EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 979-995 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Modified AES-512 Bits Algorithm for Data Encryption Kwame Owusu Bempah1,∗, Kwasi Baah Gyamfi2, Francis Ohene Boateng3, Isaac Owusu-Mensah4 1 Department of Computer Science and Information Technology, Faculty of Health and Applied Science, Christian Service University College, Kumasi, Ghana 2 Department of Mathematics, Faculty of Physical Science , Kwame Nkrumah University of Science and Technology, Kumasi, Ghana 3 Department of Mathematics Education, Faculty of Applied Science and Mathematics Education, Akenten Appiah- Menka University of Skills Training and Entrepreneurial De- velopment, Kumasi, Ghana 4 Department of Integrated Science Education, Faculty of Science Education, Akenten Appiah- Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana Abstract. Privacy is given top priority as far as communication involving transfer of confidential document or data is concerned. There is therefore a need to keep confidential data secretive from being invaded by unauthorised access, and this is done through cryptography algorithms, where the Advanced Encryption Standard algorithm has been the widely accepted symmetric block encryption algorithm for such purposes. This paper therefore focuses on developing a new AES-512 bits symmetric encryption algorithm through modification of the conventional AES-128 algorithm to be used purposefully in the classroom for document transfer.The development comes with increasing the plaintext bits of the conventional AES-128 algorithm to 512 bits plaintext which undergoes five operational transformations: STATE, SKGF, SRL, SCL and AARC through key size in the Galois field, GF (29). A numerical example is then given to explain the use of the algorithm, and finally, we provide a comparative study of this algorithm and other existing symmetric encryption models, such as the AES-128 and DES algorithms. 2020 Mathematics Subject Classifications: 68P25, 68P27, 68P30 Key Words and Phrases: Cryptography, Encryption, Symmetric Algorithm, AES ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5114 Email addresses: kowusubempah@csuc.edu.gh (K.Owusu Bempah), kbgyamfi.cos@knust.edu.gh (K.Baah Gyamfi), foboateng@aamusted.edu.gh (F.O. Boateng) iowusumensah@aamusted.edu.gh (I.Owusu- Mensah) https://www.ejpam.com 979 © 2024 EJPAM All rights reserved. K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 980 1. Introduction Cryptography involves studying and using mathematical techniques called algorithms to transform crucial messages, data or information to unclear or unreadable format [4] and comprises of encryption and decryption. Encryption simply means converting or turning a clear data, document or message into an unreadable format called ciphertext using al- gorithms or techniques whiles decryption is the process of converting the unclear message into a clear readable format called plaintext. In this era, cryptography branches into mathematics and computer science and has an alliance with information theory and security, which play a vital role in security in com- puting, communication in mobile phones, passwords in computing, and even engineering, unlike the ancient time when cryptography consisted of only the encryption and decryp- tion of messages using keys. So many applications resulted to a technique called Data Encryption Standard(DES) which is now becoming out of date due to the very small key size of 56-bits [2]. Due to the slowness of the DES algorithm implementation, the National Institute of Standards and Technology (NIST) introduced the Advanced Encryption Stan- dard (AES) with the ultimate aim of being both faster and also protected. The Advanced Encryption Standard, AES, uses plaintext block lengths of 128, 192, and 256 bits with either a key size of 128, 192, or 256 bits, respectively, through 10 rounds, 12 rounds, and 14 rounds of encryption respectively. Currently, the advanced encryption standard algo- rithm has been expanded and enhanced through development due to the numerous AES attackers. In view of this, researchers are still persisting in developing algorithms that would be able to deter intruders from invading confidential documents or data. A high speed and highly restricted image encryption algorithm in [9] was proposed by modifying the AES algorithm to improve its performance by decreasing both the com- putation cost and hardware requirement and also enhancing the security level. The only drawback in the process was that the encryption and decryption times really increased, and the attacks on the proposed algorithm could also lead to a reduction in the number of rounds of the algorithm. The research work of [5], discovered an efficient method for implementing the function of the AES byte substitution (S-box). This was focused on implementing the AES in non- volatile FPGAs. It was found that the proposed method uses minimum space and runs much faster than the one that requires using the whole S-box in the logic area, but the only drawback was that FPGAs cannot function in areas with low battery cases. In the research paper [8], an efficient implementation of the AES algorithm was adopted on an FPGA using the VirtexE family of devices (XCV812), where the results of the sequen- tial pipeline architecture and that of pipeline architecture were compared. It was found that, the design in using sequential took 2744 CLB with a throughput of 258.5M, while 2136 CLB slices with throughput of 2868Mbits were obtained for the pipeline. Using the pipeline for design can be more efficient based on area and throughput. But there could be comparison using various techniques such as parallelization, loop unrolling, or memory for implementing the AES algorithm to determine the effect on area and throughput as well. K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 981 In [6], a new version of the conventional AES algorithm was introduced using an input block of 512 bits with a key size of 512 bits and resources such as processor and mem- ory. The parameters of the AES 128 and 256 bits were compared to the new version and showed that the new version produced greater confidentiality and throughput and was resistant to linear and differential attacks. In the process, an increment of 230% in the throughput was realised compared to the AES-128 algorithm. The only drawback in the process was that there should be a performance analysis of the proposed algorithms and other symmetric algorithms to evaluate their security and other resources. In [7], an algorithm for advanced encryption standard was proposed, where sub-keys were produced differently from the main key and each sub-key is used to encrypt the AES round singly, but the proposed algorithm was very slow compared to the conventional AES algorithm and was resistant to brute force attack. Further, in [1], a 4D teserract symmetric block was redesigned to be of good territory for the manufacture of encryption keys, and four rounds and other sub rounds are engaged to shift the teserract to be able to randomly produce some of the encryption keys. The XOR operation technique is used in this method due to its speed and lightweight. A T-0,1 512 key was compared to the AES-256 to evaluate their speed performance, and the results show that the assumed key size was accepted based on the International Telecommunica- tion Union (ITU-T). All the algorithms presented above are used in industrial and multimedia settings, which involve networking and require the use of the internet to transfer or store confidential data. There is no algorithm for AES that has been developed practically for educational purposes. Motivated by this, in this paper, we develop a new AES-512 bits symmetric encryption algorithm through modification of the conventional AES-128 algorithm using five operational transformations: STATE, SKGF SRL, SCL, and AARC, to be used pur- posefully in the classroom for document transfer. 2. Preliminaries This section discusses the generation of the proposed AES-512-bit algorithm keys in Galois Field, GF (29), which are given in bit strings. 2.1. Key generation process for the proposed AES-512 bits algorithm in GF (2r) Every degree in the Galois field, GF (2r),is generally represented by the polynomial ar−1x r−1 + ar−2x r−2 + ar−3x r−3 + .... + ar−r, where r is the degree in the field. This implies that in GF (29), the corresponding polynomial with r = 9 is given as; a8x 8 + a7x 7 + a6x 6 + a5x 5 + a4x 4 + a3x 3 + a2x 2 + a1x+ a0 Where a0, a1,to a8 represent the coefficients of the polynomial and serve as bit strings.The bit strings to be used for the polynomial in our proposed algorithm to represent the keys are gotten in GF (29). K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 982 2.2. Proposed approach for generation of Bit strings in GF (pr) For any Galois field, GF (pr), where r= degree and p=2, the powers or exponents in the field are given as: 1 ≥ x ≥ r or 1 ≤ x ≤ r. 1 In our modified algorithm, GF (p9), r = 9 implies that the powers of the exponent are given as: {1 ≥ x ≥ 9} implying x = 9, 8, 7, 6, 5, 4, 3, 2, 1 2 The values of x are raised to the powers of p to get the Total Number of Strings (TNS) for each value of x as, TNS= 2x 3 The estimated Total Number of Strings (TNS) needs to have a pattern as to how the strings are written. This is given as: PTNS = TNS 2 . The division by 2 is a result of bits, which are in binary mode. Where PTNS is Pattern for Total Number of Strings. Example 1: For x = 9 as a power in the Galois field, then TNS = 2x = 29 = 512 and PTNS = TNS 2 = 256. This means we produce 256 (ones) continuously, followed by 256 (zeros), or vice versa. Example 2: Also, for x = 8 as a power in the Galois field, then TNS = 2x = 28 = 256 and PTNS = TNS 2 = 128. This means we produce 128 (ones) continuously, followed by 128 (zeros), or vice versa. The process is repeated again to make up the TNS. The following strings are produced for each value of x, which is substituted into the poly- nomial in GF (29), and the results for the polynomial have been arranged according to its degree. 000000000 000000001 degree 1= 000000010, 000000011 degree 2= 000000100, 000000101, 000000110, 000000111 degree 3= 000001000, 000001001, 000001010, 000001011, 000001100, 000001101, 000001110, 000001111 degree 4= ,000010000, 000010001, 000010010, 000010011, 000010100, 000010101, 000010110, 000010111, 000011000, 000011001, 000011010, 000011011, 000011100, 000011101, 000011110, 000011111 degree 5= 000100000,000100001,000100010,000100011, 000100100,000100101, ,000100110, 00100111,000101000,000101001,000101010,000101011,000101100, 000101101, 000101110,000101111,000110000,000110001,000110010,000110011, 000110100,000110101,000110110,000110111,000111000,000111001, 000111010,000111011,000111100,000111101,000111110, 000111111 degree 6= 001000000,001000001,001000010,001000011,001000100,001000101, 001000110,001000111,001001000,001001001,001001010,001001011,001001100, 001001101,001001110,001001111,001010000,001010001,001010010,001010011, 001010100,001010101,001010110,001010111,001011000,001011001,001011010, 001011011,001011100,001011101,001011110,001011111,001100000,001100001, K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 983 001100010,001100011,001100100,001100101,001100110,001100111,001101000, 001101001,001101010,001101011,001101100,001101101,001101110,001101111, 001110000,001110001,001110010,001110011,001110100,001110101,001110110, 001110111,001111000,001111001,001111010,001111011,001111100,001111101, 001111110,001111111 degree 7= 010000000,010000001,010000010,010000011,010000100,010000101, 010000110,010000111,010001000,010001001,010001010,010001011,010001100, 010001101,010001110,010001111,010010000,010010001,010010010,010010011, 010010100,010010101,010010110,010010111,010011000,010011001,010011010, 010011011,010011100,010011101,010011110,010011111,010100000,010100001, 010100010,010100011,010100100,010100101,010100110,010100111,010101000, 010101001,010101010,010101011,010101100,010101101,010101110,010101111, 010110000,010110001,010110010,010110011,010110100,010110101,010110110, 010110111,010111000,010111001,010111010,010111011,010111100,010111101, 010111110,010111111,011000000,011000001,011000010,011000011,011000100, 011000101,011000110,011000111,011001000,011001001,011001010,011001011, 011001100,011001101,011001110,011001111,011010000,011010001,011010010, 011010011,011010100,011010101,011010110,011010111,011011000,011011001, 011011010,011011011,011011100,011011101,011011110,011011111,011100000, 011100001,011100010,011100011,011100100,011100101,011100110,011100111, 011101000,011101001,011101010,011101011,011101100,011101101,011101110, 011101111,011110000,011110001,011110010,011110011,011110100,011110101, 011110110,011110111,011111000,011111001,011111010,011111011,011111100, 011111101,011111110,011111111 degree 8= 100000000,100000001,100000010,100000011,100000100,100000101, 100000110,100000111,100001000,100001001,100001010,100001011,100001100, 100001101,100001110,100001111,100010000,100010001,100010010,100010011, 100010100,100010101,100010110,100010111,100011000,100011001,100011010, 100011011,100011100,100011101,100011110,100011111,100100000,100100001, 100100010,100100011,100100100,100100101,100100110,100100111,100101000, 100101001,100101010,100101011,100101100,100101101,100101110,100101111, 100110000,100110001,100110010,100110011,100110100,100110101,100110110, 100110111,100111000,100111001,100111010,100111011,100111100,100111101, 100111110,100111111,101000000,101000001,101000010,101000011,101000100, 101000101,101000110,101000111,101001000,101001001,101001010,101001011, 101001100,101001101,101001110,101001111,101010000,101010001,101010010, 101010011,101010100,101010101,101010110,101010111,101011000,101011001, 101011010,101011011,101011100,101011101,101011110,101011111,101100000, 101100001,101100010,101100011,101100100,101100101,101100110,101100111, 101101000,101101001,101101010,101101011,101101100,101101101,101101110, 101101111,101110000,101110001,101110010,101110011,101110100,101110101, 101110110,101110111,101111000,101111001,101111010,101111011,101111100, 101111101,101111110,101111111,110000000,110000001,110000010,110000011, K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 984 110000100,110000101,110000110,110000111,110001000,110001001,110001010, 110001011,110001100,110001101,110001110,110001111,110010000,110010001, 110010010,110010011,110010100,110010101,110010110,110010111,110011000, 110011001,110011010,110011011,110011100,110011101,110011110,110011111, 110100000,110100001,110100010,110100011,110100100,110100101,110100110, 110100111,110101000,110101001,110101010,110101011,110101100,110101101, 110101110,110101111,110110000,110110001,110110010,110110011,110110100, 110110101,110110110,110110111,110111000,110111001,110111010,110111011, 110111100,110111101,110111110,110000000,111000000,111000001,111000010, 111000011,111000100,111000101,111000110,111000111,111001000,111001001, 111001010,111001011,111001100,111001101,111001110,111001111,111010000, 111010001,111010010,111010011,111010100,111010101,111010110,111010111, 111011000,111011001,111011010,111011011,111011100,111011101,111011110, 111011111,111100000,111100001,111100010,111100011,111100100,111100101, 111100110,111100111,111101000,111101001,111101010,111101011,111101100, 111101101, 111101110,111101111,111110000,111110001,111110010,111110011, 111110100,111110101,111110110,111110111,111111000,111111001,111111010, 111111011,111111100,111111101,111111110,111111111 3. Methodology This section describes the proposed approach for the modified AES-512 bit algorithm for the encryption of confidential documents or data to be used purposefully in the class- room. The modified AES-512 bit encryption algorithm uses plaintext of sixty-four bytes composed of a block array of 8× 8 square matrix with a key size in Galois field, GF (29). The plaintext is encrypted with a key using the algorithm,Ai = V +Mi Where V is plaintext comprising of sixty-four characters made up of a block array of sixty-four bytes put in a (8 × 8) square block matrix. M = keys in GF (29) and i= key initiation in the range 1 ≤ i ≤ 8. The modified algorithmic encryption is accomplished through the following five operational transformations: STATE, SKGF, SRL, SCL, and AARC. The justification for these steps is given alongside. STATE transformation: In this stage, the plaintext with characters of sixty-four byte is put in a (8 × 8) square block array of matrix. An initial key, say i = 1, containing sixty-four keys, is needed to initiate the algorithm, and this key is chosen randomly within GF (29) and written in a (8 × 8) square matrix of block array. It is then executed using the algorithm, A1 = V +M1. This step is purposely to put the plaintext into a square block array matrix, and by ar- ranging the plaintext in this pattern, it accelerates the ensuing encryption operations and ensures the processing of data adequately and completely. SKGF transformation: This stage is called Sub Keys from Galois Field (SKGF), and K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 985 the sub keys within the field, GF (29) for i = 2, 3, 4, 5, 6, 7, 8, represent the sub keys, which comprise of sixty-four keys each randomly chosen and initiated after the initiation of i = 1 to encrypt the same plaintext, V producing the sub algorithm, Ai = ∑8 i=2(V +Mi). This means the plaintext in this stage undergoes again seven series of encryption processes. This transformation is intended to produce round keys after the initial key, and producing the round keys is crucial for the AES encryption technique as it ensures the complexity of the algorithmic encryption, which promotes and enhances confidentiality. SRL transformation: This stage is called Shift Row Left Transformation (SRL), where the subscript of A from the output of the preceding stage would serve as the pivotal po- sition, which would represent the type of row and number of times the position would be shifted to the left outside the matrix for another matrix to be formed. This step actually involves shifting rows in the square block array matrix and thereby assisting in mixing and scrambling the data to increase diffusion, which is essential for attaining great encryption. SCL transformation: This stage is called the Shift Column Left (SCL) transformation, where we shift the entire column of the matrix of the ciphertext gotten from the SRL stage. The subscript of A from the output of the preceding stage serves as the type of column and also the number of times to be shifted to the left outside the matrix to create a different matrix. Meaning Ai for i = 1, 2, 3, 4, 5, 6, 7, 8 from the preceding stage represents the first, second, and up to eighth columns would be shifted respectively depending on the value of i to the left outside the matrix. This step is mixing of column operation which additionally increases the diffusion and con- fusion attributes of the algorithmic encryption which becomes more resistant to attacks. AARC transformation: This is the last stage in the algorithm called Add All Round Ciphertext (AARC). A bulky ciphertext, Ab, is produced where all the ciphertexts in the preceding stage are added to get a bulky ciphertext. This step involves extra operations to complete the encryption technique, and this final step is crucial for finalising the encryption process and generating the ciphertext that is protected to be ready for dissemination, propagation, and storage. Generally, each transformational step plays an essential role in the modified AES-512 bits algorithm, providing overall security and strength to the encryption technique. Numerical Example Supppose we want to encrypt a confidential data composed of plaintext characters of sixty-four bytes; Cryptography is the process of encryption and decryption of data The plaintext characters is put into an array of 8 × 8 square matrix block denoted as V to form the first stage called STATE K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 986 V =  C r y p t o g r a p h y space i s space t h e space p r o c e s s space o f space e n c r y p t i o n space a n d space d e c r y p t i o n space o f space d a t a  The characters would then be converted into their decimal mode values using the Ameri- can Standard Code for Information Interchange, (ASCII) table [3] below. Dec Chr Dec Chr Dec Chr Dec Chr Dec Chr Dec Chr 32 space 48 0 64 @ 80 P 96 , 112 p 33 ! 49 1 65 A 81 Q 97 a 113 q 34 ” 50 2 66 B 82 R 98 b 114 r 35 # 51 3 67 C 83 S 99 c 115 s 36 $ 52 4 68 D 84 T 100 d 116 t 37 % 53 5 69 E 85 U 101 e 117 u 38 & 54 6 70 F 86 V 102 f 118 v 39 ’ 55 7 71 G 87 W 103 g 119 w 40 ( 56 8 72 H 88 X 104 h 120 x 41 ) 57 9 73 I 89 Y 105 i 121 y 42 ∗ 58 : 74 J 90 Z 106 j 122 z 43 + 59 ; 75 K 91 [ 107 k 123 { 44 , 60 < 76 L 92 \ 108 l 124 | 45 - 61 = 77 M 93 ] 109 m 125 } 46 . 62 > 78 N 94 ˆ 110 n 126 ˜ 47 / 63 ? 79 O 95 111 o This produces the numerical values for the plaintext V below. V =  67 114 121 112 116 111 103 114 97 112 104 121 32 105 115 32 116 104 101 32 112 114 111 99 101 115 115 32 111 102 32 101 110 99 114 121 112 116 105 111 110 32 97 110 100 32 100 101 99 114 121 112 116 105 111 110 32 111 102 32 100 97 116 97  The plaintext V is substituted into our modified AES-512 bits algorithm, and an initial key, say i = 1, representing the first sixty-four keys, is chosen randomly from the Galois field, GF (29). This produces the al- gorithm A1 = V +M1 which is given as K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 987 A1 =  67 114 121 112 116 111 103 114 97 112 104 121 32 105 115 32 116 104 101 32 112 114 111 99 101 115 115 32 111 102 32 101 110 99 114 121 112 116 105 111 110 32 97 110 100 32 100 101 99 114 121 112 116 105 111 110 32 111 102 32 100 97 116 97  + M1 Since the keys are polynomials, the plaintext, V , is converted into the polynomial mode by first converting into binary mode, after which the binary numbers will serve as bit strings to be used as coefficients for the polynomial in GF (29) given in the table below. K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 988 V Binary polynomial,GF (29) V binary polynomial,GF (29) 67 1000011 1 + x5 + x6 116 1110100 97 1100001 1 + x+ x6 32 100000 116 1110100 1 + x+ x2 + x4 112 1110000 1 + x+ x2 101 1100101 1 + x+ x4 + x6 111 1110011 1 + x+ x2 + x5 + x6 110 1101110 1 + x+ x3 + x4 + x5 112 1110000 1 + x+ x2 110 1101110 1 + x+ x3 + x4 + x5 100 1100100 1 + x+ x4 99 1100011 1 + x+ x5 + x6 116 1110100 1 + x+ x2 + x4 32 100000 1 100 1100100 1 + x+ x4 114 1110010 1 + x+ x2 + x5 111 1110011 1 + x+ x2 + x5 + x6 112 1110000 1 + x+ x2 105 1101001 1 + x+ x3 + x6 104 1101000 1 + x+ x3 114 1110010 1 + x+ x2 + x5 115 1110011 1 + x+ x2 + x5 + x6 102 1100110 1 + x+ x4 + x5 99 1100011 1 + x+ x5 + x6 116 1110100 1 + x+ x2 + x4 32 100000 1 32 100000 1 114 1110010 1 + x+ x2 + x5 105 1101001 1 + x+ x3 + x6 111 1110011 1 + x+ x2 + x5 + x6 97 1100001 1 + x+ x6 121 1111001 1 + x+ x2 + x3 + x6 103 1100111 1 + x+ x4 + x5 + x6 104 1101000 1 + x+ x3 115 1110011 1 + x+ x2 + x5 + x6 101 1100101 1 + x+ x4 + x6 111 1110011 1 + x+ x2 + x5 + x6 115 1110011 1 + x+ x2 + x5 + x6 32 100000 1 114 1110010 1 + x+ x2 + x5 105 1101001 1 + x+ x3 + x6 97 1100001 1 + x+ x6 100 1100100 1 + x+ x4 121 1111001 1 + x+ x2 + x3 + x6 111 1110011 1 + x+ x2 + x5 + x6 102 1100110 1 + x+ x4 + x5 116 1110100 1 + x+ x2 + x4 112 1110000 1 + x+ x2 114 1110010 1 + x+ x2 + x5 121 1111001 1 + x+ x2 + x3 + x6 32 100000 1 32 100000 1 99 1100011 1 + x+ x5 + x6 32 100000 1 101 1100101 1 + x+ x4 + x6 121 1111001 1 + x+ x2 + x3 + x6 111 1110011 1 + x+ x2 + x5 + x6 110 1101110 1 + x+ x3 + x4 + x5 101 1100101 1 + x+ x4 + x6 112 1110000 1 + x+ x2 110 1101110 1 + x+ x3 + x4 + x5 32 100000 1 97 1100001 1 + x+ x6 K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 989 A1 = 1 + x5 + x6 1 + x+ x2 + x5 1 + x+ x2 + x3 + x6 1 + x+ x2 1 + x+ x6 1 + x+ x2 1 + x+ x3 1 + x+ x2 + x3 + x6 1 + x+ x2 + x4 1 + x+ x3 1 + x+ x4 + x6 1 1 + x+ x4 + x6 1 + x+ x2 + x5 + x6 1 + x+ x2 + x5 + x6 1 1 + x+ x3 + x4 + x5 1 + x+ x5 + x6 1 + x+ x2 + x5 1 + x+ x2 + x3 + x6 1 + x+ x3 + x4 + x5 1 1 + x+ x6 1 + x+ x3 + x4 + x5 1 + x+ x5 + x6 1 + x+ x2 + x5 1 + x+ x2 + x3 + x6 1 + x+ x2 1 1 + x+ x2 + x5 + x6 1 + x+ x4 + x5 1 1 + x+ x2 + x4 1 + x+ x2 + x5 + x6 1 + x+ x4 + x5 + x6 1 1 + x+ x3 + x6 1 + x+ x2 + x5 + x6 1 + x+ x2 1 + x+ x2 + x5 1 + x+ x2 + x5 + x6 1 + x+ x2 + x5 + x6 1 + x+ x4 + x5 1 1 + x+ x2 1 + x+ x2 + x4 1 + x+ x3 + x6 1 + x+ x4 1 1 + x+ x4 1 + x+ x2 + x4 1 + x+ x3 + x6 1 + x+ x2 + x5 + x6 1 + x+ x4 1 + x+ x6 1 + x+ x2 + x4 1 + x+ x2 + x5 1 1 + x+ x5 + x6 1 + x+ x4 + x6 1 + x+ x2 + x5 + x6 1 + x+ x4 + x61 + x+ x3 + x4 + x5 1 + x+ x6  + M1 This produces ciphertext in polynomials for encryption round 1. We now proceed to the SKGF transformation, where the sub keys from the GF (29) produce the sub-algorithms for i = 2 to 8. Thus Ai = 8∑ i=2 (V +Mi) produces; Ai(11) · · · Ai(18) ... . . . ... Ai(81) · · · Ai(88)  =  V(11) · · · V(18) ... . . . ... V(81) · · · V(88)  +  Mi(11) · · · Mi(18) ... . . . ... Mi(81) · · · Mi(88)  This means the same plaintext, V , undergoes seven series of encryption processes with K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 990 different keys chosen randomly from GF (29) up to encryption round 8.We then embark on the next stage for encryption round 9, which is the SRL transformation, where the output from the STATE and SKGF becomes the input, with subscripts i and j representing the matrix of the ith row and jth column, respectively, and 1 ≤ R ≤ 8 represents Row1 to Row8 Algorithm 1 SRL Transformation 1: Ai(SRL) = 8∑ i=1 q∑ j=k+1 Aij(V +Mi) Ensure: R1 ←− A1(STATE) 2: if i = 1 for q = 8 then 3: k = 0 4: else 5: if i = 1 for q = 1 then 6: k = 0 Ensure: R2 ←− A2(SKGF ) 7: if i = 2 for q = 8 then 8: k = 2 9: else 10: if i = 2 for q = 2 then 11: k = 0 Ensure: R3 ←− A3(SKGF ) 12: if i = 3 for q = 8 then 13: k = 3 14: else 15: if i = 3 for q = 3 then 16: k = 0 Ensure: R4 ←− A4(SKGF ) 17: if i = 4 for q = 8 then 18: k = 4 19: else 20: if i = 4 for q = 4 then 21: k = 0 K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 991 Algorithm 2 Continuation Ensure: R5 ←− A5(SKGF ) 1: if i = 5 for q = 8 then 2: k = 5 3: else 4: if i = 5 for q = 5 then 5: k = 0 Ensure: R6 ←− A6(SKGF ) 6: if i = 6 for q = 8 then 7: k = 6 8: else 9: if i = 6 for q = 6 then 10: k = 0 Ensure: R7 ←− A7(SKGF ) 11: if i = 7 for q = 8 then 12: k = 7 13: else 14: if i = 7 for q = 7 then 15: k = 0 Ensure: R8 ←− A8(SKGF ) 16: if i = 8 for q = 8 then 17: k = 0 We then proceed to the SCL transformation and the last stage, which is the AARC transformation for encryption rounds 9 and 10, respectively, where the output from the SRL stage becomes the input for the SCL transformation. This produces a bulky cipher- text, Ab, for encryption round 11. The algorithm below is used to realise the ciphertext. K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 992 Algorithm 3 SCL Transformation: Return Aij(SRL) from Step8 1: Ai(SCL) = 8∑ i=1 q∑ j=k+1 Aij(SRL) (V +Mi) Ensure: C1 ←− A1(SRL) 2: if i = 1 ∀ i ̸= j for 1 ≤ i ≤ 8 then 3: j = k + 1 for k = 1 4: else 5: if 2 ≤ i ≤ 8 then 6: j = k + 1 for k = 1 Ensure: C2 ←− A2(SRL) 7: if 1 ≤ 1 ≤ 2 ∀ i ̸= j for 1 ≤ i ≤ 8 then 8: j = 2k + 1 for 1 ≤ k = 2 9: else 10: if 3 ≤ i ≤ 8 then 11: j = 2k + 1 for k = 1 Ensure: C3 ←− A3(SRL) 12: if 1 ≤ i ≤ 2 ∀ i ̸= j then 13: j = 3k + 1 for k = 1 14: else 15: if i = 3 then 16: j = 3k + 1 for k = 2 17: else 18: if 4 ≤ i ≤ 8 then 19: j = 3k + 1 for k = 1 Ensure: C4 ←− A4(SRL) 20: if 1 ≤ i ≤ 3 ∀ i ̸= j then 21: j = 4k + 1 for k = 1 22: else 23: if i = 4 then 24: j = 4k + 1 for k = 0 25: else 26: if 5 ≤ i ≤ 8 then 27: j = 4k + 1 for k = 1 K.Owusu Bempah et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 979-995 993 Algorithm 4 Continuation SCL Transformation Ensure: C5 ←− A5(SRL) 1: if 1 ≤ i ≤ 4 ∀ i ̸= j then 2: j = 5k + 1 for k = 1 3: else 4: if i = 5 then 5: j = 3 6: else 7: if 6 ≤ i ≤ 8 then 8: j = 5k + 1 for k = 1 Ensure: C6 ←− A6(SRL) 9: if 1 ≤ i ≤ 5 ∀ i ̸= j then 10: j = 6k + 1 for k = 1 11: else 12: if i = 6 then 13: j = 5 14: else 15: if 7 ≤ i ≤ 8 then 16: j = 6k + 1 for k = 1 Ensure: C7 ←− A7(SRL) 17: if 1 ≤ i ≤ 6 ∀ i ̸= j then 18: j = 7k + 1 for k = 1 19: else 20: if i = 7 then 21: j = 7 22: else 23: if i = 8 then 24: j = 7k + 1 for k = 1 Ensure: C8 ←− A8(SRL) 25: if 1 ≤ i ≤ 8 then 26: 1 ≤ j ≤ 8 27: AARC transformation 28: return Aij(SCL) for 1 ≤ i ≤ 8 from step36 to step39 29: Ab = 8∑ i=1 Aij(SCL) Following the proceedings of this algorithm, we provide a comparative study of the modified algorithm and existing symmetric encryption algorithms like AES-128 and DES. Key Size: The modified AES-512 algorithm uses a key size of 512 bits, which provides better security and makes it resistant to different types of attacks, like differential and linear attacks, whereas the DES algorithm and AES-128 algorithm use a key size of 56 REFERENCES 994 bits and 128 bits, respectively. But the DES algorithm becomes vulnerable to attacks such as brute force attacks due to its smaller key size, whereas the AES-128 algorithm is less prone to attacks. Block Size: The DES algorithm and the AES-128 algorithm operate with a block size of 64 bits and 128 bits, respectively, while the modified AES-512 bits algorithm uses a block size of 64 bits to process data. The large block size in the modified algorithm enables more data to be encrypted in each block, which could lead to improved efficiency. Encryption Rounds: The DES algorithm and the AES-128 algorithm undergo 16 rounds of encryption and 10 rounds of encryption, respectively, whereas the modified AES-512 bits algorithm requires a different number of rounds to cater for the larger key size and block size. The modified algorithm’s additional rounds may enhance security but could slow down during execution. Security: The key size of the DES algorithm has been broken and becomes vulnerable to attacks due to its smaller size. Both the AES-128 and AES-512 algorithms are based on the advanced encryption standard (AES), but the modified AES-512 algorithm provides improved security features and offers resistance against several forms of attack, such as linear and differential attacks, making it resilient to protect confidential documents. Adaptability:The modified AES-512 bits algorithm was developed specifically for educa- tional purposes for document transfer, and the AES-128 bits algorithm is an established encryption algorithm used in the industry for data security and communication networks. The DES algorithm is not able to meet modern application requirements for security due to restrictions in key size and encryption strength. 4. Conclusion Due to the need to always secure confidential documents, a modified AES-512 bits al- gorithmic encryption approach has been developed specifically to be used in the classroom for the encryption of confidential documents or data. The algorithm produces keys for encryption of the data in polynomial mode, making it difficult for intruders to invade the data. The algorithm could be extended in the future to provide performance analysis of the modified algorithm with other symmetric encryption algorithms to provide insights into the efficiency and effectiveness of the modified algorithm and its potential for real-world applications. Acknowledgements The authors thank the reviewers for their contributions. We acknowledge that the inputs offered by the editor and reviewers have really helped us improve the paper. References [1] Ali M Alshahrani. Develp a 512–bit high speed symmertic algorithm. In 2016 Universal Technology Management Conference (UTMC), page 62, 2016. REFERENCES 995 [2] Ross Anderson, Eli Biham, and Lars Knudsen. Serpent: A proposal for the advanced encryption standard. NIST AES Proposal, 174:1–23, 1998. [3] Christoforus Juan Benvenuto. Galois field in cryptography. University of Washington, 1(1):1–11, 2012. [4] Albrecht Beutelspacher. Cryptology. MAA, 1994. 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