Motivated by recent research on asymmetric cryptosystems, a novel asymmetric scheme for image encryption that uses double random-decomposition technique in the fractional Fourier transform domain is proposed. The scheme endures the Special Attack as against conventional asymmetric cryptosystems based on phase-truncated Fourier transform (PTFT), and equal modulus decomposition. In the proposed scheme, an input image is bonded with a random phase mask and then it is subjected to a fractional Fourier transform. The resulting image is decomposed into two components using the random-decomposition technique. One of them will act as the first private key and the other component is subjected to the second fractional Fourier transform followed by another random-decomposition. Again, two new components are obtained, one will act as the second private key and the other is phase-truncated before subjecting it to LU decomposition followed by affine transform to get the encrypted image. The new scheme possesses enlarged key-space consisting of private keys obtained from random-decomposition, orders of fractional Fourier transform, affine transform parameters and permutation matrix of LU decomposition, thereby having a much greater capability to resist brute force attack. A sensitivity analysis has been carried out with respect to the encryption parameters. In addition to its resistance to the Special Attack, the scheme is immune to the basic attacks such as known-plaintext attack, chosen-plaintext attack, ciphertext-only attack, by virtue of its asymmetric nature. The above analysis along with statistical analysis through 3D plots and correlation distribution establish the strength of the proposed cryptosystem.