finding inverse in affine cipher

Question

I am working on an example of Affine cipher, the decryption function is:

$$ x=Dk(y)=7^{-1}(y-3) mod 26 $$

I didn't understand how 7 inverse is 15?

$$ 7^{-1} = 15 $$

Please help me to understand this.

thanks @labbhattacharjee , but how do I get the exact value 15 please explain with your answer. — manoj karru, Jun 05 '16 at 11:39
See my answer here : http://math.stackexchange.com/questions/407478/solving-a-linear-congruence — lab bhattacharjee, Jun 05 '16 at 11:40
Alternatively using http://mathworld.wolfram.com/CarmichaelFunction.html, $$7^{-1}\equiv7^{12}\pmod{26}$$ — lab bhattacharjee, Jun 05 '16 at 11:42

score 1 · Answer 1 · answered Mar 28 '17 at 06:08

$7^{-1}$ is actually $7\,\,mod\,\,26$ here.Using Euclid' Algorithm

$\Rightarrow26=7*3+5$

$\Rightarrow 7=5*1+2$

$\Rightarrow 5=2*2+1$

$\Rightarrow 2=1*2+0$

Now use Back Substitution,

$\Rightarrow 1=5-2*2$

$\Rightarrow 1=5-2*\left(7-5*1\right)$

$\Rightarrow 1=3*5-2*7$

$\Rightarrow 1=3*\left(26-7*3\right)-2*7$

$\Rightarrow 1=26*3-7*11$

using bezout coefficients

-11 is the inverse of $7\,\,mod\,\,26$

or we can write

$26-11=15$ is also inverse of $7\,\,mod\,\,26$

1 Answers1