affine cipher problem

[maronvomra]

Hi,
In affine cipher if the output is negative then how to get the keys
like
if the keys are (a,b) and if letter C is represented as 2, maps to the
letter T, represented as 19, and the letter R, represented as 17 maps
to Q , represented as 16 so we get
19 = a.2 + b mod 26
16 = a.17 + b mod 26
Subtracting these two equ. we get 3 = -15a or, 15power -1 mod 26 = 7
mod 26
now how to get k which is (5,9)

Thanks.

[Jeff Dege]

It's just plain algebra, except for the difference in how the
multiplicative inverse is calculated.

You've got 3 = -15 * a, which is 1 * 5a = -3, which is 15 * a = 23

Now 15 * 7 = 105, which is 15 * 7 = 1, so as you say, 15^-1 = 7

So going back to 15 * a = 23, you get 7 * 15 * a = 7 * 23, which gives
you 1 * a = 161, or a = 5

And then plugging a = 5 into 19 = a * 2 + b, we get 5 * 2 + b = 19, which
is 10 + b = 19, and then b = 9.

--
That's the mild form of the vilification that is the reflexive tool of
intimidation for the Left when it doesn't control the state. When it does,
you get the gulag and the firing squad.
- David Horowitz

[maronvomra]

Thanks a lot. I got it now.