Why is Ring-LWE more efficient compared to LWE? Can someone please tell me why is the Ring-LWE more efficient?
By introducing polynomials in place of matrices, what kind of optimizations do we introduce that make Ring-LWE more efficient?
If you're using LWE for cryptography, then you'll need a system of $n$ linear equations with errors (mod some $q$). So for those $n$ equations, your key size will have $O(n^2)$ coefficients. In Ring-LWE, you only need $n$ coefficients (see Oded Regev's survey, page 5).
Because you're dealing with polynomials now, you can use the Fast Fourier Transform or (even better for finite fields) the number theoretic transform to multiply them.