FAST HASH FUNCTION BASED ON BCCM ENCRYPTION ALGORITHM FOR E-COMMERCE (HFBCCM)

Transcription

1 FAST HASH FUNCTION BASED ON BCCM ENCRYPTION ALGORITHM FOR E-COMMERCE (HFBCCM) Mahmoud M. Maqableh School of Engineering and Computer Sciences, Durham University, South Road, Durham, DH1 3LE, United Kingdom, E- mail: ABSTRACT Recently, we have proposed a fast block cipher encryption algorithm based on chaotic maps (BCCM). We released that the whole encryption process is extremely sensitive to input message and secret key, which could be exploited to calculate hash value or message authentication code of input message. In this paper, we propose a fast hash function based on block cipher encryption algorithm with changeable parameters (HFBCCM). The proposed hash function has very simple and flexible design that produces different hash values length using different block sizes. Theoretical analysis and computer simulations indicate that our algorithm can satisfy all requirements of cryptography hash functions. It is simple, flexible, high secure, very fast algorithm, and extremely sensitivity to input message and secret key with strong confusion and diffusion capability. These properties confirm that HFBCCM algorithm is practical and reliable with high potential to be adopted in E-Commerce. KEYWORDS Hash Function, Hash value, Message authentication code, Chaotic maps, Encryption, E-Commerce. 1. INTRODUCTION Hash function is one of the most important of security primitives that play a fundamental role in modern cryptography (Menezes et al., 1997). It is one-way function uses to produce hash value from input message and it is very hard to find the original message from a given hash value (Schneier, 1996). Cryptography hash functions are further divided into keyed and unkeyed hash functions. The keyed hash functions depend on secret key and input message, while the unkeyed hash functions depend on input message only (Ferguson and Schneier, 2003). In general, hash functions process whole input message to produce fixed-length hash value using iterative procedure. A well designed hash functions will produce different hash value with slightly difference in input message. Thus, any change or changes in the input message will produce a completely different hash value. Chaos theory is dynamical system that display complex behaviour from simple nonlinear equation and it look like random but it completely deterministic and unpredictable (Zeng et al., 1993). Chaos theory has attracted the cryptography field due to it characteristics, such as deterministic nature, unpredictable, randomlook nature and sensitive to initial value (Maqableh et al., 2008). Cryptographers utilized dynamical chaotic maps to develop new cryptography primitives by exploiting some of chaotic maps such as: Logistic map, Henon map, Tent map and Lorenz attractors (Yang et al., 2010, Yang et al., 2009, Akhshani et al., 2009, Yi, 2005). Logistic map is one of the most popular chaotic maps, simple, one dimensional, nonlinear dynamical system, and sensitive to initial conditions. Logistic map behaviour mainly depends on values of r and x 0, by changing one of the input parameters different behaviours will be raised. It was built based on iterative function idea, where the next value (x n ) of x depends on the previous one (x n-1 ). Logistic map govern by the following equations: xn+ 1 = r xn ( 1 xn ) ; (1)

2 where x n ( 0,1 ),r [ 0,4],n N, and x 0 is the initial population. 2. DETAILS OF THE PROPOSED HASH FUNCTION (HFBCCM) In the present paper, fast hash function algorithm based on BCCM encryption algorithm is proposed that called HFBCCM. BCCM is a fast block cipher encryption algorithm based on chaotic maps with changeable parameters (Maqableh, 2010). After many experiments and analysis on BCCM, we released that it could be possible to exploit it to design fast hash function with very high security by doing some modifications. Overview of the proposed hash function algorithm with necessary modifications is illustrated in Figure 1. HFBCCM is chaotic based hash function with variable control parameters, see Table 1. The first step in proposed algorithm is to select secret key that will be used to calculate the hash value, and input message to calculate it hash value. Then, we need to generate subkeys by iterating logistic map as chaotic map 75 times with secret key as initial value and r equal After that Logistic map will be iterated [8*(R+1)] times to generate all subkeys by taking the first w-bit of x n+1 value of the same size of sub-blocks. In each iteration equations 2, 3, and 4 will be applied to generate value that will be used for subkeys order at end of each round, shifting rows and mixing columns, respectively. f i ( i subkey[ R][ i]* R)%8 1 ; (2) w i ( i subkey[ R][ i]* R)% L 1; (3) c i ( i subkey[ R][ i])%8 1; (5) where i the subkey index, R is round number, and L is half sub-block size. Now, the input message will be divided to equal blocks sizes and each block will be further divide to eight sub-blocks. Size of each block depends on final hash value size and size of each sub-block will be eighth of the full block size. If the last block size is less than full block size, then the input message will be padded by 1 followed by zeros to have full block size. Finally, input message will processed throughout the compression function by iterating each block R times and in each round applying compression function operations, see Table 2. The final output value of the previous round will be XORed with the next block after it XOR with subkeys of R+1. At the end hash value of input message will be generated. Figure 1. Overview of HFBCCM HFBCCM can be customized in different ways for different needs and applications as follow: 1- HFBCCM can be used as keyed hash function by assigning secret key value to initial value of chaotic map. 2- HFBCCM can be used as unkeyed hash function by using value of each block to generate the block subkeys.

3 3- HFBCCM can be used to process different input messages with different lengths. 4- HFBCCM can be used to calculate hash value based on several chaotic maps 5- HFBCCM can be used to produce different length of the hash value in bits, such as 64, 128, 192, 256, and HFBCCM has R number of rounds, R[4, N], with larger number of rounds a higher secure hash function will be. Table 1. HFBCCM Parameters Parameter Meaning Values Range R Number of rounds 1,2,3, w Word size in bits 8,16,32,64 s Secret key in bits 128,160,192, Table 2. Primitive operations for HFBCCM Notation Meaning X+Y The 2 nd complement addition of words X-Y The 2 nd complement subtraction of words X>>>Y Circle right shift Y bits X<<<Y Circle left shift Y bits X Y Bit wise exclusive OR of words X%Y The modulus, or remainder, operator Figure 2. HFBCCM compression function 3. PERFORMANCE ANALYSIS

4 In this section we will study performance and statistical analysis of HFBCCM hash function. Several tests and simulations will be performed on the computed hash values. The hash value size is 265, the sub-block size and subkeys is 32 bits size, and the number of rounds is Hash value distribution Uniform distribution is very important condition in hash functions security (Xiao et al., 2009). We performed the uniform distribution test on the following paragraph: The computer security term refers to the protection of data, networks, computer programs, computer power and other elements of computerized information systems. (a) ASCII distribution of the original message, (b) Hash values distribution in hexadecimal format, Figure 3. Distribution of original message and hash value Difference between ASCII distribution of original message and hexadecimal distribution of hash value are shown in Figure 3. The experiment result shows that original message ASCII distribution focus in very small space, on the other hand hash value distribution is irregular and spread over large hash space. The simulation result confirms that the proposed hash function is secure enough against statistical attack and it is difficult to retrieve information about input message from calculated hash value. 3.2.Hash value result of text input message In order to evaluate HFBCCM hash value sensitivity to input message as text, simulation tests have been performed on the following text: The computer security term refers to the protection of data, networks, computer programs, computer power and other elements of computerized information systems. with k= 2F050FE938943ACC45F65567FFFFFFFFE. Then, several changes have been performed and the correspondence hash values have been calculated as follow: Scenario 1: The original Message.

5 Scenario 2: Change the first character T in the original message into U. Scenario 3: Change the first character s small letter in the word "security" to "S" capital letter. Scenario 4: Remove the full stop from the end of the statement. Scenario 5: Remove the first blank space The computer. Scenario 6: Change the word to in the original message into do. Scenario 7: Change the word programs in the original message into programmes. Scenario 8: Add a blank space to the end of the original message. Scenario 9: Add number zero at the beginning of the original message. Scenario 10: Remove the comma after the word data. Scenario 11: Use keyed hash function with k= 2F050FE938943ACC45F65567FFFFFFFFF. Scenario 12: Use keyed hash function with k= 2F050FE938943ACC45F65567FFFFFFFFD. Correspondence hash values in hexadecimal format for each of previous scenarios as follows: Scenario 1: 1DB403A7BA175B5110E251FC53CE0BCEAB5596F799B2E71FF7C07DD9896AA0FD. Scenario 2: F4589F14E0B306216BDEDF6D556EDC3A0D3AEEA60175AC C49467B2603. Scenario 3: BA54C028337C73A2BE76675B738A6B2C5DCCA8613B5F068FC BF07F091. Scenario 4: EB63912F0519FC1FF9ACCCFEE99F4B48FF5B185A16E1DFA1B0E F5A15E. Scenario 5: 48E47BE03338D835A001390E4871B479BC361688EA8F4FB14096ED53223F163F. Scenario 6: EC4B817E7322A0E15ACE084B8AC8BDC101B0D9EFC53CD54AE9B0A56127C5EF09. Scenario 7: B391A4E60A7981D3DCF155BE7A5BCEFC8AB7592E2E3E0C45FB89C6412A672B7E. Scenario 8: CFD7EA002FA6F258D0FA924E45794C322BB4C7FF1010B2FC2500C8E4. Scenario 9: EE6C0077B EADED EEEF0D9560F44E0AC5DB6C7EDFA5186EF. Scenario 10: BB0875DC8D1B6A075D29C370748D89D9C629BFAC811CBDA95C227463F Scenario 11: 8F293F24D39C88CCEF5578F648ACA091B3A40D881DAD62D D8319BA224. Scenario 12: 1261EC2372C976A5FB8D3DFB2C95EAEF895D0C4A5E8F1533FFA0F7F5B30371C9. The simulation results show that HFBCCM are very sensitive to any slight (bit or bits) changes in the input text message or used key and that will cause significant changes in the final hash value. 3.3.Hash value result of Images input message In order to evaluate HFBCCM hash value sensitivity to input message as image, simulation tests have been performed on Ayham image of size pixels with 8-bit greyscale, as shown in Figure 4, and used key is k= 2F050FE938943ACC45F65567FFFFFFFFF. Then, several changes have been performed then the correspondence hash values have been calculated as follow: Scenario 1: The Ayham image with 8-bit greyscale, as shown in Figure 3, is used. Scenario 2: Add 1 to the value of the pixel located at the upper left corner. Scenario 3: Subtract 1 from the value of the pixel located at the upper left corner. Scenario 4: Add 1 to the value of the pixel located at index x=250, y=188. Scenario 5: Add 1 to the value of the pixel located at the lower right corner. Scenario 6: Subtract 1 from the value of the pixel located at the lower right corner. Scenario 7: Change the last digit in the key from number F to E. Scenario 8: Change the last digit in the key from number F to 0. Calculated 256-bit hash values in hexadecimal format for each of previous scenarios as follow: Scenario 1: E790B2C1A8F1A57E8BC1530AAEBB2571D A74FF2D2A6B25101B81B44DC. Scenario 2: 55DFF319A02FA4585A37C34C4DA6B5B8EDBBEFC4484FA1CF04A4577D389D2792. Scenario 3: B190868F7445B92D0E6CE582BF B409CBA17AD AF8. Scenario 4: 53DB4D C B370F2C4CC120EC045FA361B9F6F5BBFFC9. Scenario 5:09EF BB7CF6C497ADDE2C0E1041A716DDE D56D2E2D179. Scenario 6: DB7D134FB4020ED386E65C35D70035C654D A80A37AA863FD9C6. Scenario 7: 21833CA7CA870E32CBDD7DC7F3EDBB8FFA7D0DA7C EA1DC236B935EFCA. Scenario 8: 62084EB69C0583AE232E35EC198E122EE5A4A87F13A572298FCB2A092A89B92E. The simulation results show that HFBCCM are very sensitive to any slight (bit or bits) changes in input message or used key will cause significant changes in final hash value.

6 Figure 4. Ayham 8-bit greyscale input image 3.4.Statistical analysis of diffusion and confusion Confusion and diffusion properties are very important for many cryptography algorithms such as block cipher encryption algorithms and hash functions (Baris Coskun, March- 2006). Hash function requires spreading influence of whole input message into hash value space. Well designed hash functions should have complex relation between bits in input message and correspondence bits in hash value. Consequently, changing any bit or bits in input message should lead to affect at least half hash value bits and each bit of hash value bits has 50% probability to be changed. A message has been chosen randomly and correspondence hash value has been generates. Then, one of input message bit selected and toggled randomly and calculates binary hash value. Now, a comparison between original and new hash value has been performed and counting number changed bits in the same locations that called Bi. This test has been performed 3000 times and correspondence distributions numbers of bits changes are shown in Figure 4. (a) Plot of B i (b) Histogram of B i

7 Figure 4: Distribution of changed bit number B i Similar to references (Xiao et al., 2005, Zhang et al., 2007, Akhavan et al., 2009),the following are six statistics for HFBCCM: Minimum bit number changed: N Bmin = min ( B i ) ; (2) 1 Maximum bit number changed: N Bmax = max ( B i ) ; (3) 1 Mean bit number changed: B = 1 Bi ; (4) N Mean probability changed: B P = ( ) ; (5) Standard variance of the changed bit number: 1 ΔB = (Bi N 1 Standard variance: B ) 2 ; (6) 1 ΔP = (B / 256 P ) 2 i 100 ; (7) N 1 where N is total number of statistics and Bi is number of changed bits in ith test, ΔB and ΔP indicate the stability of diffusion and confusion. Table 3 shows the statistical results for different number of tests N=3072, 2048, 1024, and 512, respectively. As shown in table 3, the mean changed number of bit and the mean changed bit probability are and 50.05, respectively, which is very close enough to the ideal values 128 and 50%. Whereas ΔB and ΔP indicate the stability of diffusion and confusion and show that HFBCCM algorithms are very stable. In conclude HFBCCM shows that it is very difficult to attack and secure enough against statistical attacks and can be adopted in many different applications. Table 3. Statistic of number of changed bit Bi N=3072 N=2048 N=1024 N=512 Mean B P (%) ΔB ΔP (%) Bmin Bmax Uniform distribution on hash space A message has been generated randomly and calculated the correspondence hash value to ensure hash space distribution. One bit toggled randomly and calculated the correspondence hash value. Late on, comparing and counting number of bits toggled in calculated hash space for the two hash values. This test performed 3000 times by fix first input message and toggle one bit randomly to calculate number of toggled bit in same location in hash space, as shown in Figure 6. The mean of toggled bits is , which is very close to the ideal mean value (1500). Theref ore, HFBCCM will be strong enough against the statistical attacks and collision resistance.

8 Figure 5. Hash value distribution in hash space with N= 3000 and mean Analysis of collision resistance HFBCCM collision resistance capability has been tested as follow. First, message has been selected randomly and calculated it associated hash value in ASCII format. Then, one bit selected and toggled randomly to generate new hash value in ASCII format. Now, difference between two ASCII hash values in the same position has been calculated. Finally, calculate absolute difference (d) by calculating summation of all differences of all characters for two hash values, see equation 8. Figure 6: Distribution of number of ASCII characters with the same value at same location in hash value with number of tests 3000, 2000, and 1000 d N i 1 t ( e i ) t ( e `i ) (8) N i1 f ( t( e ), t( e` )), i i 1, f ( x, y) 0, x y x y (9) where e i and e`i is the ith entry of original input message and ith entry of new message after toggled, respectively, and t(.) is a function convert the entry to decimal value equivalent of ASCII. This kind of test has been performed 3000, 2000, and 1000 times and maximum, minimum, mean and mean/c, c is number of hexadecimal character of hash value, for each case has been calculated, as shown in table 4. Figure 6, shows the distribution of number of ASCII characters with same value at same location in the hash value with number of tests 3000, 2000, and 1000, see equation 9. It is noticed that the maximum number of equal ASCII characters in same location is only three and the collision possibility is very low and most of the entries are different in ASCII format.

9 Table 4. Absolute differences of two hash values N=3000 N=2000 N=1000 Maximu Minimum Mean Mean/C Execution time of HFBCCM with different parameters The main idea behind designing a hash function is to provide very fast security algorithm compare with encryption algorithm for security application. An ideal hash function algorithm should provide high security with high speed. Several speed tests have been performed on HFBCCM algorithm execution time with different parameters, such as number of rounds and size of the input text/image. This test have been performed for the proposed hash algorithm on Laptop Intel Core(TM) 2 Duo 2.00 GHz CPU with 3 GB RAM running on Linux Operating System, using GMP (GNU Multi-Precision Library) and OpenCV Speed average is very small for calculating hash value using HFBCCM algorithm with different parameter, see Tables 5 and 6. Therefore, the proposed algorithm demonstrated that calculating execution time of generating hash value is very small and it very good candidate to be used for E-Commerce applications over broadband networks. Table 5. Execution times for HFBCCM to generate hash value of Images Image size (in Image size on HFBCCM Encryption/Decryption Time(s) in Seconds pixels) disk (in bytes) R=4 R=8 R=12 R=16 64 x bytes x bytes x bytes x bytes x bytes Table 6. Execution times for HFBCCM to generate hash value of texts Text size (in bits) HFBCCM Encryption/Decryption Time(s) in Seconds R=4 R=8 R=12 R= CONCLUSION In this paper we have proposed new hash function based on block cipher encryption algorithm that called BCCM. HFBCCM is very simple, flexible, and sensitive to any change that process different length of input messages to generate different hash values with length based on different chaotic maps. It can be used to calculate the correspondence hash value of input messages as texts or images with high confusion and diffusion properties. Several computer simulations and theoretical analysis have been performed that confirm the new hash function algorithm is satisfying the characteristics and conditions of cryptography hash

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