Enter Modular Exponentiation

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Solve 3²⁰ mod 23 using:

Modular exponentiation

Build an algorithm:

n is our exponent = 20

y = 1 and u ≡ 3 mod 23 = 3

See here

n = 20 is even

Since 20 is even, we keep y = 1

Determine u² mod p

u² mod p = 3² mod 23

u² mod p = 9 mod 23

9 mod 23 = 9
Reset u to this value

Cut n in half and take the integer

20 ÷ 2 = 10

n = 10 is even

Since 10 is even, we keep y = 1

Determine u² mod p

u² mod p = 9² mod 23

u² mod p = 81 mod 23

81 mod 23 = 12
Reset u to this value

Cut n in half and take the integer

10 ÷ 2 = 5

n = 5 is odd

Since 5 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(12) mod 23

(y)(u) mod p = 12 mod 23

12 mod 23 = 12
Reset y to this value

Determine u² mod p

u² mod p = 12² mod 23

u² mod p = 144 mod 23

144 mod 23 = 6
Reset u to this value

Cut n in half and take the integer

5 ÷ 2 = 2

n = 2 is even

Since 2 is even, we keep y = 12

Determine u² mod p

u² mod p = 6² mod 23

u² mod p = 36 mod 23

36 mod 23 = 13
Reset u to this value

Cut n in half and take the integer

2 ÷ 2 = 1

n = 1 is odd

Since 1 is odd, calculate (y)(u) mod p

(y)(u) mod p = (12)(13) mod 23

(y)(u) mod p = 156 mod 23

156 mod 23 = 18
Reset y to this value

Determine u² mod p

u² mod p = 13² mod 23

u² mod p = 169 mod 23

169 mod 23 = 8
Reset u to this value

Cut n in half and take the integer

1 ÷ 2 = 0

Because n = 0, we stop

We have our answer

Final Answer

3²⁰ mod 23 ≡ 18

You have 1 free calculations remaining

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What is the Answer?

3²⁰ mod 23 ≡ 18

How does the Modular Exponentiation and Successive Squaring Calculator work?

Free Modular Exponentiation and Successive Squaring Calculator - Solves xⁿ mod p using the following methods:
* Modular Exponentiation
* Successive Squaring
This calculator has 1 input.

What 1 formula is used for the Modular Exponentiation and Successive Squaring Calculator?

Successive Squaring I = number of digits in binary form of n. Run this many loops of a² mod p

For more math formulas, check out our Formula Dossier

What 6 concepts are covered in the Modular Exponentiation and Successive Squaring Calculator?

exponent

The power to raise a number

integer

a whole number; a number that is not a fraction
...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...

modular exponentiation

the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus)

modulus

the remainder of a division, after one number is divided by another.
a mod b

remainder

The portion of a division operation leftover after dividing two integers

successive squaring

an algorithm to compute in a finite field

Example calculations for the Modular Exponentiation and Successive Squaring Calculator

Modular Exponentiation and Successive Squaring Calculator Video

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