Empirical and Statistical Complexity Analysis of AES-128

doi:10.21203/rs.3.rs-1418564/v1

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Empirical and Statistical Complexity Analysis of AES-128

https://doi.org/10.21203/rs.3.rs-1418564/v1

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Analysis of algorithms is an interesting topic in the theoretical computer science area of research. An algorithm analyzed with respect to its run time complexity gives an overall idea about the working of the algorithm in the practical scenario. Empirical complexity is a comparatively newer analysis approach which focuses on running the algorithm practically with varying and increasing input and then predicting the complexity of the algorithm. This is then followed by a statistical analysis of the empirical estimate to give a clearer bound on the empirical complexity.

Cryptographic algorithms are the sole bearer of providing security for any application and hiding confidential information of national or personal importance. In today’s fast running world, besides security, the run time of the algorithm is also of utmost importance. Cryptographic algorithm constitutes of many complex mathematical functions. In this paper, we have analyzed the most sought after cryptographic algorithm AES-128 in terms of its computational complexity and then practically drawn an asymptotical bound through the empirical complexity approach. Then, a regression analysis on the concluded bound is done to formally settle on the empirical complexity of AES-128.

Algorithm Analysis

Empirical Complexity

Statistical Analysis

Cryptography

AES-128

The sole reason behind analyzing an algorithm in terms of its complexity is to make a prior decision about the run time behavior of the algorithm in order to make a clearer picture of the real-time working of the algorithm. Algorithm analysis is done so as to characterize the algorithm in accordance with its suitability for various applications or in a more general perspective in order to compare with another similar algorithm for the same applications. Analyzing an algorithm can help one understand it better and make room for improvement as during the analysis process one tends to replace a more complex part of the algorithm with a simpler version and hence resulting in a better, simpler and compact algorithm. Theoretical computer scientists term the analysis of algorithms as its computational complexity where the main goal is to classify the algorithms according to its complexity and computational problems according to the difficulty inherited by them. Developing an algorithm with minimal asymptotic execution time or with least possible number of operation for formulating any given method are two different approaches incorporated to compute the overall complexity. In general, calculation of the complexity order of an algorithm can be done by taking a count of the least number of operations required to compute it. In a practical scenario, such classifications make no sense as the sole focus is on order-of-growth worst-case behavior of the algorithm. A better and novel approach to compute the real time performance of an algorithm is to compute the empirical computational complexity of the algorithm. The empirical analysis is done by running the program (constituting the algorithm) on a computer and measures its run time complexity for inputs ranging in several magnitudes and henceforth, fitting the observed performance into a statistical model that predicts performance as a function of input size.

The two main factors that differentiate one cryptographic algorithm from others are its capability to secure the confidential data and the time and efficiency with which it does so. One of the most sought after algorithm for securing information is Advanced Encryption Standards (AES). AES follows a symmetric key structure which means, for the processes of encryption and decryption, a single key is required. In the following sections, we have given a brief description of AES-128 and analyzed it in terms of both its computational complexity and empirical computational complexity. Finally, we have tried to fit a statistical model that predicts its performance in terms of varying inputs.

Empirical and statistical complexity of an algorithm is quite a novel idea and as such not much attention has been given to this field in terms of research. [1] is a detailed book on measuring the computational complexity of an algorithm. Through the method of linear regression, [2] suggests some ways of measuring the complexity of an algorithm statistically. Two compounding operations, i.e. operations working in a group are compared in [3] to check their statistical similarity or dissimilarity and [4] is a novel approach proposed in which the order of complexity of two dense nxn matrix multiplication proved to be O (n²). For some statistical concepts such as regression analysis, probability distribution, correlation etc. readers are referred to [5], [6], [7] and [8]. [9] is a detailed explanation on the empirical computational complexity of an algorithm. For approximating the empirical complexity through a computer based approach readers are suggested to go through [10].

All about Advanced Encryption Standards (AES) can be found in the standard paper from National Institute of Standards and Technology (NIST) in [11] and the proposal sent to NIST in [12].

Cryptographic algorithms can be ordered into two unique classes dependent on the key they use for encryption and decryption. The class of cryptographic algorithms that utilizes two distinct keys for both encryption and decoding are known as asymmetric key algorithms. RSA is a case of such a class. The cryptographic algorithms that depend on utilizing a similar key for both the processes are known as symmetric key algorithms. Data Encryption Standards (DES), Blowfish, Advanced Encryption Standards are a few instances of this class. AES works by encoding a block of size 128 bits and has a key length of 128 bits, 192 bits, and 256 bits so giving the AES principles the name as AES-128, AES-192 and AES-256 with every standard utilizing a series of 10, 12 and 14 separately. In AES, the whole information is handled in equal during each round. At present AES-128 is most commonly in use. So, the description of AES is based on the implementation of this particular standard.

Three most important criteria based on which the AES standards were formulated are:

The algorithm should resist all known attacks.
The algorithm should be designed in a simple way.
The algorithm should be effective in terms of speed and code compactness on different platforms.

The general working progression of AES is shown in Fig. 1 underneath. A solitary 128-piece block is taken care of as contribution for both encryption and decryption and is known as the in matrix. This block is duplicated into a state array and afterward replicated to a output matrix. The state array is changed at each phase as the calculation advances. The key is then ventured into a variety of the key array of words which is known as the w matrix.

3.1. Inner Working of Each Round in AES-128

AES starts with an AddRoundKey stage followed by four sub-stages for the first 9 rounds and three sub-stages for the tenth round. This process is the same for both encryptions is decryption with the only difference being that in decryption each of the sub-stages is inversed. The four stages of encryption are: Substitute Bytes, Shift Rows, Mix Columns, Add Round Key

The tenth round is left to work only with three sub-stages and hence the Mix column step is not formulated. Similarly, the inverse all these steps is used in decryption.

The four steps of the encryption process are briefed herein.

Substitute Bytes

This stage ordinarily known as SubBytes is basically a table query from a s-box which is a 16X16 network of bytes. The originator of AES concentrated on planning the s-box which is impervious to all conceivable cryptanalytic attacks. Thus, a low relationship between's the info bits and the output bits was of most extreme significance. This was made conceivable by limiting both the relationship between's straight changes of information/output bits and contrast proliferation likelihood simultaneously. The reverse SubByte utilizes a converse s-box.

Shift Rows Transformation

This stage known as ShiftRows is a basic change where the first row of the state is left unaltered. A roundabout move by 1 byte to one side is done in the subsequent row followed by a round move by 2 bytes and 3 bytes to one side for the third and the fourth row individually. The backward ShiftRow activity is completed along with these rows by moving every one of the last three columns the other way.

Mix Columns Transformation

This progression known as MixColumn is fundamentally a replacement that utilizes arithmetic of the Galois Field regularly known as GF (2⁸). The state containing the matrix works on the singular columns of the matrix independently. Every byte in a column is mapped into another column which is an element of each of the four bytes in the matrix. Every component of the resultant product matrix is the total of results of components of one row and one column where the individual augmentations and duplications are acted in GF (2⁸).

Add Round Key Transformation

In this step, the bitwise xor activity is performed on the 128 bits of state and 128 bits of the round key.

3.2. Modes of Operations

Block ciphers mode of operations allows encrypt/decrypt of messages larger than the length of the block. Two of the modes namely Cipher Block Chaining (CBC) and Counter (CTR) are explained below

Cipher Block Chaining (CBC)

One of the most sought after characteristic of any encryption algorithm is that it should be capable of producing a different cipher text each time the same plain text is encrypted. The CBC mode of operation allows this. The first round is XORed using an Initialization Vector (IV). The same key is used for each block. The property of XOR operation is used herein which supports getting back the decrypted message by XORing the cipher texts. The CBC mode of operation for the process of encryption is shown in Fig. 2 below

Counter (CTR): In the CTR method of activity, a counter equivalent to the length of the plain content block size is utilized. The counter worth must be distinctive for each plain text block being encoded. During encryption, the counter is encrypted and afterward XORed with the plain content block to get the cipher text. For decoding, a similar counter an incentive for each block is XORed with the cipher text to get back the message in its original structure.

Computational Complexity of AES-128

The block cipher algorithms work normally on fixed block size. AES-128 works by encrypting a fixed block of length 128 bits and takes approximately the same time despite varying input. Thus AES-128 can be termed to be O (1) for both encryption and decryption. If one wants to encrypt messages larger than 128 bits then they need to put them into one of the modes of operations. In such case, as we have O (n) blocks of data to encrypt/decrypt, this brings the order of complexity to O (n), where n is considered to be the size of the message to be encrypted/ decrypted. So, the time complexity of AES-128 for both encryption and decryption in any of the standard modes of operations like CBC or CTR is polynomial or to be more precise is linear with respect to the size of the message. This can be justified in the following paragraph.

This concludes that AES-128 with any standard modes of operation has a run time complexity of O (n), where n is the length of the message.

Empirical complexity is considered to be a real time complexity of an algorithm an algorithm and is generally represented by Empirical O. It is the experimental gauge of the non-insignificant and applied weight-based factual bound. This work is mainly an extension of our previous work published in Soft Computing [9]

First running a program constituting the algorithm and then fitting a statistical model gives a practical view of the algorithm’s complexity in empirical form. We implemented AES-128 with CBC mode of operation in MATLAB version 2015a in a system with the following specifications:

Processor: Intel (R) Core ™ i3

RAM: 4.00 GB

Operating System: 64 bit Windows 7

The run time of the algorithm was noted with increasing the input. Several trials for both encryption and decryption of same message length but with a different message, each time was taken and a mean of all the trials was the execution time of the algorithm for both the processes. Taking of several trials is justified because the empirical run time of an algorithm can be affected due to the cache memory. Same message run again and again can be executed from the cache itself and hence may give a vague measure of the run time. Table 1 and Table 2 below gives the execution time.

Table 1

Runtime for Encryption Process of AES-128
Messages as Input	Trail 1	Trial 2	Trial 3	Trial 4	Trial 5	Mean of Execution Time
16	0.01520	0.0276	0.0222	0.0262	0.0226	0.0228
32	0.04899	0.0589	0.0542	0.0550	0.0541	0.0542
48	0.08279	0.0855	0.0842	0.0863	0.0841	0.0846
64	0.11658	0.1361	0.1141	0.1135	0.1171	0.1195
80	0.15037	0.1564	0.1359	0.1480	0.1401	0.1462
96	0.18416	0.1705	0.1768	0.1595	0.1780	0.1738
112	0.21795	0.1889	0.2047	0.2095	0.1948	0.2032
128	0.25175	0.2496	0.2663	0.2276	0.2340	0.2459
144	0.28554	0.2778	0.2910	0.2494	0.2525	0.2712
160	0.31933	0.3353	0.2939	0.3011	0.4244	0.3348
176	0.35312	0.3306	0.3232	0.3071	0.3238	0.3276
192	0.38691	0.4291	0.3590	0.3322	0.4942	0.4003
208	0.42071	0.4242	0.3949	0.4776	0.4619	0.4359
224	0.45450	0.3783	0.5521	0.4052	0.5823	0.4745
240	0.48829	0.5632	0.4454	0.4185	0.6181	0.5067
256	0.52208	0.4790	0.5316	0.5218	0.4742	0.5057

Table 2

Runtime for Decryption Process of AES-128
Messages as Input	1st Trail	2nd Trail	3rd Trial	4th Trial	5th Trail	Mean of Execution Time
16	0.17313	0.2363	0.2067	0.2047	0.1774	0.1996
32	0.33777	0.3604	0.3783	0.3468	0.3831	0.3613
48	0.50241	0.4806	0.4332	0.4955	0.4752	0.4774
64	0.66705	0.7515	0.6874	0.6745	0.7159	0.6993
80	0.83169	0.7712	0.7419	0.8467	0.7231	0.7829
96	0.99633	0.8648	0.9904	0.8522	0.9655	0.9338
112	1.16097	1.0421	1.0794	1.1493	1.1003	1.1064
128	1.32561	1.3728	1.4374	1.3054	1.1962	1.3275
144	1.49025	1.8960	1.7968	1.3858	1.3142	1.5766
160	1.65489	1.9702	1.5415	1.6108	1.8512	1.7257
176	1.81953	1.6213	1.5834	1.5682	1.7367	1.6658
192	1.98417	2.4075	1.7271	1.7448	2.3954	2.0518
208	2.14881	2.0665	1.8660	2.1969	2.6174	2.1791
224	2.31345	1.8920	2.1872	1.9555	2.5175	2.1731
240	2.47809	2.6679	2.1026	2.0659	2.7080	2.4045
256	2.64273	2.7228	3.1249	2.4156	2.2378	2.6288

Let x be the length of the message to be encrypted/decrypted and y be the execution time in seconds. Then we have to determine a function f of the form y = f (x). The estimated run time empirically can be found by using the Fundamental Theorem of Finite Difference and more precisely the inverse of it which states that if the ${n}^{th}$ difference of a tabulated function is constant with higher differences being almost zero, then the function is a ${n}^{th}$ degree of the polynomial

Fundamental Theorem of Finite Difference

$${\varDelta }^{n}{P}_{n}\left(x\right)=C=Constant$$

The ${n}^{th}$ difference of a ${n}^{th}$ degree of the polynomial is Constant.

$${\varDelta }^{n+r}{P}_{n}\left(x\right)=0, r=\text{1,2},3,\dots \dots$$

Higher differences of a ${n}^{th}$degree of the polynomial is zero.

Its converse is also true. No function has this unique property except a polynomial.

For the difference Table 3, we found that the first difference is almost constant. Hence a 1st -degree polynomial will approximate the execution time for both the processes. So, the empirical complexity of AES-128 for encryption/decryption can be termed to be empirical O (n).

Now we shall fit a statistical model to the above formulated empirical complexity.

Table 3

Empirical Estimate of Encryption and Decryption Time
Encryption (Estimation)			Decryption (Estimation)
x	y	Δy	X	y	Δy
16	0.0228	0.030 >	16	0.1996	0.16 >
32	0.0542	0.030 >	32	0.3613	0.16 >
48	0.0846	0.030 >	48	0.4774	0.11 >
64	0.1195	0.030 >	64	0.6993	0.11 >
80	0.1462	0.034 >	80	0.7829	0.22 >
96	0.1738	0.034 >	96	0.9338	0.22 >
112	0.2032	0.026 >	112	1.1064	0.08 >
128	0.2459	0.026 >	128	1.3275	0.08 >
144	0.2712	0.027 >	144	1.5766	0.15 >
160	0.3348	0.027 >	160	1.7257	0.15 >
176	0.3276	0.029 >	176	1.6658	0.17 >
192	0.4003	0.029 >	192	2.0518	0.17 >
208	0.4359	0.042 >	208	2.1791	0.22 >
224	0.4745	0.042 >	224	2.1731	0.22 >
240	0.5067	0.025 >	240	2.4045	0.24 >
256	0.5057	0.025 >	256	2.6288	0.24 >

The empirical approach deals with specific data set as input in hand. One may be interested in the whole universe (population) from where the data comes. So, adopting a linear regression approach which deals with overfitting of data can be used to deal with the whole population of input. Use of nonlinear regression model is not suggested in this case as the run time of AES-128 depends only on single independent variable i.e. input in terms of the length of the message which in this case is linear. Even though we square a predictor attached to the independent variable, the final function used for fitting the model will be linear. Using the principle of least squares, we will fit a first degree polynomial of the form y = a + bx’.

Where the variables ‘a’ and ‘b’ are to be estimated using the following two equations:

$\sum y$ = na + b $\sum x\text{'}$

$\sum x\text{'}y$ = a $\sum x\text{'}$ + b $\sum {x\text{'}}^{2}$ (1)

These equations are obtained by setting

$\frac{\delta s}{\delta a}$ = 0 and $\frac{\delta s}{\delta b}$ = 0

Where S = $\sum ({y-\widehat{y)}}^{2}$ is the residual sum of squares which estimates:

$\widehat{y}$ = $\widehat{a}$ + $\widehat{b}$ x’

On solving Eq. 1 we obtain

$\widehat{a}$ = -0.01859 and $\widehat{b}$ = 0.002112 (for encryption) and

$\widehat{a}$ = 0.00849 and $\widehat{b}$ = 0.01029 (for decryption)

So we have

$\widehat{y}$ = -0.01859 + 0.002112 x’ (for encryption) and

$\widehat{y}$ = 0.00849 + 0.01029 x’ (for decryption) (2)

Table 4 below shows the residual table. The residual table is constructed in order to check the goodness of fit of the model.

Table 4

Residual Table
Encryption (Residuals)				Decryption (Residuals)
x	y	$\widehat{y}$	y - $\widehat{y}$ (Residuals)	x	y	$\widehat{y}$	y - $\widehat{y}$ (Residuals)
16	0.02912	0.015202	0.013918	16	0.23594	0.17313	0.06281
32	0.05924	0.048994	0.010246	32	0.38242	0.33777	0.04465
48	0.08768	0.082786	0.004894	48	0.48154	0.50241	-0.02087
64	0.12452	0.116578	0.007942	64	0.74878	0.66705	0.08173
80	0.14292	0.15037	-0.007450	80	0.77696	0.83169	-0.05473
96	0.17168	0.184162	-0.01248	96	0.93154	0.99633	-0.06479
112	0.20656	0.217954	-0.01139	112	1.10732	1.16097	-0.05365
128	0.2421	0.251746	-0.00965	128	1.32954	1.32561	0.00393
144	0.26446	0.285538	-0.02108	144	1.5378	1.49025	0.04755
160	0.32652	0.31933	0.00719	160	1.67366	1.65489	0.01877
176	0.32386	0.353122	-0.02926	176	1.62256	1.81953	-0.19697
192	0.39026	0.386914	0.003346	192	1.98286	1.98417	-0.00131
208	0.44104	0.420706	0.020334	208	2.15716	2.14881	0.00835
224	0.47304	0.454498	0.018542	224	2.2934	2.31345	-0.02005
240	0.50372	0.48829	0.01543	240	2.50796	2.47809	0.02987
256	0.5117	0.522082	-0.01038	256	2.76656	2.64273	0.12383

The goodness of fit of the fitted model can be tested using a residual plot method. The residuals are plotted against the corresponding estimates and if an almost horizontal pattern is observed then, we can say that the model predicted is actual empirical complexity of the run time. The fitted line plot and the residual plot for the encryption process are shown below in Fig. 3 and Fig. 4 respectively. These were plotted using a statistical tool ‘MINITAB’.

The fitted line plots in Fig. 3 show the regression result graphically. It can be seen that the fitted line plot shows a linear line which tells that the statistical analysis approach gives a linear dependence of execution time on input for the encryption process.

The residual plot in Fig. 4 has four sub-plots which are explained below:

Normal Probability Plot

The normal probability plot is a graphical technique for addressing the distribution of data i.e. whether the data is distributed normally or not. The points of the data should form an approximately straight line for the data to be considered as normally distributed. This is the case in our plot and so the data seems to be coming from a normal distribution.

Versus Fit

Residual versus fits plot is a scatter plot with y-axis containing the residuals and the x-axis the corresponding estimates (fitted value). Our versus fit plot suggests that there is no unusual data point in the data set. It also suggests that the residuals and the fitted values are correlated which means that the fitted model which we predicted in our case to be linear is actually the case as the residuals are cantered on zero throughout the range of the fitted values.

Histogram

Histograms of the residuals suggest whether the data are skewed or include outliers. If we get one long tail in one direction this means that the residuals are skewed and if one bar is far away from others, it means the residuals include outliers. These both conditions are not seen in our Histogram.

Versus Order

This plot displays the order of collection of residual data. This is used to investigate whether the residuals are independent of each other or not. In an ideal scenario, the residuals on the plot should be distributed randomly around the normal line. This is what we observe in our encryption residual plot.

From the above discussion, it can be concluded that the empirical estimate of AES-128 for encryption which was to be O (n), is what we prove statistically. The fitted line plot of Fig. 4 shows the execution time in the y-axis and number of input in the x-axis which is linear.

Similarly, we can say for the process of decryption that, the empirical run time is O (n).

Theoretical computational complexity may give a vague asymptotic approximation of an algorithm as sometimes it is unwise to consider one specific operation as dominant in case of compound operations. A better approach is to run the algorithm practically and calculate the complexity with respect to input by gradually increasing the size of the input. This methodology known as the empirical analysis is although system-dependent but is more robust. Then, fitting of a statistical model to prove what was estimated empirically is true; a model fitting which defines the equation of the empirical complexity of the algorithm can be done.

We have analyzed AES-128 through all three approaches. Although the bound that was imposed theoretically on the execution time of AES-128 i.e. O(n) with any standard modes of operations is what we achieve empirically and prove statistically, the analysis done gives a robust execution complexity of AES-128. The defined Eqs. 2 and 3 through the statistical approach can be used to predict the run time of the algorithm with input size of any magnitude as this is proved statistically that the empirical complexity of AES-128 is O (n).

In the future, other cryptographic algorithms in use such as Blowfish, RSA, and other growing trends in the field will be analyzed and a comparative study of different cryptographic algorithms will be done in order to draw a line between their effectiveness in terms of employment for real-time applications and security purposes.

Ethical approval: The authors declare that the manuscript complies with the ethical standards of the journal.

Funding Details: The authors declare that they have not received any funding for this research.

Conflict of Interest: The authors declare that they not have any conflict of interest.

Informed Consent: The authors gives right to the journal for all the process related to the handling of the manuscript.

Author Contributions: All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Prashant Pranav, Sandip Dutta and Soubhik Chakraborty. The first draft of the manuscript was written by Prashant Pranav and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Competing Interest: The authors have no relevant financial or non-financial interests to disclose.

Data Availability: Data may be made available upon request

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Empirical and Statistical Complexity Analysis of AES-128

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Abstract

Figures

1. Introduction

2. Related Work

3. Advanced Encryption Standards

3.1. Inner Working of Each Round in AES-128

3.2. Modes of Operations

4. Empirical Computational Complexity Of Aes-128

5. Fitting Of A Statistical Model To Empirical Result For Aes-128

6. Testing The Statistical Model

7. Conclusion And Future Scope

Declarations

References

Status:

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Encryption (Residuals)				Decryption (Residuals)
x	y	\(\widehat{y}\)	y - \(\widehat{y}\) (Residuals)	x	y	\(\widehat{y}\)	y - \(\widehat{y}\) (Residuals)
16	0.02912	0.015202	0.013918	16	0.23594	0.17313	0.06281
32	0.05924	0.048994	0.010246	32	0.38242	0.33777	0.04465
48	0.08768	0.082786	0.004894	48	0.48154	0.50241	-0.02087
64	0.12452	0.116578	0.007942	64	0.74878	0.66705	0.08173
80	0.14292	0.15037	-0.007450	80	0.77696	0.83169	-0.05473
96	0.17168	0.184162	-0.01248	96	0.93154	0.99633	-0.06479
112	0.20656	0.217954	-0.01139	112	1.10732	1.16097	-0.05365
128	0.2421	0.251746	-0.00965	128	1.32954	1.32561	0.00393
144	0.26446	0.285538	-0.02108	144	1.5378	1.49025	0.04755
160	0.32652	0.31933	0.00719	160	1.67366	1.65489	0.01877
176	0.32386	0.353122	-0.02926	176	1.62256	1.81953	-0.19697
192	0.39026	0.386914	0.003346	192	1.98286	1.98417	-0.00131
208	0.44104	0.420706	0.020334	208	2.15716	2.14881	0.00835
224	0.47304	0.454498	0.018542	224	2.2934	2.31345	-0.02005
240	0.50372	0.48829	0.01543	240	2.50796	2.47809	0.02987
256	0.5117	0.522082	-0.01038	256	2.76656	2.64273	0.12383