8×8 S-boxes over Klein four-group and Galois field ${GF}\left({2}^{4}\right)$: AES redesign

1.   Introduction

Cancer is a dangerous disease and belongs to the family of genetic diseases. It is the uncontrolled magnification of abnormal blood cells in the body that stops normal functions and is prone to infection. The cancer that affects blood cells is known as blood cancer. Leukemia is an example of blood cancer and a mutation in the DNA of blood cells is its main cause, resulting in abnormal behavior of blood cells. This will not only prone to infection but also in some cases becomes chronic and creates tumors in bones. Throughout the world, nearly 1.24 million people are affected by blood cancer annually. Medicos and scientists are always probing for better ways to care for people fighting cancer. One way to do this is to develop and study incipient drugs. Drug revelation is not an easy task, as it is expensive, time consuming, and challenging in some cases. Numerous ways have been discovered to treat cancer. Drug therapy is one of them. Drug therapy is used to stop the growth of cancer cells and eliminate them from the body, as well as to restore healthy cells. Anticancer drugs are also used to kill and halt this malignant disease and many drugs test are accompanied to fight the fatal disease. This requires timely diagnosis, screening, and medication that benefits patients to control the deadly disease in the future. For further detail see ^[1,2,3,14].

Topological Indices (TIs) are termed as numeric descriptors that are obtained from molecular graphs to describe chemical system and are mostly used to investigate the physiochemical properties of several drugs. There are several kinds of polynomials and topological indices which are extensively calculated, represent chemical structure and have vital position in chemical graph theory. Among these classes, degree-based topological indices are of great importance and particularly in chemistry. The use of graph invariants (TIs) in QSPR and QSAR studies has been of key interest in recent years. Topological indices have application in various areas of biology, mathematics, bioinformatics, mathematics, informatics, biology etc., but their utmost significant use to date is in the non-empirical Quantitative Structure- Property Relationships (QSPR) and Quantitative Structure -Activity Relationships (QSAR) ^[5,18,26]. The ABC index, Wiener index and Randic index are helpful for predicting the bioactivity of drugs. The QSPR models help to determine the optimal relationship between the topological indices and psychochemical characteristic. These psychochemical qualities are being studied because they have a big impact on bioactivity and drug transit in the human body. In this paper, we compute degree-based TIs related to blood cancer drugs. In the same way, anticancer drugs represent chemical compounds on which given topological indices are well defined and discussed in QSPR analysis. The corresponding characteristic calculated in this way is highly correlated with characteristic of blood cancer drugs by the use of linear regression.

2.   Material and method

In drugs, structure elements denote vertices, and corresponding bonds connecting the atoms are termed edges. Graph G (V, E) is considered as simple, finite, and connected, whereas V and E represented in chemical graph are termed as vertex and edge sets, respectively. Degree of vertex in graph G is number of vertices adjacent to it and is denoted by d_u. Valence of a compound in chemistry and the degree of vertex in the graph are closely related concepts, for more details see ^[4,10,11,13]. Degree-based topological indices used are defined below:

Def. 2.1 The ABC index ^[13] of a molecular graph G is defined as

Def. 2.2 The first-degree-based topological index is Randic index $X\left(G\right)$ introduced by Milan Randic in 1975 ^[15]. Randic index is defined as:

Def. 2.3 The sum connectivity index ^[16] of a molecular graph G is defined as

Def. 2.4 The GA index ^[17] of a molecular graph G is defined as:

Def. 2.5 The first and second Zagreb indices ^[19] of a molecular graph G are defined as follows:

Def. 2.6 The harmonic index ^[20] of a molecular graph G is defined as:

Def. 2.7 The hyper Zagreb index ^[21] of a molecular graph G is defined as:

Def. 2.8 The forgotten index ^[23] of a molecular graph G is defined as:

The values of physical properties are taken from Chem Spider. It is observed from data Tables 2 and is found these data values are normally distributed. So, linear regression model is most adequate to test and adopt for said analysis. For more information on degree-based topological indices, we refer the reader to visit the following articles ^{[4,5,6,7,8,9,22,27,28]}.

The molecular formula for bulasan is C₈H₁₄N₆S₂. Busulfan is an antineoplastic alkylating agent and is used for many kinds of cancer. Alkylating agents have the ability to add alkyl groups to several electronegative groups under conditions present in cells. They prohibit tumor development by crosslinking guanine bases in DNA double-helix strands, directly attacking DNA. The strands are unable to separate and uncoil. It is mandatory in DNA replication and cells are no longer divide. The molecular formula of clofarabine is C₁₀H₁₁ClF N₅ O₅. Clofarabine interfere in growth of cancer cells. Clofarabine prevents cells from making DNA and RNA by interfering with the synthesis of nucleic acids, thus stopping the growth of cancer cells. The chemical formula of azacitidine is C₈H₁₂N₄O₅. Azacytidine has been used as an antineoplastic agent. The molecular formula of meracptopurine is C₅H₄N₄S. Mercaptopurine is one of a large series of purine analogues that interfere with nucleic acid biosynthesis and have been found active against human leukemias. The molecular formula of Tioguanine is C₅H₅N₅S. Antineoplastic compound which also has antimetabolite action. The drug is used in the therapy of acute leukemia.

The molecular formula of nelarabine is C₁₁H1₅N₅ S₅. Nelarabine is a purine nucleoside analog and antineoplastic agent used for the treatment of acute T-cell lymphoblastic leukemia and T cell lymphoblastic lymphoma with inadequate clinical response to prior chemotherapeutic treatments. The molecular formula of cytarabine is C₁₁H1₅N₅ S₅. Cytarabine is an antineoplastic antimetabolite used in the treatment of several forms of leukemia, including acute myelogenous leukemia and meningeal leukemia. The molecular formula of bosutinib is C₂₆H₂₉Cl₂N₅ O₅. It is used to treat a certain type of chronic myeloid leukemia (a cancer of white blood cells). The molecular formula of dasatinib is C₂₂H₂₆ClN₇S.Dasatinib is a tyrosine kinase inhibitor used for the treatment of lymphoblastic or chronic myeloid leukemia. The molecular formula of melphala is C₁₃H₁₈Cl₂N₂ O₂. Melphalan is an antineoplastic in the class of alkylating agents and is used to treat various forms of cancer. Alkylating agents are so named because of their ability to add alkyl groups to many electronegative groups under conditions present in cells. They stop tumor growth by crosslinking guanine bases in DNA. The molecular formula of dexamethasone is C₂₂H₂₉FO₅. Dexamethasone is a glucocorticoid available in various modes of administration that is used for the treatment of various inflammatory conditions, including bronchial asthma, as well as endocrine and rheumatic disorders. The molecular formula of doxorubicine is C₂₇H₂₉NO₁₁. Doxorubicin is an antineoplastic in the anthracycline class. Anthracyclines are among the most important antitumor drugs available. Doxorubicin is widely used for the treatment of several solid tumors while daunorubicin and idarubicin are used exclusively for the treatment of leukemia. The molecular formula of carbopalatin is C₆H₁₂N₂O₄Pt. Carboplatin is a alkylating agent used to treat advanced ovarian cancer.

3.   Results and discussions

In this section, degree-based TIs are imposed on blood cancer drugs. The relation between QSPR analysis and topological indices depicts that they are highly correlated as regards physicochemical properties use to cure blood cancer. The thirteen medicines azacitidine, buslfan, mercaptopurine, tioguanine, nelarabine, cytarabine, clofarabine, bosutinib, dasatinib, melphala, dexamethasone, doxorubicine, carboplatin are used for this analysis of said disease. The chemical structure for given drugs is shown in Figure 1. In drugs, structure elements denote vertices, and corresponding bonds connecting the atoms are termed edges. Hence, the study used regression analysis for the calculation purpose.

3.1.   Regression model

In this article, quantitative structure analysis about nine topological indices is calculated for QSPR modeling purpose. Five physical properties, boiling point (BP), molar volume (MV), molar refractivity (R), complexity and flash point (FP), for 13 medicines arranged in Figure 1, are investigated. We execute the regression analysis for the drugs and tested linear regression model is tested with the help of equation as under:

Here, P is the physicochemical property of the candidate drug. The TI, A, and b represent topological index, constant and regression coefficient, respectively. All data tables are calculated by the use of SPSS software version-26 to obtain accurate results. The nine TIs of candidate blood cancer drugs and their physical property are investigated with the help of linear QSPR model. By applying Eq 1, we calculate linear regression model for degree based TIs of candidate drugs given as under.

Theorem 1. Let ${G}_{1}$ be the graph of Azacitidine, the various topological indices of G are given as follows.

i) $ABC\left({G}_{1}\right) = 22.04$

ii) $RA\left({G}_{1}\right) = 13.12$

iii) $S\left({G}_{1}\right) = 13.46$

iv) $GA\left({G}_{1}\right) = 27.72$

v) $M1\left({G}_{1}\right) = 162$

vi) $M2\left({G}_{1}\right) = 209$

vii) $F\left({G}_{1}\right) = 516$

viii) $H\left({G}_{1}\right) = 11.95$

ix) $HM\left({G}_{1}\right) = 934.00$

Proof. Let ${G}_{1}$ be the graph of Azacitidine with edge set E, Let ${E}_{m, n}$ represents the class of edges of G joining vertices of degrees $m$ and $n$. With $\left|{E}_{\mathrm{1, 2}}\right| = 3,$ $\left|{E}_{\mathrm{1, 3}}\right| = 4,$ $\left|{E}_{\mathrm{1, 4}}\right| = 6,$ $\left|{E}_{\mathrm{2, 3}}\right| = 4,$ $\left|{E}_{\mathrm{2, 4}}\right| = 5,$ $\left|{E}_{\mathrm{3, 3}}\right| = 3,$ $\left|{E}_{\mathrm{3, 4}}\right| = 1,$ $\left|{E}_{\mathrm{4, 4}}\right| = 4.$

i) By using definition 2.1 and above given edge partitions ${E}_{m, n}$ we get,

ii) By using Definition 2.2 and above given edge partitions ${E}_{m, n}$ we get,

iii) By using Definition 2.3 and above given edge partitions ${E}_{m, n}$ we get,

iv) By using definition 2.4 and above given edge partitions ${E}_{m, n}$ we get,

v) By using Definition 2.5 and above given edge partitions ${E}_{m, n}$ we get,

vi) By using Definition 2.5 and above given edge partitions ${E}_{m, n}$ we get,

vii) By using definition 2.6 and above given edge partitions ${E}_{m, n}$ we get,

viii) By using definition 2.7 and above given edge partitions ${E}_{m, n}$ we get,

ix) By using definition 2.8 and above given edge partitions ${E}_{m, n}$ we get,

Theorem2. Let ${G}_{2}$ be the graph of Buslfan, The various Topological indices of ${G}_{2}$ are given as follows.

i) $ABC\left({G}_{2}\right) = 20.77$

ii) $RA\left({G}_{2}\right) = 11.31$

iii) $S\left({G}_{2}\right) = 11.04$

iv) $GA\left({G}_{2}\right) = 22.23$

v) $M1\left({G}_{2}\right) = 148$

vi) $M2\left({G}_{2}\right) = 176$

vii) $F\left({G}_{2}\right) = 526$

viii) $H\left({G}_{2}\right) = 9.45$

ix) $HM\left({G}_{2}\right) = 878.00$

Proof. Let ${G}_{2}$ be the graph of Buslfan with edge set ${E}^{'}$, Let ${{E}^{'}}_{(m, n)}$ represents the class of edges of ${G}_{2}$ joining vertices of degrees $m$ and $n$. With $\left|{{E}^{'}}_{\left(\mathrm{1, 4}\right)}\right| = 18,$ $\left|{{E}^{'}}_{\left(\mathrm{2, 4}\right)}\right| = 3,$ $\left|{{E}^{'}}_{\left(\mathrm{4, 4}\right)}\right| = 5.$

i) By using definition 2.1 and edge partitions ${{E}^{'}}_{(m, n})$ we get,

ii) By using def 2.2 and edge partition ${{E}^{'}}_{(m, n})$ we get,

iii) Definition 2.3 and edge partition ${{E}^{'}}_{(m, n})$ gives

iv) Using def 2.4 and edge partition ${{E}^{'}}_{(m, n})$ we get,

v) Using def 2.5 and edge partition ${{E}^{'}}_{(m, n})$ we get,

vi) Using def 2.5 and edge partition ${{E}^{'}}_{(m, n})$ we get,

vii) Using def 2.6 and edge partition ${{E}^{'}}_{(m, n})$ we get,

viii) Using def 2.7 and edge partition ${{E}^{'}}_{(m, n})$ we get,

ix) By using def 2.8 and edge partition ${{E}^{'}}_{(m, n})$ we get,

One can calculate the topological indices of the remaining drugs by adopting a similar procedure applied in Theorem 1, Theorem 2 and using definitions 2.1 to 2.8. Also, the calculated values of all drugs are listed in Table 2.

Using (1), we have calculated the following diverse linear models for all degree-based topological index, which are given as under:

1. Regression models for atom bond connectivity index ABC (G):

Boiling point = 395.921 + 5.802 [ABC (G)]

Refractive index = 6.034 + 2.468 [ABC (G)]

FP = 197.699 + 3.486 [ABC (G)]

MV = -10.083 + 7.258 [ABC (G)]

Complexity = -107.244 + 18.352 [ABC (G)]

2. Regression models for the atom-bond connectivity index RA (G)]:

Boiling point = 385.002+ 10.833[RA (G)]

Refractive index = 4.535 + 4.408 [RA (G)]

FP = 191.561+ 6.451[RA (G)]

MV = -14.889 + 12.947 [RA (G)]

Complexity = -125.985+ 33.119 [RA (G)]

3. Regression models for atom bond connectivity index S (G):

Boiling point = 384.612+ 10.664 [S (G)]

Refractive Index = 5.102 + 4.293 [S (G)]

FP = 191.365+ 6.348 [S (G)]

MV = -11.972 + 12.540 [S (G)]

Complexity = -125.538 + 32.402 [S (G)]

4. Regression models for atom bond connectivity index GA (G):

Boiling point = 386.718+ 5.013 [GA (G)]

Refractive Index = 6.258 + 2.010 [GA (G)]

FP = 192.642 + 2.985 [GA (G)]

MV = -7.333 + 5.841 [GA (G)]

Complexity = 121.577 + 15.270 [GA (G)]

5. Regression models for atom bond connectivity index M1 (G):

Boiling point = 405.004+.738 [M1 (G)]

Refractive Index = 9.310 +.318[M1 (G)]

FP = 202.599 + 0.448 [M1 (G)]

MV = 1.253 +.932 [M1 (G)]

Complexity = -97.937 + 2.432 [M1 (G)]

6. Regression models for atom bond connectivity index HM (G):

Boiling point = 419.068 +.115 [HM (G)]

Refractive index = 13.368 + 0.052 [HM (G)]

FP = 210.170 + 0.071 [HM (G)]

MV = 13.491 +.151 [HM (G)]

Complexity = -78.956 +.404 [HM (G)]

7. Regression models for atom bond connectivity index M2 (G):

Boiling point = 414.903+.535 [M2 (G)]

Refractive index = 13.461 +.233 [M2 (G)]

FP = 208.190 +.327 [M2 (G)]

MV = 15.185 +.677 [M2 (G)]

Complexity = -87.855 + 1.849 [M2 (G)]

8. Regression models for the atom-bond connectivity index F (G):

Boiling point = 422.835+.200 [F (G)]

Refractive Index = 13.494 +.092 [F (G)]

FP = 212.010 +.124 [F (G)]

MV = 13.736 +.273 [F (G)]

Complexity = -71.482 +.717 [F (G)]

9. Regression models for atom bond connectivity index H (G):

Boiling point = 377.349 + 12.647 [H (G)]

Refractive Index = 4.240 + 4.943 [H (G)]

FP = 187.144+ 7.500 [H (G)]

MV = -14.817 + 14.434 [H (G)]

Complexity = -136.607 + 37.543 [H (G)]

3.2.   Quantitative structure analysis and comparison between topological indices and correlation coefficient of physicochemical properties

The physicochemical properties of 13 blood cancer drugs are presented in Table 2. Their TI values are listed in Table 1 and computed from their molecular structure. The correlation coefficient between five physicochemical properties and TIs is listed in Table 3. The graph between the correlation coefficient of the physicochemical properties of the drug and the topological index is drawn in Figure 2.

3.3.   Calculation of statistical parameters

In this section, QSPR modeling is done to find a relation between physicochemical properties of blood cancer drugs such as medicines azacitidine, buslfan, mercaptopurine, tioguanine, nelarabine, cytarabine, clofarabine, bosutinib, dasatinib, melphala, dexamethasone, doxorubicine, carboplatin and their calculated degree based TIs. whereas TIs, b, r, and N are independent variable, regression model constant, correlation coefficient and sample size respectively. We perceive the correlation coefficient come to be one the experimental and theoretical calculation are close that are marked bold in tables. This type of test can helpful to compare and decide the improvement of model. It is noted the value of r is greater than 0.6 and p value is less than 0.05. Hence, it decides all properties are significant. Tables 4–12 Represent the Statistical parameters used in QSPR models of TIs.

3.4.   Standard error of estimate (SE) and comparison

Measure of variation for an observation calculated around the computed regression line is said to be standard error estimate. It measures the amount of accuracy of predictions made around computed regression line and is mentioned in Table 13. We also compare the physicochemical properties of the experimental and theoretical calculated values of the models and are presented in Tables 14–18.

4.   Conclusions

It is obvious from statistical parameters used in linear QSPR models and topological indices that: ABC (G) index provides high correlated value for molar volume r = 0.953. HM index offers maximum correlated value of complexity i.e. r = 0.954. GA index depicts utmost correlation coefficient of refractive index r = 0.966. Harmonic H (G) provides maximum correlated value of flash point r = 0.772.

In this paper, we have computed topological indices and relate it with linear QSPR model for the drugs used to cure blood cancer. The results obtained in this way will helpful for designing some new drugs to obtain preventive measure for the said disease in pharmaceutical industry. The correlation coefficient has significant contribution to the range of topological indices for these drugs. The results are an eye-opener for the researcher working on drugs science in pharmaceutical industry and provide a way to estimate physicochemical properties for novice discoveries of anticancer medicines to cure other specific cancer disease.

Conflict of interest

We declare no conflict of interest.