Equivalance Operator - Perfect Secrecy

Question

The equivalance operator is the inverse of the XOR operator, it's symmetric. Would this mean that it would also provide theoretical perfect secrecy just like XOR?

XOR

x ⊕ y = (x ∨ y ) ∧ ¬ (x ∧ y )

EQUIVALENCE

x ≡ y = ¬ (x ⊕ y)

There is no information leakage in theory if you just take the inverse of a perfectly symmetric function, it has to be perfect in both ways in theory.

What are you thoughts?

Answer 1

Actually, for any group operator $\odot$, we see that $x \odot y$ gives perfect security, assuming that $y$ is uniformly random over all possible group members.

The $\equiv$ operation is a group operation, with the group members being $\{0, 1\}$, with $1$ being the identity. Hence, yes, it gives perfect security.