Enter Modular Exponentiation

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Solve 3³⁸ mod 43 using:

Modular exponentiation

Build an algorithm:

n is our exponent = 38

y = 1 and u ≡ 3 mod 43 = 3

See here

n = 38 is even

Since 38 is even, we keep y = 1

Determine u² mod p

u² mod p = 3² mod 43

u² mod p = 9 mod 43

9 mod 43 = 9
Reset u to this value

Cut n in half and take the integer

38 ÷ 2 = 19

n = 19 is odd

Since 19 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(9) mod 43

(y)(u) mod p = 9 mod 43

9 mod 43 = 9
Reset y to this value

Determine u² mod p

u² mod p = 9² mod 43

u² mod p = 81 mod 43

81 mod 43 = 38
Reset u to this value

Cut n in half and take the integer

19 ÷ 2 = 9

n = 9 is odd

Since 9 is odd, calculate (y)(u) mod p

(y)(u) mod p = (9)(38) mod 43

(y)(u) mod p = 342 mod 43

342 mod 43 = 41
Reset y to this value

Determine u² mod p

u² mod p = 38² mod 43

u² mod p = 1444 mod 43

1444 mod 43 = 25
Reset u to this value

Cut n in half and take the integer

9 ÷ 2 = 4

n = 4 is even

Since 4 is even, we keep y = 41

Determine u² mod p

u² mod p = 25² mod 43

u² mod p = 625 mod 43

625 mod 43 = 23
Reset u to this value

Cut n in half and take the integer

4 ÷ 2 = 2

n = 2 is even

Since 2 is even, we keep y = 41

Determine u² mod p

u² mod p = 23² mod 43

u² mod p = 529 mod 43

529 mod 43 = 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 = (41)(13) mod 43

(y)(u) mod p = 533 mod 43

533 mod 43 = 17
Reset y to this value

Determine u² mod p

u² mod p = 13² mod 43

u² mod p = 169 mod 43

169 mod 43 = 40
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

3³⁸ mod 43 ≡ 17

You have 1 free calculations remaining

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What is the Answer?

3³⁸ mod 43 ≡ 17

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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