The need to hide sensitive information has existed since ancient times. Nowadays, it has become even more important due to the rapid development of technology and multimedia. Transmissions of images and videos play an important role in medical systems, military image databases, and other areas (Li, 2006). Furthermore, our everyday electronic devices are used to transmit images, video clips, or music over wireless networks. Therefore, various algorithms based on mathematical calculations were proposed to securely hide this information, which gave birth to modern cryptography.

One of the simplest ideas was introduced in 1919 by Vernam. In modern terms, his proposal can be formalized as follows: given the message bit string μ, the sender can use the key string k of the same length and add it to the original message bit-wise to obtain the ciphertext c = μ ⊕ k. Here, we used ⊕ to denote the XOR operation. The receiver uses the ciphertext c to restore the original message by adding k to it bit-wise, i.e. we have μ = c ⊕ k. The presented idea is now known as the Vernam cipher.

Modern cryptography has gone a long way since then. In this work, we focus on the symmetric cryptography branch, which covers block and stream ciphers to encrypt messages. For implementation purposes, block ciphers are usually converted into stream ciphers by applying them in various modes of encryption to obtain the ciphertext.

One of the most popular block ciphers was standardized in Dworkin et al. (2001) and is now known as the Advanced Encryption Standard (AES). It uses a 128, 192, or 256-bit secret key to encrypt a 4 × 4 block of 16-bit entries. Depending on the size of the secret key, either 10, 12 or 14 rounds of four steps are performed.

Symmetric encryption is also a tool for hiding visual information. Over the years, many image encryption techniques have been proposed. One of the most popular ways to encrypt an image is by applying some chaotic mapping, e.g. the logistic map (Zia et al., 2022; Zhang and Liu, 2023). We use one of such maps, namely Arnold’s cat mapping, in our paper. These mappings can produce satisfactory results when used with statistically independent sampling and binary key derivation (Dinu and Frunzete, 2025). Many researchers recommend chaos-based encryption techniques for images due to their computational efficiency (Zia et al., 2022). However, generating large chaotic sequences is time-consuming, and, therefore, the real-time application of such schemes is problematic. For this reason, parallelization of computations is used in Daoui et al. (2023) to speed up the running time. This is where the matrix computations can be helpful due to the natural parallelization of matrix operations. We view it as one of the advantages of our scheme over other schemes.

Other researchers incorporate elliptic curves into their work (Hernández-Díaz et al., 2021; Chillali and Oughdir, 2023; Nagaraj et al., 2015; Singh et al., 2024). For example, in Hernández-Díaz et al. (2021), elliptic curves are used to generate encryption keys, while in Singh et al. (2024), elliptic curve points combined with a hash function are utilized to create dynamic S-boxes. However, calculations in elliptic curves are more complex than the structures discussed in this paper. Additionally, our approach relies solely on a highly non-linear mapping called the matrix power function (MPF), rather than a combination of two different methods.

In our paper (Dindiene et al., 2022), we presented a block cipher based on MPF. We have shown how to encrypt a single block of data using MPF as opposed to the multiple rounds approach used in AES. Later, in our papers (Levinskas and Mihalkovich, 2021; Mihalkovich et al., 2022a), we analysed the statistical properties and performance of our proposal implemented in various modes of encryption. The results obtained in Mihalkovich et al. (2022a) have shown that by carefully choosing the parameters of the system, our proposal can be executed in time comparable to the AES cipher. Moreover, in Mihalkovich et al. (2022c) and Mihalkovich et al. (2022b), we have extended our research to non-commuting algebraic structures.

While the obtained results have shown some promise, in this paper, we make several changes as compared to the original idea. The most significant change comes from applying an extra mapping to the original data prior to encrypting it with our cipher. We demonstrate how this additional step affects the quality of encryption and compare our results to AES. The proposed changes can be summarized as follows:

The rest of the paper is organized as follows: in Section 2 we present the key definitions of the algebraic structures and functions we use throughout the paper; in Section 3 we introduce the changes to the original idea; in Section 4 we investigate the quality of encryption in two modes, where the finite field of integers Z p is used as a platform group; in Section 5 we implement Galois Fields in our proposal and compare the quality of encryption to the results of the previous section. As mentioned previously, we also compare our results to analogous modes of AES in both sections. Conclusions are presented at the end of the paper.

First, we consider the finite field of integers Z p , where p is a prime.

In this paper, we use a special case of Sylow q-subgroup, where k = 1 and two primes p and q are linked via a relation p = 2 q + 1. Therefore, the Sylow subgroup has the following representation: (1) G q = ⟨ g 2 i | i = 0 , 1 , … , q − 1 ⟩ , where g is a generator of the multiplicative group Z p ∗ . Notably, each non-identity element of G generates the whole group. We also define a mapping γ : Z q ↦ G q as follows: γ ( x ) = g 2 x . Clearly, γ is an invertible mapping. In this paper, we use this mapping to interchange between the elements of Z q and G q to fit our needs.

Note that the Sylow subgroup can be defined in a more general case. However, our research does not require considering it.

Using the multiplicative group G q and a ring of integers Z q , we can define the following mapping:

In this paper, we compare two platform groups: the Sylow q-subgroup G q and the Galois Field GF ( q k ).

In our previous paper (Dindiene et al., 2022), we considered a symmetric block cipher based on the MPF, where we used the Sylow q-subgroup G q as a platform group along with an additional one-to-one mapping Γ (an entry-wise application of γ), which was used to map the initial values of the pixels to the elements of G q . However, as we have shown in Levinskas and Mihalkovich (2021), the function Γ does not help to achieve good statistical properties of our cipher. Therefore, here we use a so-called Arnold’s cat map defined below:

In this paper, we use the parameter values a = 1 and b = 1. Note that Arnold’s cat map is invertible since the determinant of the transformation matrix is equal to 1.

Furthermore, in our research, we use the Hadamard product, otherwise known as the entry-wise multiplication of two matrices, and its inverse, denoted by an upper index H. In other words, we have:

Also, we define the 3 × 3 correlation matrix Σ avg , where each row corresponds to a distinct channel, and each column represents a different direction of a neighbouring pair: horizontal, vertical, or diagonal. Each entry in the matrix corresponds to the correlation coefficient between neighbouring pixel intensities for the respective pair type.

We use this correlation matrix as one of the tools to investigate relations between the entries of the encrypted blocks.

In this section, we revise the block cipher considered in Dindiene et al. (2022). We present here a modified version of our cipher, since we implement Arnold’s cat map together with the previously used function Γ. Therefore, we introduce an additional step applied to the whole image to improve the quality of encryption.

Let μ be a string of bits representing the initial message. We cut this string into m 2 8-bit chunks and therefore obtain its representation in the following form: μ = μ 11 ‖ μ 12 ‖ … ‖ μ 1 m ‖ μ 21 ‖ μ 22 ‖ … ‖ μ 2 m ‖ … μ m m , where ‖ is the concatenation operation. Furthermore, if the message is too short, we append the appropriate amount of random bits at the end of the message. We can now construct the matrix representation M of the initial message, where the entries of M are integers μ i j ∈ [ 0 , 255 ]. We apply Arnold’s cat map to the matrix M a fixed number of times and denote the obtained result by M ′ = ACM ( τ , M ), where τ is the number of iterations. We now encrypt M ′ using a secret key – a triplet of matrices ( X , Y , Z ), where X , Y ∈ Z q m × m , Z ∈ G q m × m , q > 255, matrix X does not contain any zero entries and Y is invertible. For application purposes, we want to keep the values of q and p as small as possible. Hence, in this paper, we set q = 281 and p = 563.

The encryption algorithm consists of the following steps: S 1 ≡ ( X + M ′ ) mod q ; S 2 ≡ ( Z ⊙ Y Γ ( S 1 ) Y ) mod p ; S ≡ ( Γ − 1 ( S 2 ) + X ) mod q , where Γ ( X ) : Z q m × m ↦ G q m × m is a publicly known one-to-one mapping which replaces entries of matrix X with elements from G q – a Sylow subgroup of Z p . Clearly, Γ − 1 ( S 2 ) is the inverse transformation. We use ⊙ to denote the Hadamard product of two matrices.

Note that we use both q and p to perform modular operations. Therefore, we specify the modulus for each step of the encryption algorithm. Clearly, the modular arithmetic is applied entry-wise.

The decryption algorithm works similarly in reverse, i.e.: D 1 ≡ ( S − X ) mod q ; D 2 ≡ Y − 1 ( Γ ( D 1 ) ⊙ Z H ) Y − 1 mod p ; M ′ ≡ ( Γ − 1 ( D 2 ) − X ) mod q . Now all that remains is to apply Arnold’s cat map to M ′ in reverse to obtain the initial message in the matrix form M and then concatenate its entries as presented above to restore the message μ.

Previously, in Dindiene et al. (2022), we have shown that the considered block cipher is perfectly secure, i.e. it satisfies the following property: (5) Pr [ c = c 0 | μ = μ 0 ] = Pr [ c = c 0 ] , where c 0 and μ 0 are some fixed ciphertext and plaintext values. In other words, the ciphertext is statistically independent of the original message.

Perfect secrecy property was previously proven for the Vernam cipher. However, due to one-time secret keys and several other disadvantages, the Vernam cipher remains a purely theoretical algorithm: while easy to understand, it is never used in the real world.

One disadvantage of any perfectly secure cipher is the size of the secret key, since it has to be at least as long as the plaintext. This, together with one-time use of the key, prevents applying any modes of encryption to long messages. On the other hand, we have shown in Dindiene et al. (2022) that in our case, the secret key can be reused. Therefore, the perfectly secure block cipher we consider can be transformed into a stream cipher by applying modes of symmetric encryption. We consider two of these modes in this paper.

Like AES, our cipher operates on blocks. Therefore, to convert our idea to a stream cipher, we have to apply one of the encryption modes. Two of such modes considered in this paper are the electronic codebook (ECB) mode and the cipher block chaining (CBC) mode. A well-known fact is that the ECB mode is considered insecure even for such ciphers as AES. In this paper, we explore how Arnold’s cat mapping affects the quality of the ECB mode of our block cipher. Moreover, we perform a comparison of CBC modes for AES and our proposal.

We can also use the Galois fields in our approach for better comparison with AES. In this paper, we consider the Galois field GF ( 2 9 ) as the platform group. We use the irreducible polynomial ϕ ( x ) = x 9 + x 4 + 1 due to the adjusted default parameter in the Python library for the Galois field. As in the case of AES, the private key matrices X , Y , Z have size 4 × 4 with their entries x i j ∈ [ 1 , 256 ], y i j ∈ [ 1 , 510 ], z i j ∈ GF ( 2 9 ) ∖ { 0 }. Moreover, Y is an invertible matrix. The entries of the message block matrix M are integers in the range [ 0 , 255 ]. As previously, we apply Arnold’s cat map to the matrix M τ times and denote the obtained result by M ′ = ACM ( τ , M ). We now use the secret key K → = ( X , Y , Z ) to encrypt a single block M ′ as follows: S 1 = X + M ′ ; S 2 ≡ ( Z ⊙ Y Γ ( S 1 ) Y ) mod ϕ ( x ) ; S ≡ ( Γ − 1 ( S 2 ) + X ) mod 2 9 , where Γ ( S 1 ) : ( 1 + Z 511 ) 4 × 4 ↦ GF 4 × 4 ( 2 9 ) is a publicly known one-to-one mapping which replaces entries of matrix S 1 with elements from GF ( 2 9 ). Clearly, Γ − 1 ( S 2 ) is the inverse transformation. Of course, the simplest case of Γ is to interpret entries of S 1 as polynomials in GF ( 2 9 ). However, Γ can also permute elements of GF ( 2 9 ).

Note that the first step of encryption does not use modular arithmetic, i.e. since the entries of key matrix X are in the range [ 1 , 256 ], whereas the entries of M ′ are in the range [ 0 , 255 ] their sum S 1 contains entries in the range [ 1 , 511 ] and therefore can be interpreted as an element of ( 1 + Z 511 ) 4 × 4 .

The decryption algorithm is as follows: D 1 ≡ ( S − X ) mod 2 9 ; D 2 ≡ Y − 1 ( Γ ( D 1 ) ⊙ Z H ) Y − 1 mod ϕ ( x ) ; M ′ = Γ − 1 ( D 2 ) − X , where we see that at the last step, no modular arithmetic is used.

The number of changing pixel rates (NPCR) and the unified average changed intensity (UACI) are the two most common statistical characteristics used to evaluate the strength of image encryption ciphers with respect to differential attacks (Wu et al., 2011). The results of the numerical analysis using these metrics need to be compared to the critical values that these statistics are theoretically expected to reach.

Strict avalanche criterion resides in different fields of cryptography, and it is generalized by the avalanche effect (Castro et al., 2005). It measures the amount of non-linearity in substitution boxes (also known as s-boxes), and it is a key component in AES (Castro et al., 2005).

Let us explore these characteristics for our cipher defined over the Galois field. We consider two images: ‘Skittles.png’ and the Linux penguin. The experiment is based on comparing two ciphertexts of 4 × 4 block size and is performed as follows:

The first comparison is whether the specific coordinate pixel value changed or not – this difference is captured by the NPCR. The expected value for this statistic is given by the following expression: (15) E ( NPCR ) = L − 1 L · 100 % , where L − 1 is the upper bound of the uniform distribution U ( 0 , L − 1 ). In our case, we have L = 512 and therefore E ( NPCR ) ≈ 99.8 %.

Also, we keep track of the absolute difference in ciphertexts’ pixel intensities and use the UACI metric to measure it. The expected value is: (16) E ( UACI ) = L + 1 3 L · 100 % , and therefore in our setup we have E ( UACI ) = 33.4 %.

We resize both considered images to a 4 × 4 block size and perform 50 iterations, and calculate the average NPCR and UACI values for all iterations. During the experiment, we do not use Arnold’s cat mapping, since otherwise we would not evaluate the encryption scheme appropriately. We output the values in Table 1.

We can observe that the obtained values of both characteristics are close to the expected ones. Therefore, we do not reject the hypothesis that the ciphertexts are generated independently of each other.

Now we calculate the avalanche effect of our cipher. Since we consider 4 × 4 blocks, each entry has 8 bits that can be changed in the initial image, and the block consists of three channels, we perform exactly 4 · 4 · 8 · 3 = 384 iterations. For both images, we output the results as heatmaps for each channel and for each coordinate separately.

We can observe that for both images, the results are similar (see Fig. 19). The variance from the expected value of 0.5 is approximately 0.02, which shows an insignificant difference in value variation during the experiment, despite the coordinate or the channel. Therefore, from the avalanche effect perspective, the theoretically expected value coincided with the experimental result.

We can also observe that there is no clear pattern of higher or lower values. The results are relatively random, although the confidence interval, which includes the value of 0.5, is narrow.

In this paper, we have introduced improvements to our original block cipher, presented in Dindiene et al. (2022), e.g. we now use individual keys for each of the RGB colours, and we implemented Arnold’s cat mapping to the initial digital image to improve the quality of encryption. Also, we implemented Galois Fields in our proposal, making it more convenient for practical applications.

We considered both versions of our block cipher in ECB and CBC modes of encryption and analysed the quality of encrypted images using statistical characteristics. The obtained results have shown that the implementation of Arnold’s cat mapping plays an important part in the overall quality of encryption. For example, we have demonstrated in Fig. 9c and 9d that the considered plaintexts are statistically indistinguishable given the ciphertext, which is evidence for the semantic security of our proposal. We calculated the correlation matrices for both images. The maximum absolute correlations of 0.005 and 0.009, respectively, have shown the lack of statistically significant linear relations between the pixels, which is a desirable result for a secure cipher. Also, we saw that the entropy for both images is 8.131, which is close to the theoretical maximum. Therefore, the statistical results obtained for the CBC mode of our block cipher are indistinguishable from the AES.

Implementing Arnold’s cat mapping proved especially useful for monochromatic images. We have shown in Fig. 7 that our proposal, together with Arnold’s cat mapping, surpasses the AES cipher in ECB mode. Moreover, the maximum absolute correlation of the ciphertext was 0.009, which is similar to the result obtained for the CBC mode of encryption. However, as can be seen from Fig. 14b, the ECB mode is still not recommended for our cipher, since Arnold’s cat mapping cannot remove the dominance of one colour in this case. On the other hand, while ciphertexts in Figs. 4 and 14a are still low quality, they seem more chaotic than the ciphertext obtained from the ECB mode of AES (see Fig. 6a), even without additional scrambling.

We also considered the avalanche effect of our proposal. The obtained results have shown that we can expect an average of 99.6 % value of NPCR and a 33.5 % value of UACI statistical characteristics. These results are fairly close to the desired values of these characteristics. Furthermore, we can expect the avalanche effect in the range [ 0.48 ; 0.52 ], which is close to the ideal value of 0.5.

Note, however, that in our paper we have fixed the parameters of Arnold’s cat mapping, as well as the number of its iterations. In the future, it may be interesting to explore the effect of scrambling with other parameter values.

Also, to shorten this paper, we have left out the performance analysis for the enhanced version of our cipher. Though from our previous work, there are indications that our proposal can be executed reasonably fast, it may be necessary to consider the influence of Arnold’s cat mapping and parallelization on the performance of our cipher.