I was taught that two random variables $X,Y$ are independent when the induced sigma fields $X^{-1}(\mathscr{B})$ and $Y^{-1}(\mathscr{B})$ are made up with independent events.
Now I'm following some course in cryptography where they seem to take the definition of independence as being $P[X=x,Y=y] = P[X=x]P[Y=y]$
My question is are these equivalent?
Also I wanted to know what's the situation for mutual independence. Should I work with induced sigma field or with points.
Since, $X=x$ and $Y=y$ do represent the atomic events within the sigma fields of two discrete random variables, and $\forall x\in\Bbb R\,\forall y\in\Bbb R:P[X=x,Y=y]=P[X=x]P[Y=y]$ is the statement that all such events are independent, then yes, your two definitions are essentially equivalent.
To be more complete your cryptography textbook should say that any two random variables are independent iif: $$\forall\mathrm A\subseteq \Bbb R~ \forall\mathrm B\subseteq \Bbb R~:~ \mathsf P(X\in\mathrm A, Y\in\mathrm B)=\mathsf P(X\in\mathrm A)\,\mathsf P(Y\in\mathrm B)$$