Power Modulo Calculator

What is Modular Arithmetic?

Modular arithmetic is often called "Clock Arithmetic." It is a system of math where numbers "wrap around" after they reach a certain value (the modulus).

Think of a clock. If it is 10:00 and you add 4 hours, it doesn't become 14:00 (on a standard 12h clock). It becomes 2:00.

The Math: $14 \pmod{12} = 2$.

This simply means: Take 14, divide it by 12, and the Remainder is 2.

Why "Power Modulo" is Difficult

Calculating $5 \pmod 3$ is easy. (The remainder is 2).

But what if you need to calculate $5^{100} \pmod 7$?

If you try to type $5^{100}$ into a standard calculator, you will get an error or a massive number like $7.8 \times 10^{69}$. Standard computers cannot handle numbers this big accurately.

To solve this, computer scientists use Modular Exponentiation algorithms. These methods calculate the remainder at every step of the multiplication, keeping the numbers small and manageable.

The Secret of Internet Security (RSA)

Why do we need to calculate such huge numbers? To keep your passwords safe.

The entire internet relies on the RSA Encryption Algorithm. This system uses massive prime numbers to create "Public" and "Private" keys.

How the Algorithm Works (Square and Multiply)

Our calculator doesn't actually calculate the full huge number (which might have billions of digits). It uses a "Divide and Conquer" strategy called Square and Multiply.

To calculate $3^{45} \pmod 7$:

1. Convert the exponent (45) into binary: 101101.
2. Iterate through the bits from left to right.
3. Square the base at every step.
4. Multiply by the base only when the bit is '1'.
5. Apply $\pmod 7$ after every single operation.

This allows the computer to solve the equation in milliseconds, even if the exponent has hundreds of digits.

Modulo Operations Table

Here are some simple examples to check your understanding.