I was watching this video, and was curious how they were able to do the following:
$$m^e\ modN = c$$ $$c^d\ modN = m$$
Therefore,
$$m^{ed}modN = m$$
It's all simple algebra, but I wasn't sure how they were able to substitute $m^e$ for $c$ without including the $modN$. Is there a theory that allows us to do this?
I have never taken a course on number theory or modular arithmetic, just curious about the topic.
Assume $m^{ed} mod N = x$ i.e. $m^{ed} = Nk+x$
$m^e = Nk_1 + c$
$c^d = Nk_2+m$
$m^{ed} = (Nk_1+c)^d = Nk'+c^d = Nk'+Nk_2+m$
Substituting $k'+k_2=k$ we get $ m^{ed}= Nk+m$
So $m^{ed} mod N = m$