Image encryption scheme based on the commutation and anti-commutation rules

A novel image encryption algorithm based on the commutation and anti-commutation rules is proposed. An image can be encrypted into two images by superposing its two one-dimensional Fourier transforms or fractional Fourier transforms in horizontal and vertical directions according to the commutation and anti-commutation rules. The image gained with commutation rule can be regarded as the encrypted image and the other gained with anti-commutation rule as the decryption key. Numerical results have demonstrated its effectiveness and shown the enhancement in security with the fractional orders as the extra keys. An optical setup is also designed to implement the proposed scheme.