10 modulo 1000. First, we convert the exponent 10 to binary, which is 1010. We initialize result to 1 and base to 2. Since 10 is even, we square the base to get 4 and halve the exponent to 5. Now 5 is odd, so we multiply result by base getting 4, then square base to 16 and halve exponent to 2. Continue this process until we get our final answer of 24.
Here's the complete C++ implementation of binary exponentiation. The function takes three parameters: base, exponent, and modulus. We initialize the result to 1 and reduce the base modulo the modulus. In the main loop, we check if the current exponent is odd, and if so, multiply the result by the current base. We then square the base and halve the exponent. This continues until the exponent becomes zero. The function efficiently computes large powers in logarithmic time.
Binary exponentiation has numerous practical applications and offers significant performance improvements. While naive exponentiation has linear time complexity, binary exponentiation achieves logarithmic complexity. This makes it essential for cryptographic systems like RSA, competitive programming contests, and mathematical computations. For example, computing 2 to the power of one million would require about one million operations naively, but only about 20 operations using binary exponentiation.