Keywords: cellular automata, cryptography, image encryption, image processing, secret key cryptosystem.

In this work we present a new graphic symmetrical cryptosystem in order to encrypt and decrypt colored images. This cryptosystem is based on a reversible memory cellular automata of dimension and uses a cryptographically secure pseudorandom bit generator (CSPRBG) in the encryption phase.

Let I be a colored image defined by pixels and a palette of c colors. Then, the image I can be represented as a linear array of n elements, M, with coefficients in , where , and , for black & white, grey-level and colored images, respectively.

The proposed cryptosystem consists of three phases. In the first phase, the setup phase, the two users agree the key to be shared and the cellular automaton (CA) to be used is defined. In the second phase, the encryption phase, the sender chooses a truly random generator or a CSPRBG and encrypts the secret image to be sent to the receiver. Finally, in the third phase, the decryption phase, the receiver uses the inverse CA to the one considered in the setup phase and decrypts the received image.

Specifically, the protocol is as follows:

Before encrypting an image, the sender and the receiver, Alice and Bob, respectively, agree to use a 128-bit secret key for the cryptosystem:

The one-dimensional reversible memory cellular automaton proposed to be used in the cryptosystem for encrypting is , and its inverse CA, , is used for decrypting, where:

1. The cellular space, , is a linear array of size .

2. The state set is given by .

3. The set of indices is selected so that each cell has the following neighborhood : .

4. The transition function of , f, is defined as follows
   
   where is defined from the 128 bits secret key, K as follows:
   
   and where , , are the bits of the key K given in the expression (66). Hence, the new state of the cell at time t+1 is determined by the following expression

   (67)

   
   

Note that the transition function of , f, permits to define its CA inverse by defining the transition function of by the expression

For encrypting a black & white, grey-level or colored image, I, Alice considers its representation as a linear array, M, of coefficients in . Moreover, she chooses either a truly random or a CSPRBG and develops the following protocol:

1. Alice considers the linear array M as the initial configuration of :

2. Alice uses the CSPRBG in order to obtain a sequence of n integers in : , for each image. This set can be considered as the configuration of the cellular automaton , by taking that is, , . Once the protocol has finished, this configuration, ,can be destroyed.

3. Alice encrypts each pixel of I, that is, determines the new states of each cell of the configuration by means of determining two evolutions of the cellular automaton , and , by using the formula (67). In this way, Alice obtains the encrypted array of M, , which is formed by the concatenation of the two evolutions computed:
   

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