Domain Examination of Chaos Logistics Function As A Key Generator in Cryptography

The use of logistics functions as a random number generator in a cryptography algo-rithm is capable of accommodating the diffusion properties of the Shannon principle. The problem that occurs is initialization x 0 was static and was not affected by changes in the key, so that the algorithm will generate a random number that is always the same. This study design three schemes that can providing the ﬂexibility of the input keys in conducting the examination of the value of the domain logistics function. The results of each schemes do not show a pattern that is directly proportional or inverse with the value of x 0 and relative error x 0 and successfully fulﬁll the properties of the butterﬂy effect. Thus, the existence of logistics functions in generating chaos numbers can be accommodated based on key inputs. In addition, the resulting random numbers are distributed evenly over the chaos range, thus reinforcing the algorithm when used as a key in cryptography.


INTRODUCTION
The logistic function f (x) = rx(1 − x) or in the iterative form x i+1 = rx i (1 − x i ) was usually used as a generator in the generation of chaos-based random numbers [1]. This function is able to accommodate the diffusion properties of Shannon's principle on cryptography algorithms, since they have a sensitivity to initial values.
Research [2], [3], [4], uses numbers as static initialization on logistics functions in cryptography algorithms. This means that changes to the key don't have an effect on random numbers, thus the algorithm will still acquire a random number that is always the same. Research [5], also uses numbers as input values that directly fill in by user, nevertheless becomes inefficient since the user needs more input other than the key and plaintext. This research not only combines algorithms such as [6], [7], [8], but designs a new algorithm to obtain an unique key. Additionally, the function will generate a random number, if the initialization domain r and x 0 as seed is limited to a certain value, 0 < x 0 < 1, and r = 4 .
The value of r is constant, making the strength of the algorithm rests on the value of x 0 . Moreover it needs an examination process by using any scheme that can increase x 0 complexity space, on the contrary remain in logistics function domain. This study provides the flexibility of key inputs that are efficient and can generate different random numbers of logistics functions. 4578 ISSN: 2088-8708

PROPOSED EXAMINATION SCHEME
The examination process of the logistics function domain conducted to a wide variety of inputs that allow it to be used as a key. Suppose the key k 1 , k 2 , . . . , k 8 is the result of a conversion of eight ASCII characters. Key input is set up to eight characters, considering the user's ability to recall key and also consider the complexity of the key guessing space when the use of the characters too little. Each key input of less than eight characters, then do the padding process with character " §" that equivalent to 167 in ASCII.
This study provides three algorithms that are used for the examination process of initialization value in domain 0 < x 0 < 1, as shown in the general scheme in Figure 1. The ratio of r a is designed as a comparison of two values to accommodate the domain from initialization. Let r a = p a /q a where q a > p a , for a = 1, 2, 3 and p a , q a ∈ R.

The First Scheme
Given each k i ∈ Z 256 for i = 1, 2, . . . , 8 is the decimal base number of ASCII conversion results. To be able to make changes at every turn of the input, given index value d i ∈ Z 256 for j = 1, 2, . . . , 8 as a constant value which is multiplied by the value k i . Scheme-1 is determined by using r 1 = p 1 /q 1 where, Determination of p 1 value is obtained from the sum of multiplication k i and d i which is performed based on position, it is only for k 1 and d 5 there is crossing position and at k 1 squared. Multiplication of position difference is also done to acquire the q 1 value, however in sum with the average value of each k i . The multiplication combination is done as a variation to gain a unique ratio value, taking into account the requirement q 1 > p 1 .

Second scheme
Scheme-2 also uses the same decimal number and index value, where each k i , d i ∈ Z 256 for i, j = 1, 2, . . . , 8. The ratio value is determined by the equation r 2 = p 2 /q 2 , (3) The p 2 value is obtained using the average of the first six values of k i , whereas to earn the q 2 value is the sum of the multiplication of k i and d i values according to the order of each value.

Third scheme
Every k i , d j ∈ Z 256 for i, j = 1, 2, . . . , 8. Scheme-3 is obtained based on r 3 = p 3 /q 3 , where the determination of the numerator and denominator is given in Equation (5) and Equation (6). The p 3 value is the IJECE ISSN: 2088-8708 4579 average of the multiplication k i with d i only at the first value, the third value, and the sixth value with each index.
The value of q 3 is the sum of the k i and d i multiplications based on the index with the constants selected by different increments.

Analysis of Examination Process
Domain examination of the logistics function is performed based on the three schemes given in the previous section. Referring to the general scheme in Figure 1, then testing related variations of inputs that allow to be used as a key. Each possible key is an ASCII character whose decimal basis is in the range 0 to 255. The problem that occurs is not all numbers have a character. Consequently the test for the lowest number can't start from decimal 0 instead in decimal 32 which is proportional to the space character, whereas testing for the largest number at decimal 255 is equivalent to the character . In addition to character testing for minimum and maximum decimals, key tests are also tested with one bit difference, so that it can be seen how sensitive each scheme is to generating initialization values. Table 1 is the simulation result of each scheme in obtaining the value of x 0 . Testing with one bit difference (numbers 1 through 6) shows the changes in values that begin to occur in the 4th or 5th mantissa of the x 0 value in each schema. Accordingly, the process of domain examination for each scheme succeeds to generate different initialization. This condition corresponds to the need of logistics functions in obtaining random numbers based on chaos. Tests with the same eight-character input based on the smallest decimal, the medium decimal, and the largest decimal are given successively in the numbers 7, 8, and 9 in Table 1. The initialization values in scheme-2 and scheme-3 obtain the same value, although the input is very different. This condition occurs since the determination ratio of r 2 and r 3 using the average process to get the numerator, while for the denominator using the addition and multiplication of characters with the index value with the same position. On the other hand, scheme-1 uses a combination of multiplicity of position difference, squared and mean process. In that case, scheme-1 keeps generating different initialization values.

Relative Error Test
Relative error testing [9] was conducted to see whether linear character reduction would yield also linear results on the x r values with proportional or inversely proportional.
Use E R = |c a − p a | /c a · 100%, the keyÿÿÿÿÿÿÿÿ is chosen as the reference value c a (number 1 in Table 2), and the approximation value as the key input p a less than eight charactersÿ (iterated from number 2 to number 8).
ISSN: 2088-8708 The relative error results in each scheme do not show a proportional or inversely proportional pattern. So, it will complicate cryptanalyst to see the pattern of input changes on the same character.

Linear Regression Test
One-to-one correspondence between the input-output is also important so that cryptanalyst difficult to reconstructing scheme and make prediction of the key used as input. This relationship can be seen through a linear regression test based on the rate of change on the resulting value of x 0 . Figure 2, is the result of each scheme visualized using the Scatter plot. The diagram of each scheme has no linear relationship, because when the curve matching process is used, the coefficient of determination (R 2 ) is close to zero. This test illustrates that any changes to the key characters is done patterned, will not provide an initial value x 0 linearly patterned, either proportional or inversely.

Butterfly Effect Test
The butterfly effect test is used to indicate the change in bits in the key input, whether it gives a large change to the output. Suppose that as a comparator key ZZZZZZ and ZZZZZY are selected which has a difference of one bit. The result of the two keys with scheme-1 is obtained by a random number of the first 500 iterations, shown in Figure 3. Visually, the random number generated is very different although the difference in initialization value x 0 is only 0.0000151538 ≈ 1.538 × 10 −6 or in an absolute relative of 0.063%. A minor change in input and a major change in output proves that the 1st scheme has successfully fulfilled the butterfly effect. The initial value difference of x 0 is 0.0487% in scheme-2. But the rate of change occurs very significantly that appears in the Scatter diagram in Figure 4. So, the scheme-2 has also fulfilled the butterfly effect test.  Significant changes also occur at random values with scheme-3, as shown in Figure 5, although the difference is 0.0399% at input value x 0 . So, the scheme-3 also meets the properties of the butterfly effect. The three schemes have succeeded in fulfilling the properties of the butterfly effect, thus the existence of the logistics function in generating chaos numbers can be accommodated based on key inputs. Each scheme can be used as a complement to a cryptography algorithm to meet the diffusion properties of Shannon's principle.

Analysis Of The Algorithm Ability
Each scheme is tested in correlation [10], to detect the connectedness of the random number generated based on the input. While MAPE [11], it is used to find out how massive the difference of random value of key change. Used three variations of the input [12], the first is the same character input, a second input alphabetic characters that revolve around the 26-character alphabet. While the last test, used alphabet, symbols, and numbers.
Calculation of correlation in Table 3 show that there are two values on scheme-1 and scheme-3 which correlation is negative, besides the rest is positive. Cryptographically, the negative value is not too influential, hence it seen how close the value to zero indicating the unrelated two random numbers are generated. In the context of the relations, this same analogy can be used to test the difference of two random numbers generated. Overall the correlation value generated by each scheme is within the range of 0.00 -2.99. Based on [13], the interval shows the strength of a very weak relationship. This condition provides information that 1 bit key input difference, can generate different random numbers on each scheme. ISSN: 2088-8708 In addition to the correlation and MAPE analysis, we also tested the distribution of random number data using box-plot diagrams. Based on Figure 6, each box has almost the same size in which the upper and lower whisker lines vary slightly, but the maximum value is always close to one and the minimum is near zero. The distribution of data in the chaos range will strengthen the cryptography algorithm if used as a key, this condition will certainly complicate cryptanalyst to be able to search for infinitely many numbers although limited.

CONCLUSION
Each designed scheme is capable of providing key input flexibility that can execute domain logistics function values. A 1 bit difference in the key character affects every random number generation, so each key will generate a different random number sequence. Under the key input conditions of the same eight characters, the 1st scheme is better at generating different initialization values than the scheme-2 and schema-3. In addition, the one character reduction of the eight identical characters in the key input does not show a proportional or reverse pattern with the initial x 0 values and the relative error x 0 . The resulting random numbers distributed evenly over the chaos range will amplify the algorithm when used as a key in cryptography. This condition will certainly complicate cryptanalyst to be able to search for infinitely many numbers although limited. The three schemes have succeeded in fulfilling the nature of the butterfly effect, thus the existence of the logistics function in generating chaos numbers can be accommodated based on key inputs. Each scheme can be used as a complement to a cryptography algorithm to satisfy the diffusion properties of the Shannon principle.