## Enter Modular Exponentiation

Solve 28633 mod 8633 using:

Modular exponentiation

##### Build an algorithm:

n is our exponent = 8633

y = 1 and u ≡ 2 mod 8633 = 2

See here

##### n = 8633 is odd

Since 8633 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(2) mod 8633

(y)(u) mod p = 2 mod 8633

2 mod 8633 = 2
Reset y to this value

##### Determine u2 mod p

u2 mod p = 22 mod 8633

u2 mod p = 4 mod 8633

4 mod 8633 = 4
Reset u to this value

8633 ÷ 2 = 4316

##### n = 4316 is even

Since 4316 is even, we keep y = 2

##### Determine u2 mod p

u2 mod p = 42 mod 8633

u2 mod p = 16 mod 8633

16 mod 8633 = 16
Reset u to this value

4316 ÷ 2 = 2158

##### n = 2158 is even

Since 2158 is even, we keep y = 2

##### Determine u2 mod p

u2 mod p = 162 mod 8633

u2 mod p = 256 mod 8633

256 mod 8633 = 256
Reset u to this value

2158 ÷ 2 = 1079

##### n = 1079 is odd

Since 1079 is odd, calculate (y)(u) mod p

(y)(u) mod p = (2)(256) mod 8633

(y)(u) mod p = 512 mod 8633

512 mod 8633 = 512
Reset y to this value

##### Determine u2 mod p

u2 mod p = 2562 mod 8633

u2 mod p = 65536 mod 8633

65536 mod 8633 = 5105
Reset u to this value

1079 ÷ 2 = 539

##### n = 539 is odd

Since 539 is odd, calculate (y)(u) mod p

(y)(u) mod p = (512)(5105) mod 8633

(y)(u) mod p = 2613760 mod 8633

2613760 mod 8633 = 6594
Reset y to this value

##### Determine u2 mod p

u2 mod p = 51052 mod 8633

u2 mod p = 26061025 mod 8633

26061025 mod 8633 = 6631
Reset u to this value

539 ÷ 2 = 269

##### n = 269 is odd

Since 269 is odd, calculate (y)(u) mod p

(y)(u) mod p = (6594)(6631) mod 8633

(y)(u) mod p = 43724814 mod 8633

43724814 mod 8633 = 7302
Reset y to this value

##### Determine u2 mod p

u2 mod p = 66312 mod 8633

u2 mod p = 43970161 mod 8633

43970161 mod 8633 = 2292
Reset u to this value

269 ÷ 2 = 134

##### n = 134 is even

Since 134 is even, we keep y = 7302

##### Determine u2 mod p

u2 mod p = 22922 mod 8633

u2 mod p = 5253264 mod 8633

5253264 mod 8633 = 4400
Reset u to this value

134 ÷ 2 = 67

##### n = 67 is odd

Since 67 is odd, calculate (y)(u) mod p

(y)(u) mod p = (7302)(4400) mod 8633

(y)(u) mod p = 32128800 mod 8633

32128800 mod 8633 = 5407
Reset y to this value

##### Determine u2 mod p

u2 mod p = 44002 mod 8633

u2 mod p = 19360000 mod 8633

19360000 mod 8633 = 4814
Reset u to this value

67 ÷ 2 = 33

##### n = 33 is odd

Since 33 is odd, calculate (y)(u) mod p

(y)(u) mod p = (5407)(4814) mod 8633

(y)(u) mod p = 26029298 mod 8633

26029298 mod 8633 = 803
Reset y to this value

##### Determine u2 mod p

u2 mod p = 48142 mod 8633

u2 mod p = 23174596 mod 8633

23174596 mod 8633 = 3624
Reset u to this value

33 ÷ 2 = 16

##### n = 16 is even

Since 16 is even, we keep y = 803

##### Determine u2 mod p

u2 mod p = 36242 mod 8633

u2 mod p = 13133376 mod 8633

13133376 mod 8633 = 2583
Reset u to this value

16 ÷ 2 = 8

##### n = 8 is even

Since 8 is even, we keep y = 803

##### Determine u2 mod p

u2 mod p = 25832 mod 8633

u2 mod p = 6671889 mod 8633

6671889 mod 8633 = 7213
Reset u to this value

8 ÷ 2 = 4

##### n = 4 is even

Since 4 is even, we keep y = 803

##### Determine u2 mod p

u2 mod p = 72132 mod 8633

u2 mod p = 52027369 mod 8633

52027369 mod 8633 = 4911
Reset u to this value

4 ÷ 2 = 2

##### n = 2 is even

Since 2 is even, we keep y = 803

##### Determine u2 mod p

u2 mod p = 49112 mod 8633

u2 mod p = 24117921 mod 8633

24117921 mod 8633 = 5952
Reset u to this value

2 ÷ 2 = 1

##### n = 1 is odd

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

(y)(u) mod p = (803)(5952) mod 8633

(y)(u) mod p = 4779456 mod 8633

4779456 mod 8633 = 5407
Reset y to this value

##### Determine u2 mod p

u2 mod p = 59522 mod 8633

u2 mod p = 35426304 mod 8633

35426304 mod 8633 = 5105
Reset u to this value

1 ÷ 2 = 0

##### Because n = 0, we stop

28633 mod 8633 ≡ 5407

28633 mod 8633 ≡ 5407
##### 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