I’m trying to learn how to do a hill cipher encryption / decryption by hand. For decryption, I calculate the derterminant, but when I need to invert it, I don't understand how I can get it ... I follow this example:
How did he get the 9 ??? for me you calculate something like 0,33333 mod 26 ...
I tried another example with a determinant 21 but no good result. Can I have a explanation about it
Thank you very much !
On
Fractions and negative exponents work very differently in modular arithmetic. We still have $3^{-1} = \frac13$ and $3\cdot \frac13 = 1$, but the result of trying to calculate it is wildly different.
By definition, $\frac13$ is the unique number such that if you multiply it with $3$, you get $1$. In other words, we have $\frac13\cdot 3 = 1$. This is true regardless of whether we're on the real number line, or in modular arithmetic. This is the definition of the fraction symbol.
However in modular arithmetic (say, modulo $26$), we don't have access to decimals. We only have access to the $26$ numbers $0, 1, 2, \ldots, 25$. And we have to identify $\frac13$ (defined by $\frac13\cdot 3 = 1$) from among these. Now, it turns out that $9\cdot 3 = 27 = 1$ (and just as importantly, no number other than $9$ satisfies this), so we get $\frac13 = 9$.
All that he is saying is that $3\times9\equiv1\pmod{26}$, which is true, since $3\times9=27=26+1$.