International Journal of Network Security, Vol.21, No.2, PP.250-258, Mar. 2019 (DOI: 10.6633/IJNS.201903_21(2).09) 250
Construction and Analysis of Key Generation Algorithms Based on Modi ed Fibonacci and Scrambling Factors for Privacy Preservation
Amiruddin Amiruddin1,2, Anak Agung Putri Ratna1, and Riri Fitri Sari1 (Corresponding author: Amiruddin Amiruddin)
Department of Electrical Engineering, Universitas Indonesia1 Jl. Margonda Raya, Pondok Cina, Beji, Kota Depok, Jawa Barat 16424, Indonesia Sekolah Tinggi Sandi Negara, Bogor, Jawa Barat, Indonesia2 (Email: [email protected]) (Received Sept. 24, 2017; revised and accepted Apr. 12, 2018)
Abstract protect from interception and to ensure the data con den- tiality, many wireless systems use cryptographic systems Cryptographic key is the most important factor for sup- with secret keys that are only available to the legitimate porting encryption of con dential data before it is trans- senders and recipients. mitted in a communication network. A good crypto- Various methods or approaches have been proposed to graphic key has properties of random sequence and long generate long and random encryption keys [36]. Each period. For these purposes, a randomness capable and method or approach has advantages and disadvantages lightweight computing algorithm is required. The ran- and cannot be applied to all dierent kinds of applica- domness capability and computation time of such an algo- tions. Therefore, a key generation function should be rithm can be measured by using randomness test and al- tailored and adjusted to the characteristics of the appli- gorithmic complexity analysis, respectively. In this paper, cations that will use it. One important consideration in two models of key generation algorithm using the mod- designing a key generation algorithm is its algorithmic i ed Fibonacci and scrambling factor were constructed. complexity [24]. For applications in low-capacity devices Such modi cation and scrambling factor are intended to for IoT, low complexity algorithms are required. Unfor- support the randomness capability and low algorithmic tunately, the existing key generation algorithms lack the complexity. The proposed key generation algorithms have measurement of their complexity. been simulated and analyzed. The key generation algo- Fibonacci sequence [10] which is a very famous series rithm Model 2 (called hereinafter "Scrambled Fibonacci- function in the eld of mathematics can be used to gener- based") is better than Model 1 in term of randomness, ate encryption keys. It is a sequence of numbers where a despite both having similar linear algorithmic complex- number is found by adding up the two preceding numbers. ity, denoted by . O(n) Beginning with 0 and 1, the sequence goes as 0, 1, 1, 2, Keywords: Cryptography; Key Generation; Randomness; 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the Scrambled Fibonacci; Scrambling Factor expression is xi = xi−1 + xi−2. Its lightweight operation and ability to save computing time are of the reasons for 1 Introduction using it in the key generation function. Several crypto- graphic methods used Fibonacci sequence or its behavior Wireless communication system and its services have be- for encryption application [9, 12]. Applying Fibonacci is come an important component of modern life and society. suitable for common areas that do not involve data pri- An example of such a wireless network is the Internet vacy. However, for con dential data-based applications, of Things (IoT) that grows rapidly, nowadays, to sup- it is necessary to make improvement on the Fibonacci port human beings need on information. However, due to function. Moreover, the operation used in Fibonacci pro- the nature of the Radio Frequency (RF) spectrum used duces a regular pattern (ascending or descending) that as shared transmission medium, wireless communications can be used as an entrance point to analyze the resulted are essentially vulnerable and prone to interception [5]. key sequence. Therefore, it is necessary to modify the The next generation of wireless communication systems Fibonacci operation so that the resulted pattern becomes should support applications with very low communication random and hardens the eorts to analyze it. latency, availability, high reliability and security [27]. To In this paper, a key generation algorithm based on the International Journal of Network Security, Vol.21, No.2, PP.250-258, Mar. 2019 (DOI: 10.6633/IJNS.201903_21(2).09) 251 modi ed Fibonacci and scrambling factor is constructed a two phase approach to achieve secret key. In the rst and analysed. The key generation function will be ap- step, the sender estimates his state and sends this along plied to the Internet of Things network with constrained with other information which is obtained from his obser- devices that have limited storage, low computing power vation to the recipient. In the second step, the recipient and energy. The contribution of this work is a construc- uses this information including the estimated state of the tion and analysis of key generation algorithm based on sender to generate the secret key. However, the protocol Fibonacci and scrambling factor which satis es the re- has not been reported whether it has been implemented quirement for random and long period of key sequences. or not. The remaining of the paper is organized as follows. Sec- Al-Moliki et al. [2] enhanced the con dentiality of Vis- tion 2 gives a brief overview of the previous related works ible Light Communication (VLC) networks by suggest- regarding cryptographic key generation methods. Section ing a new key generation protocol for optical Orthog- 3 describes the key generation de nition and its perfor- onal Frequency Division Multiplexing (OFDM) schemes mance measurement. The proposed method is described in an indoor environment. The keys are extracted from in Section 4. Section 5 discusses the simulation and the the bipolar OFDM samples produced from optical OFDM result and Section 6 closes the paper with the conclusion. schemes. This approach which emphasizes the source of key generation diers from our proposed approach which emphasizes the process of the key generation. 2 Related Works Karimian et al. [14] proposed a novel approach of key generation that extracts keys from real-valued ECG fea- As technology grows, research on cryptographic key man- tures. However, this approach is only suitable for ECG- agement [17,20,22,25,32] continues to be done in various based applications although it can also be modi ed for aspects including key generation [14, 18, 28, 29, 34, 35, 37], other eld applications. agreement [3, 6, 7, 13, 15, 33] or exchange [4], distribu- In this research, we proposed new key generation algo- tion [8], assignment [19], authentication [16], and update. rithm based on modi ed Fibonacci and scrambling factor However, we focus on eorts for key generation to be used to support long periodicity and randomness of the gener- on symmetric cryptosystems. Verma et al. [31] proposed a ated key sequence. The research position of our proposed method for generating cryptographic key using biometrics key generation algorithm among other algorithms is sum- with the help of ngerprint pattern. The algorithm gen- marized in Table 1. erates key by extracting minutiae points and core point and the nal key is obtained from the ngerprint image. Turakulovich et al. Turakulovich et al. [30] discussed com- 3 Key Generation and Perfor- parative factors of the key generation techniques include mance Measurement randomness, key space, key space of biometric, entropy, measured entropy of biometric, convenience and cost, se- 3.1 Key Generation cure saving, update. However, they mostly discussed the factor for biometric-based key generation technique which In the eld of cryptography, key is the most urgent pa- is dierent from our proposed method. rameter for data encryption or decryption. By de nition, Hossain et al. [11] proposed a One Time Key (OTK) a key is a sequence of a random string of bits created ex- generation technique based on User ID (UID) and pass- plicitly for scrambling and unscrambling data. Instances word. The key generation method involves a server-side of cryptographic processes demanding the usage of keys process to check for possible collisions between a new UID include, inter alia, the transformation of plain text data and a given UID to the previous client. This process takes into cipher text data (encryption) and vice versa (decryp- a long time so it is not applicable for devices with limited tion), the computation and veri cation of a digital signa- storage and energy. Torre et al. [29] studied the practical ture, the computation and veri cation of an authentica- performance of an enhanced channel-based key generation tion code from data, the computation of a shared secret system with a very short roundtrip delay, allowing recip- that is used to obtain keying material, and the derivation rocal channel assessment with increased accuracy. The re- of additional keying material from a key-derivation key. ciprocal channel measurements performed by body-worn There are two types of key, i.e. symmetric and asym- sensor nodes is used to extract encryption keys. Although metric key. Symmetric key is a key used in a symmetric- the performance is slightly increased due to the shorter key cryptographic algorithm which requires that the key round-trip delay, further apparent non-reciprocity in the must be kept secret. Asymmetric key is a key used with channel measurements can probably be attributed to in- a public-key algorithm. In asymmetric key cryptography, accuracy of the received signal strength indication in the there are two corresponding keys, i.e. private and pub- transceiver chip. lic keys. A private key is a cryptographic key used with Tavangaran et al. [27] studied the secret key generation a public-key algorithm that must be kept secret and is protocol for a compound Discrete Memoryless Multiple uniquely associated with an entity that is authorized to Sources (DMMS) with one-way communication in pres- use it. Public key is a key used with a public-key algo- ence of an eavesdropper. The key generation protocol uses rithm that may be made public and is associated with a International Journal of Network Security, Vol.21, No.2, PP.250-258, Mar. 2019 (DOI: 10.6633/IJNS.201903_21(2).09) 252
Table 1: Research position of key generation Author Method Source Implementation Verma et al., 2016 extraction Biometric with nger-print pattern Simulated on Matlab Hossain et al., 2016 generation UID and password Simulated on mobile phone Torre et al., 2017 extraction Reciprocal channel measurements Not reported Tanvangaran et al., 2017 generation Compound: information, estimated state Not yet Al-Moliki et al., 2017 extraction Bipolar OFDM samples Simulated on Monte Carlo Karimian et al., 2017 extraction ECG value - Amiruddin et al., 2017 generation User input Simulated on Matlab
private key and an entity that is authorized to use that Algorithm 1 Key Generation (Model 1) private key. 1: Begin Key generation is the process of generating keys for 2: Initialize the parameters: a, b, n, c cryptographic purpose such as encryption [1,21]. A cryp- 3: Derive the 1st element of the key tographic key can be generated through a function in soft- 4: K(1) ← mod(a, c) ware or hardware with input parameters that can be ob- 5: Derive the 2nd element of the key tained from various sources, e.g. channels used by each 6: K(2) ← mod(b, c) pair of users [28], phase uctuations in ber links, and 7: Derive the 3rd to n-th element of the key values entered by the user. 8: for i = 3 to n do 9: K(i) ← mod(K(i − 1) + K(i − 2), c) end for 3.2 Measurement 10: 11: Output the key sequence, K To measure the performance of the proposed key genera- 12: End tion algorithm, several tests were used, i.e. key generation speed, key randomness, key periodicity, and algorithmic The simulation result showed that this model pro- complexity analysis. The key generation speed was mea- duces key sequences that have a better randomness level sured by recording the start time and nish time of the than those produced by the original Fibonacci. However, key generation process and then subtracting the start time the randomness of the key sequences is not yet satis ed from the nish time. The key randomness was measured the randomness test. Therefore, we constructed another by using autocorrelation function, while the key period- model of key generation algorithm, Model 2, by adding icity was manually analyzed. All of these kinds of key a scramble factor to the previous model, Model 1. This performance measurement were also used in [26]. The addition of scrambling factor was expected to make the analysis of algorithmic complexity was measured by fol- generated key sequences more random to satisfy the ran- lowing the description and calculation example presented domness test. in [24]. 4.2 Model 2 4 Proposed Methods In Model 2, Fibonacci is modi ed to generate random and long period key sequences. Input parameters for the func- We have constructed two models of new key generation tion are a, b, n and d as the rst number, second number, algorithm based on the modi ed Fibonacci, described in key length, and modulus number, respectively. Through detail as follows. this model, the rst key,K1, is generated from the formula (a∗b−a)%d. K2 is generated similarly from (a∗b−b)%d. 4.1 Model 1 For i = 3 : n, Ki is obtained from the addition of the two previous key, Ki−1 + Ki−2, and a scrambling factor, 3 ∗ i In Model 1, modi cation made on Fibonacci series is the in formula of Ki−1 + Ki−2 + 3 ∗ i%d. Actually, the origi- application of modulo number (annotated with c). In nal Fibonacci is presented in the addition marked with a this model, the rst key, K1, is obtained by the result of + symbol. However, such an addition can cause the re- a%c. K2 is obtained by the result of b%c. Ki is obtained sulted key sequences to have a low randomness and easy by the result of the addition of (Ki−1 + Ki−2)%c. By to be guessed (as described in Model 1). Therefore, in applying a modulus number in this model, the ascending Model 2, a scrambling factor using a multiplication, 3 ∗ i, or descending pattern of the generated key sequences is is added to improve the modi ed Fibonacci in generat- reduced in periodicity. The key generation function of ing random key sequences. The scrambling factor with Model 1 is given in pseudocode form in Algorithm 1. the use of multiplication (*) involving an ever-changing International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx. 20xx used with a public-key algorithm. In asymmetric key The simulation result showed that this model produces key cryptography, there are two corresponding keys, i.e. private sequences that have a better randomness level than those and public keys. A private key is a cryptographic key used produced by the original Fibonacci. However, the randomness of the key sequences is not yet satisfied the with a public-key algorithm that must be kept secret and is randomness test. Therefore, we constructed another model uniquely associated with an entity that is authorized to use of key generation algorithm, Model 2, by adding a it. Public key is a key used with a public-key algorithm that scramble factor to the previous model, Model 1. This may be made public and is associated with a private key addition of scrambling factor was expected to make the International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx. 20xx and an entity that is authorized to use that private key. generated key sequences more random to satisfy the random. Due to the unsatisfactory result of randomness of Key generation is the process of generating keys for randomness test. Algorithm 2: Key Generation (Model 2) proposed Model 1 algorithm, we then proposed Model 2 cryptographic purpose. A cryptographic key can be 4.2 Model 2 1 //Input parameters: a, b, n, d; algorithm. generated through a function in software or hardware with 2 //Derive the 1st element of the key In Model 2, Fibonacci is modified to generate random and 3 K(1)ß mod(a*b-a,d); input parameters that can be obtained from various sources, nd long period key sequences. Input parameters for the 4 //Derive the 2 element of the key 5.2 Model 2 e.g. channels used by each pair of users [21], phase 5 K(2)ß mod(a*b-b,d); function are (, *, . and / as the first number, second rd th fluctuations in fiber links, and values entered by the user. 6 //Derive the 3 to n element of the key The key generation simulation used the Algorithm number, key length, and modulus number, respectively. 7 For i from 3 to n 2 by varying the parameters i.e. the first number ((), ß Through this model, the first key, &', is generated from the 8 K(i) mod(K(i-1)+K(i-2)+3*i,d); 3.2 Measurement 9 Next i second number (*), and the key length (key size) (.) and formula (( ∗ * − () % /. & is generated similarly from ) 10 //Output the key sequence, K let the modulus number (/) be remains in 256, as the To measure the performance of the proposed key (( ∗ * − *) % /. For 4 = 3: #, &+ is obtained from the maximum value of alphabet characters. In this simulation, addition of the two previous key, &(i-1)+&(i-2, and a generation algorithm, several tests were used, i.e. key Table 2 Different operations among Fibonacci-based algorithms we measured the key generation speed and randomness generation speed, key randomness, key periodicity, and scrambling factor, 3 ∗ 4 in formula of &(i-1)+&(i-2)+3 ∗ 4 % level, and compared with other algorithm. /. Actually, the original Fibonacci is presented in the Original Modified Fibonacci Modified Fibonacci algorithmic complexity analysis. 5.3 Key generation speed The key generation speed was measured by addition marked with a + symbol. However, such an Fibonacci Model 1 Model 2 addition can cause the resulted key sequences to have a low For measuring the speed of key generation, 1000 recording the start time and finish time of the key : randomness and easy to be guessed (as described in Model : = ( : = ;4#4?ℎ A4BC − ?ADEA A4BC), then using autocorrelation function, while the key periodicity : = * : = ;
By applying a modulus number in this model, the µ 200,00 ascending or descending pattern of the generated key sequences is reduced in periodicity. The key generation 150,00 function of Model 1 is given in pseudocode form in 100,00 Algorithm 1. 50,00 Algorithm 1: Key Generation (Model 1) 0,00 1 //Input parameters: a, b, n, c; Processing Time ( 10 20 30 40 50 60 2 //Derive the 1st element of the key Key Size (Byte) 3 K(1)ß mod(a,d); 4 //Derive the 2nd element of the key 5 K(2)ß mod(b,c); ADD MULTIPLY POWER rd th 6 //Derive the 3 to n element of the key 7 For i from 3 to n 8 K(i)ß mod(K(i-1)+K(i-2), c); Fig. 1 Processing time comparison of several operations Figure 1: Processing time comparison of several opera- 9 Next i Fig. 2 Key element plot of three key sequences with different Fig. 3 Comparison of key generation processing time (ms) for 10 //Output the key sequence, K tions The proposed generation algorithm of Model 2 is Figureinitial 2: value Key of element parameter plot a of of proposed three keyalgorithm sequences Model 1 with varying key length (byte) given in pseudocode form in Algorithm 2. dierent initial value of parameter a of proposed algo- rithm ModelThe performance1 of the proposed key generation algorithms Model 1 and Model 2 were evaluated by number, i, produces a non-patterned or random number. simulation using Matlab software. The key generation The scrambling factor with the use of power () oper- algorithm of Model 1 does not meet the randomness test as ation can also generate random numbers, but has a high 5in Fig.1, Results indicated and by similar Discussion pattern of the three generated computational overhead. The scrambling factor of the ad- key sequences. By varying the value of parameter a with dition (+) and subtraction (−) produces a number that is 5.119 (Key1), Model 20 1 (Key2), and 21(Key3), the three key ascending or descending pattern, while the division ( ) can sequences for up to 10 Bytes length have similar plot of key : The performance of the proposed key generation algo- cause in nite numbers if the divisor is a zero. In Figure 1 elements and this means that the key sequences are not rithms Model 1 and Model 2 were evaluated by simulation we show the experiment result of processing time com- using Matlab software. The key generation algorithm of parison among the addition, multiplication, and power Model 1 does not meet the randomness test as in Fig- operations. Based on this result and the previous expla- ure 1, indicated by similar pattern of the three generated nation, we use the multiplication in our scrambling factor key sequences. By varying the value of parameter a with in modifying the Fibonacci sequence. 19 (Key1), 20 (Key2), and 21(Key3), the three key se- The proposed generation algorithm of Model 2 is given quences for up to 10 Bytes length have similar plot of key in pseudocode form in Algorithm 2. elements and this means that the key sequences are not random. Due to the unsatisfactory result of randomness Algorithm 2 Key Generation (Model 2) of proposed Model 1 algorithm, we then proposed Model 2 algorithm. 1: Begin 2: Initialize the parameters: a, b, n, c 3: Derive the 1st element of the key 5.2 Model 2 4: K(1) ← mod(a ∗ b − a, c) 5: Derive the 2nd element of the key The key generation simulation used the Algorithm 2 by 6: K(2) ← mod(a ∗ b − b, c) varying the parameters i.e. the rst number (a), second 7: Derive the 3rd to n-th element of the key number (b), and the key length (key size) (n) and let the 8: for i = 3 to n do modulus number (c) be remains in 256, as the maximum 9: K(i) ← mod(K(i − 1) + K(i − 2) + 3 ∗ i, c) value of alphabet characters. In this simulation, we mea- 10: end for sured the key generation speed and randomness level, and 11: Output the key sequence, K compared with other algorithm. 12: End 5.3 Key Generation Speed The use of ∗b − a, ∗b − b, and 3 ∗ i in Algorithm 2 is For measuring the speed of key generation, 1000 (one intended to satisfy the randomness test and long period- thousand) key sequences were generated. Each key se- icity of the generated key sequences as the requirements quence was then recorded at the start and the end time of good key as stated by Shannon [34]. to get the processing time ( nish time - start time), then The dierence of operations among the original Fi- looked for the average processing time by summing up all bonacci, modi ed Fibonacci Model 1, and Model 2 is pre- processing time and then divided by 1000. We can see sented in Table 2. on Figure 2 the time it took to generate key sequences of International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx. 20xx Algorithm 2: Key Generation (Model 2) random. Due to the unsatisfactory result of randomness of proposed Model 1 algorithm, we then proposed Model 2 1 //Input parameters: a, b, n, d; algorithm. 2 //Derive the 1st element of the key 3 K(1)ß mod(a*b-a,d); 4 //Derive the 2nd element of the key 5.2 Model 2 5 K(2)ß mod(a*b-b,d); 6 //Derive the 3rd to nth element of the key The key generation simulation used the Algorithm 7 For i from 3 to n 2 by varying the parameters i.e. the first number ((), 8 K(i)ß mod(K(i-1)+K(i-2)+3*i,d); 9 Next i second number (*), and the key length (key size) (.) and 10 //Output the key sequence, K let the modulus number (/) be remains in 256, as the maximum value of alphabet characters. In this simulation, Table 2 Different operations among Fibonacci-based algorithms we measured the key generation speed and randomness level, and compared with other algorithm. Original Modified Fibonacci Modified Fibonacci Fibonacci Model 1 Model 2 5.3 Key generation speed For measuring the speed of key generation, 1000 : : = ( : = ;4#4?ℎ A4BC − ?ADEA A4BC), then : = * : = ;
Table 3 Correlation test between two key sequences generated by the proposed Model 2 algorithm Fig. 2 Key element plot of three key sequences with different Fig. 3 Comparison of key generation processing time (ms) for Figure 3: Comparison of key generation processing time initial value of parameter a of proposed algorithm Model 1 varying key length (byte) (ms) for varying key length (byte) Correlation between 2 key sequences Fig. 4 Plot of key elements of three key sequences of 64 Bytes The performance of the proposed key generation Figurewith different 4: Plot value of key of parameter elements a: of19 three(Key1), key 219 sequences(Key2), and of K1, K2 K2, K3 K3, K4 K4, K5 K5, K6 119 (Key3) of the proposed algorithm Model 2 algorithms Model 1 and Model 2 were evaluated by 64 Bytes with dierent value of parameter a: 19 (Key1), Pearson simulation using Matlab software. The key generation varying key length sizes with the same initial parameter. 219 (Key2), and 119 (Key3) of the proposed algorithm Coefficient -0.117 -0.085 -0.080 0.078 -0.222 algorithm of Model 1 does not meet the randomness test as 5.4 Randomness of key sequence Significa- It showed that the key generation time increases along Model 2 in Fig.1, indicated by similar pattern of the three generated nce test 0.535 0.653 0.673 0.681 0.237 with the increase of the key length. However, the pro- Randomness tests are involved in a measuremnet to key sequences. By varying the value of parameter a with analyze the distribution of a set of data to see if it is Result Pass Pass Pass Pass Pass 19 (Key1), 20 (Key2), and 21(Key3), the three key posed Model 1 algorithm has the fastest processing time among all, while the proposed Model 2 has the lowest tocorrelationuncorrelated function or random. (ACF), To test Rk, the is calculated randomness using of the the Correlation between 2 key sequences sequences for up to 10 Bytes length have similar plot of key generated key sequences, we have simulated key processing time in certain time but still has a relatively Equation (1) described in [52], when given a measurement K6, K7 K7, K8 K8, K9 K9,K10 K10,K1 elements and this means that the key sequences are not generation by varying the value of variable (, i.e. 19, 219, same processing time to the original Fibonacci algorithm. of the variables on , by shifting 119 for Key1, Key2Y1,Y, 2and, ..., Key3 Yn , asX given1,X2 in, ..., Fig. X n3. It appears Pearson However, this can be explained as a consequence of using (lag) of . Coefficient -0.062 -0.157 0.056 -0.418 0.050 that allk the key sequences have different and irregular more functions in the proposed algorithm Model 2 than Significa- patterns indicating that all of the key sequences are the number of functions in the Model 1 and the origi- PN−k nce test 0.744 0.406 0.765 0.022 0.790 uncorrelated or random.i=1 Yi − Y Yi+k − Y Rk = (1) nal Fibonacci. The use of more functions is intended to By varying only thePN parameter values2 b from 119 to Result Pass Pass Pass Pass Pass i=1 Yi − Y support the randomness of the generated key sequences. 124, we generated six key sequences and obtained their plot As shown in Figure 4, the autocorrelation value of all The increase in key length size is not linear with increas- of key autocorrelation values as shown in Fig. 4. The The Pearson correlation is calculated using equation ing generation time, since the ratio between the two is 64-Byte key sequences with the de ned lags of 1 to 20 is (2), autocorrelation function (ACF), Rk, is calculated using the relatively decreasing as the key length increases. in the range between the upper boundary (0.2) and the lowerequation boundary (1) described (-0.2) in indicating [52], when that given all a ofmeasurement the key se- n of the variables F1, F2, ..., F# on I1, I2, . . . , I#, by å (Xi - X)(Yi - Y) quences are uncorrelated or random. The autocorrelation i=1 5.4 Randomness of Key Sequence valueshifting for ( lagslag) of of J 0. is 1 which means that the two com- rP = (2) n 22 pared key sequences areN-k not random, since there is no key å Randomness tests are involved in a measuremnet to ana- å (Yi-Y)(Yi+k-Y) (Xi-X) (Yi-Y) element shift (lag = 0) so there is no dierence between I=1 lyze the distribution of a set of data to see if it is uncorre- i=1 the two key sequences.Rk = By using Pearson's correlation lated or random. To test the randomness of the generated N 2 (1) where EK, I, F, X̄ , Ȳ, i, n are Pearson correlation, value of key sequences, we have simulated key generation by vary- test, and varying the parameterå (Yi-Y) of b, we have simulated variable L, value of variable M, mean value of variable L, ing the value of variable a, i.e. 19, 219, 119 for Key 1, the key generation for 200i=1 key sequences and yielded the mean value of variable M, nth iteration, and number of Key 2, and Key 3, as given in Figure 3. It appears that correlation coe cient between two key sequences as in elements. all the key sequences have dierent and irregular patterns Table 3. indicating that all of the key sequences are uncorrelated The Pearson correlation is calculated using Equa- or random. tion (2), By varying only the parameter values b from 119 to 124, PN−k we generated six key sequences and obtained their plot of Xi − X Yi+k − Y rP = i=1 (2) key autocorrelation values as shown in Figure 4. The au- PN 2 2 i=1 Xi − X Yi − Y
Fig. 5 Plot of autocorrelation value of 64 Byte key sequences Fig. 6 Comparison of key element plot among three key with different value of parameter b generated by the proposed sequences generated by (a) proposed algorithm (b) Raphael’s Model 2 algorithm algorithm International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx. 20xx As shown in Fig. 4, the autocorrelation value of all 64-Byte key sequences with the defined lags of 1 to 20 is in the range between the upper boundary (0.2) and the lower boundary (-0.2) indicating that all of the key sequences are uncorrelated or random. The autocorrelation value for lags of 0 is 1 which means that the two compared key sequences are not random, since there is no key element shift (lag = 0) so there is no difference between the two key sequences. By using Pearson's correlation test, and varying the parameter of b, we have simulated the key generation for 200 key sequences and yielded the correlation coefficient between two key sequences as in Table 3.
Table 3 Correlation test between two key sequences generated by the proposed Model 2 algorithm
Correlation between 2 key sequences Fig. 4 Plot of key elements of three key sequences of 64 Bytes with different value of parameter a: 19 (Key1), 219 (Key2), and K1, K2 K2, K3 K3, K4 K4, K5 K5, K6 119 (Key3) of the proposed algorithm Model 2 Pearson Coefficient -0.117 -0.085 -0.080 0.078 -0.222 5.4 Randomness of key sequence Significa- Randomness tests are involved in a measuremnet to nce test 0.535 0.653 0.673 0.681 0.237 analyze the distribution of a set of data to see if it is Result Pass Pass Pass Pass Pass uncorrelated or random. To test the randomness of the Correlation between 2 key sequences generated key sequences, we have simulated key K6, K7 K7, K8 K8, K9 K9,K10 K10,K1 generation by varying the value of variable (, i.e. 19, 219, 119 for Key1, Key2, and Key3, as given in Fig. 3. It appears Pearson Coefficient -0.062 -0.157 0.056 -0.418 0.050 International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx.that 20xx all the key sequences have different and irregular Significa- patterns indicating that all of the key sequences are nce test 0.744 0.406 0.765 0.022 0.790 uncorrelatedAs shown or random. in Fig. 4, the autocorrelation value of all 64-ByteBy keyvarying sequences only the with parameter the defined values lags b from of 1 1to19 20 to is Result Pass Pass Pass Pass Pass 124in , thewe generated range between six key the sequences upper boundaryand obtain (0.2)ed the andir plot the oflower key autocorrelation boundary (-0.2) values indicating as shown that in all Fig of. 4the. The key The Pearson correlation is calculated using equation sequences are uncorrelated or random. The autocorrelation autocorrelation function (ACF), R , is calculated using the (2), value for lags of 0 is 1 which meansk that the two compared equationkey sequences (1) described are notin [52], random, when sincegiven therea measurement is no key n International Journal of Network Security, Vol.21, No.2, PP.250-258,of the Mar. variables 2019 (DOI: F1, 10.6633/IJNS.201903_21(2).09)F2, ..., F# on I1, I2, . . . , I#, by255 å (Xi - X)(Yi - Y) element shift (lag = 0) so there is no difference between the i=1 shifting (lag) of J. rP = two key sequences. By using Pearson's correlation test, and n (2) varying the parameterN-k of b, we have simulated the key 22 å (Xi-X) (Yi-Y) generation for 200å key(Yi-Y)(Yi+k-Y) sequences and yielded the I=1 Table 3: Correlation test (Pearson coe cient and signif- correlation coefficientR = i=1 between two key sequences as in icant test) between two key sequences generated by the k N (1) Table 3. 2 where EK, I, F, X̄ , Ȳ, i, n are Pearson correlation, value of proposed Model 2 algorithm å (Yi-Y) i=1 variable L, value of variable M, mean value of variable L, Table 3 Correlation test between two key sequences generated mean value of variable M, nth iteration, and number of Correlation between 2 key sequences by the proposed Model 2 algorithm K1,K2 K2,K3 K3,K4 K4,K5 K5,K6 elements. Prson Coef -0.117 -0.085 -0.080 0.078 -0.222 Correlation between 2 key sequences Fig.Sig. 4 Plot test of key 0.535elements of0.653 three key0.673 sequences0.681 of 64 Bytes0.237 with Resultdifferent valuePass of parameterPass a: 19 (Key1),Pass 219Pass (Key2), andPass K1, K2 K2, K3 K3, K4 K4, K5 K5, K6 119 (Key3) of theCorrelation proposed algorithm between 2 Model key sequences 2 Pearson K6,K7 K7,K8 K8,K9 K9,K10 K10,K1 Coefficient -0.117 -0.085 -0.080 0.078 -0.222 Prson Coef -0.062 -0.157 0.056 -0.418 -0.050 5.4 Randomness of key sequence Significa- Sig. test 0.744 0.406 0.765 0.022 0.790 ResultRandomnessPass tests arePass involvedPass in a measuremnetPass toPass nce test 0.535 0.653 0.673 0.681 0.237 analyze the distribution of a set of data to see if it is Result Pass Pass Pass Pass Pass uncorrelated or random. To test the randomness of the Correlation between 2 key sequences generated key sequences, we have simulated key K6, K7 K7, K8 K8, K9 K9,K10 K10,K1 generationwhere rP by, X varying, Y , ( Xthe), valueY , i ,ofn variableare Pearson (, i.e. 19, correlation, 219, 119value for Key1, of variable Key2, andx, valueKey3, as of given variable in Fig.y, 3. mean It appears value of Pearson variable , mean value of variable , th iteration, and Coefficient -0.062 -0.157 0.056 -0.418 0.050 that all thex key sequences have differenty n and irregular Significa- patternsnumber indicating of elements. that all of the key sequences are nce test 0.744 0.406 0.765 0.022 0.790 uncorrelatedA comparison or random. of the randomness of several key se- Result Pass Pass Pass Pass Pass quencesBy betweenvarying only the proposedthe parameter algorithm values b and from the 119 Raphael to 124algorithm, we generated is given six inkey Figure sequences 5, Figure and obtain 6, anded the Tableir plot 4. In Fig. 5 Plot of autocorrelation value of 64 Byte key sequences Fig. 6 Comparison of key element plot among three key ofFigure key autocorrelation 5, it can be seen values that as shownthe plot in of Fig key. 4. elements The Figurewith different 5:The Plot Pearson value of of autocorrelation correlation parameter b is generated calculated value by of theusing 64 proposed Byteequation key sequences generated by (a) proposed algorithm (b) Raphael’s (2), Model 2 algorithm algorithm autocorrelationbetween thethree function key (ACF) sets generated, Rk, is calculated by the using proposed the al- sequences with dierent value of parameter b generated equationgorithm (1) is moredescribed random in [52], than when that given generated a measurement by Raphael's by the proposed Modeln 2 algorithm ofalgorithm the variables which yieldsF1, F2, relatively ..., F# on similar I1, I2 plots, . . . , I# of, keyby ele- å (Xi - X)(Yi - Y) i=1 shiftingments. (lag) of J. rP = n (2) As shown in FigureN-k 6, by varying only the value of pa- å 22 rameter b, the correlationå (Yi-Y)(Yi+k-Y) coe cient plot and the P-value I=1(Xi-X) (Yi-Y) i=1 of the key sequenceRk = generated by the proposed algorithm indicate a random sequenceN of keys because the value (1) of where EK, I, F, X̄ , Ȳ, i, n are Pearson correlation, value of å (Yi-Y)2 P-value in average is greateri=1 than the correlation coe - variable L, value of variable M, mean value of variable L, cient. While in contrast, Raphael's algorithm produces mean value of variable M, nth iteration, and number of a key sequence that has P-value and a competing corre- elements. lation coe cient, indicating that the key sequence has a correlation or not random. Further, as shown in Table 4, the comparison of two adjacent key sequences using the Spearman correlation test, indicating that the proposed algorithm passed all randomness tests, while the Raphael algorithm failed on all random tests. All of the above comparisons show the advantages of proposed Model 2 key generation algorithm over Raphael's algorithm.
Table 4: Correlation comparison of two adjacent key se- quences of the proposed algorithm
Proposed algorithm Raphael's algorithm Adj.Key Rho Pval Test Rho Pval Test
K1,K2 -0.118 0.5356 Pass 0.253 0.178 Fail -0.085 0.6533 Pass 0.453 0.012 Fail K2,K3 Fig. 6 Comparison of key element plot among three key Fig.K 35,K Plot4 of autocorrelation-0.080 0.6739 valuePass of 64 Byte0.375 key sequences0.041 Fail 0.078 0.6816 Pass 0.005 0.980 Pass Figure 6: Comparison of key element plot among three withK 4different,K5 value of parameter b generated by the proposed sequences generated by (a) proposed algorithm (b) Raphael’s -0.222 0.2375 Pass 0.484 0.007 Fail K5,K6 Model 2 algorithm key sequences generatedalgorithm by (a) proposed algorithm (b) K6,K7 -0.062 0.7445 Pass 0.252 0.180 Fail Raphael's algorithm K7,K8 -0.157 0.4064 Pass 0.234 0.214 Fail K8,K9 0.057 0.7659 Pass 0.390 0.033 Fail K9,K10 -0.419 0.0222 Pass 0.103 0.588 Pass K10,K1 0.051 0.7901 Pass 0.002 0.002 Fail International Journal of Network Security, Vol. xxx, No.xx, PP.xxx-xxx, xxx. 20xx A comparison of the randomness of several key 5.5 The periodicity of key sequences sequences between the proposed algorithm and the Raphael Suppose a given key sequence A is 1 3 4 6 5 8 1 3 4 algorithm is given in Fig.5, Fig.6, and Table 4. In Fig. 5, it 6 5 8 1 3 4 6 5 8 .… Then the period of key sequence A is can be seen that the plot of key elements between the three 6 (the number of elements from 1 3 4 6 5 8 before key sets generated by the proposed algorithm is more experiencing the exact same loop). The longer the period, International Journal of Network Security, Vol.21, No.2, PP.250-258,random Mar. than 2019 that (DOI: generated 10.6633/IJNS.201903_21(2).09) by Raphael's algorithm which256 the better a key sequence. Since the proposed algorithm yields relatively similar plots of key elements. uses four parameters a, b, d, and n, and there is a 5.5 The Periodicity of Key Sequences scrambling factor that depends on the process iteration, the key sequences generated by the proposed algorithm has a Suppose a given key sequence A is 1 3 4 6 5 8 1 3 4 6 5 8 1 long period equal to n (the key length to be generated, 3 4 6 5 8 ... Then the period of key sequence A is 6 (the determined by the user) . This character is called One Time number of elements from 1 3 4 6 5 8 before experiencing Key or One Time Pad. Meanwhile, the algorithm used by the exact same loop). The longer the period, the better Raphael et al.[35] also satisfies a long period, but since it a key sequence. Since the proposed algorithm uses four does not use the modulus function, the resulting key parameters and , and there is a scrambling factor sequence has a regular pattern of ascending or descending. a, b, d, n So it does not meet the randomness test and it requires a that depends on the process iteration, the key sequences larger memory to sum the number that will enlarge as the generated by the proposed algorithm has a long period key length increases. equal to n (the key length to be generated, determined by the user). This character is called One Time Key or One 5.6 Complexity of algorithm Time Pad. Meanwhile, the algorithm used by Raphael et In the Key Generation algorithm Model 1 (see al. [23] also satis es a long period, but since it does not use the modulus function, the resulting key sequence has Algorithm1), there are 10 Line of Codes (LOCs). a regular pattern of ascending or descending. So it does However, LOCs beginning with a double forward slash (//) not meet the randomness test and it requires a larger Fig. 7 Comparison of correlation coefficient with different value are only descriptions and not executed, and hence these Figure 7: Comparisonof of parameter correlation b coe cient with dif- memory to sum the number that will enlarge as the key parts are ignored and not counted in the analysis of the ferent value of parameter b algorithmic complexity. LOC 3, 5, 7-9 are the processing length increases. As shown in Fig.6, by varying only the value of parameter b, the correlation coefficient plot and the P-value part of the algorithm that will be calculated on its of the key sequence generated by the proposed algorithm complexity. In LOC 3, there are 2 instructions i.e. 5.6 Complexity of Algorithm asindicate(n + 1 −a random3) or ( nsequence− 2) times of keys with because the variable the value i asof theP- BNO (D, O) and assignment (=) of the variable P(1). In counter. The assignment instruction is executed LOC 5, similar to LOC 3, there are 2 instructions, i.e. In the Key Generation algorithm Model 1 (see Algorithm value in average is greater than the correlation(i = 3) coefficient. before the loop. and assignment ( ) of the variable . LOC 1), there are 10 Line of Codes (LOCs). However, LOCs While in contrast, Raphael's algorithm produces a key BNO (Q, R) = P(2) sequenceInside the that loop, has P there-value are and 1 a comparison competing corrinstructionelation beginning with a double forward slash (//) are only de- 7-9 is an incremental loop as many as (# + 1 − 3) or (# − coefficient, and indicating 1 incremental that the counter key sequence . has In ad- a scriptions and not executed, and hence these parts are (i < n + 1) (i + +) 2) times with variable 4 as the counter. The assignment dition,correlation there or are not also random. an assignment Further, as operationshown in Table of 4, the ignored and not counted in the analysis of the algorith- (K(i) = instruction (4 = 3) is executed before the loop. comparison of two adjacent key sequencesinvolving using2 addition the mic complexity. LOC 3, 5, 7-9 are the processing part of mod (K(i − 1) + K(i − 2) + 3 ∗ i, d)) operationsSpearman (+), correlation 1 multiplication test, indicating operation that the (*), proposed 1 modu- Inside the loop, there are 1 comparison instruction the algorithm that will be calculated on its complexity. lusalgorithm operation passed (mod) all andrandomn 1 assignmentess tests, while operation the Raphael (=) of (4 < # + 1) and 1 incremental counter (4 + +). In In LOC 3, there are 2 instructions i.e. and mod (a, d) thealgorithm variable failed. on Thus, all in random LOC 5-7,tests. there All of are the 7 instruc- above assignment (=) of the variable K(1). In LOC 5, similar K(i) addition, there are also an assignment operation (P(4) = tionscomparisons repeated show (n-2) the times advantages and 1 non-repeated of proposed Model instruction. 2 key to LOC 3, there are 2 instructions, i.e. and BNO (P (4 − 1) + P (4 − 2), R)) involving 1 sum mod (b, c) Thus,generation in detail, algorithm the number over Raphael’s of instructions algorithm in. , assignment (=) of the variable . LOC 7-9 is an in- LOC3 = 4 operation (+), 1 modulus operation (mod) and 1 assignment K(2) , and LOC 7-9 . Thus the com- cremental loop as many as or times LOC5 = 4 = 7(n − 2) + 1 (=) of the variable P(4). Thus, in LOC 7-9 there are 5 (n + 1 − 3) (n − 2) plexityTable of4 Comparison the Model of 2 correlation algorithm of istwo key sequences of the with variable as the counter. The assignment instruction proposed algorithm7( n − 2) + 1 + 4 + 4 instructions repeated (# − 2) times and 1 non-repeated i or or or . is executed before the loop. 7(n − 2) + 9 7n − 5 O(n) instruction. Thus, in detail, the number of instructions in (i = 3) From the above analysis, it appears that both mod- Inside the loop, there are 1 comparison instruction ( LOC 3 = 2, LOC 5 = 2, and LOC 7-9 = 5 (# − 2) + 1. i < elsAdjacent of the proposedProposed keyalgorithm generation Raphael’s algorithms algorithm have the ) and 1 incremental counter ( ). In addition, there Thus the complexity of the Algorithm 1 is 5 (# − 2) + n+1 i++ sameKeys algorithmic complexity that is linear complexity ex- are also an assignment operation ( Rho Pval Result Rho Pval Result 1 + 2 + 2 or 5(# − 2) + 5 or 5# − 5 or !(#). K(i) = mod (K(i − pressed by . However, only the Model 2 satis es the ) involving 1 sum operation (+), 1 modulus O(n) 1)+K(i−2), c) randomnessK1, K2 test.-0.118 0.5356 Pass 0.253 0.178 Fail operation (mod) and 1 assignment (=) of the variable In the Key Generation algorithm Model 2 (see K2, K3 -0.085 0.6533 Pass 0.453 0.012 Fail Algorithm 2), there are 10 LOCs. LOCs beginning with K(i). Thus, in LOC 7-9 there are 5 instructions repeated K3, K4 -0.080 0.6739 Pass 0.375 0.041 Fail double forward slash are ignored and not counted in the (n − 2) times and 1 non-repeated instruction. Thus, in 6 Conclusions detail, the number of instructions in LOC 3 = 2, LOC 5 K4, K5 0.078 0.6816 Pass 0.005 0.980 Pass analysis of the algorithmic complexity. LOC 3, 5, 7-9 is the = 2, and LOC 7-9 = 5(n − 2) + 1. Thus the complexity WeK5, have K6 utilized-0.222 the0.2375 Fibonacci Pass sequence0.484 0.007 in construct-Fail processing part of the algorithm that will be calculated on its complexity. In LOC 3, there are 4 instructions i.e. of the Algorithm 1 is 5(n − 2) + 1 + 2 + 2 or 5(n − 2) + 5 ingK6, two K7 proposed-0.062 models0.7445 ofPass key generation0.252 0.180 algorithm, Fail or 5n − 5 or O(n). Model 1 (without scrambling factor) and Model 2 (with multiplication (*), subtraction (-), modulus (mod) and K7, K8 -0.157 0.4064 Pass 0.234 0.214 Fail In the Key Generation algorithm Model 2 (see Algo- scrambling factor). The modi cation by adding a scram- assignment (=) of the variable P(1). In LOC 5, similar to rithm 2), there are 10 LOCs. LOCs beginning with double blingK8, factorK9 is0.057 intended 0.7659 to generatePass 0.390 key sequences0.033 Fail that LOC 3, there are 4 instructions, i.e. multiplication (*), forward slash are ignored and not counted in the analy- satisfyK9,K10 randomness -0.419 tests0.0222 and Pass long periodicity0.103 0.588 which Pass have subtraction (-), modulus (mod) and assignment (=) of the sis of the algorithmic complexity. LOC 3, 5, 7-9 is the notK10, been K1 used0.051 in measuring 0.7901 Pass the existing0.535 key0.002 algorithms Fail variable P(2). LOC 7-9 is an incremental loop as many as processing part of the algorithm that will be calculated found in recent literature. Simulation results of the two (# + 1 − 3) or (# − 2) times with the variable 4 as the on its complexity. In LOC 3, there are 4 instructions i.e. models indicate that the key generation time increases multiplication (*), subtraction (-), modulus (mod) and along with the increase of the key length, but the ratio assignment (=) of the variable K(1). In LOC 5, similar between key generation time and key length relatively de- to LOC 3, there are 4 instructions, i.e. multiplication (*), creases. The randomness test results indicate that the key subtraction (-), modulus (mod) and assignment (=) of the sequences generated by Model 1 do not meet the random- variable K(2). LOC 7-9 is an incremental loop as many ness test despite having relatively fast computation time. International Journal of Network Security, Vol.21, No.2, PP.250-258, Mar. 2019 (DOI: 10.6633/IJNS.201903_21(2).09) 257
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