@TedShifrin I am solving problems in abstract algebra what do you think of my argument. If gcd(a,c) = 1 and gcd(b,c) = 1, prove that gcd(ab,c) = 1. Here is my argument.
Proof:
Let d = gcd(ab,c) $\rightarrow$ d | c and d | ab. Since gcd(a,c) = 1 = gcd(d$q_1$,c) and let d` = gcd(d,c). This means exists x,y $\in$ Z such that $dq_1x_1$ + cy = 1$
since d` | d and d | c so d` | d$x_1$ + cy = 1 hence d` = 1 . so gcd(d,c) = 1. So by euclid's lemma d | b however gcd(b,c) = 1 so d must be 1 hence gcd(ab,c) = 1.
Construct an encryption system ElGamal at the group $\mathbb{Z}_{19}^{\star}$. As base, we can use $g=\overline{2}$, that generates the group.
Describe the construction of a pair private/public key and encrypt the message $m=\overline{5}$.
Unfortunately I have no attempt. Can you help me with ...
