Parallel Implementation of Proposed One Way Hash Function

Transcription

1 UDC: : Parallel Implementation of Proposed One Way Hash Function Berisha A 1 1 University of Prishtina, Faculty of Mathematics and Natural Sciences, Kosovo artan.berisha@uni-pr.edu Abstract: One ay hash functions are idely used for data integrity, digital signatures and other forms of authe ntication. In this paper, e propose a non linear function for one-ay hash function [1] and implementation of its parallel version. An existing sequential implementation of the algorithm [1] is parallelized on multi-core microprocessors. At the end comparative analyses are done against existing sequential algorithm and its parallel version in different numbers of cores and different amount of data. Key ords : Parallel algorithms, Hill cipher technique, Non-invertible matrix, hash algorithm, One-ay hash function, plaintext, integrity. Introduction The need for data security and data integrity led to development of encryption algorithms, hash functions. This as the reason that Cryptography is developed in many phases: Classical Cryptography (Egyptian hieroglyph, Caesar cipher), Middle Age Cryptography (Frequency analysis), Cryptography from 19-th century to World War II (Polyalphabetic cipher), World War II Cryptography (Enigma), Modern Cryptography (Data Encryption Standard, Public Key, Hashing) [2]. Cryptography is a area of Cryptology (Figure 1), here thus itself is divided to symmetric cryptography, asymmetric cryptography, cryptographic protocols and hash functions. Cryptology Cryptography Symmetric Asymmetric Protocols Hash Functions Figure 1. Cryptography main areas Cryptography intersects disciplines of Mathematics Computer Science and Technical Sciences. The main purpose of Cryptography is securing and enabling communication beteen to parties and protecting the data, sensitive data or information from outside attacks. In this context cryptography is based on four specific security requirements: authentication, integrity, pr ivacy and non-repudiation. So the role of cryptography is not only data protection but also provide authentication, there are generally three cryptosystems used to achieve this: symmetric algorithms, asymmetric algorithms and hash algorithms. While symmetric and asymmetric cryptosystem is used for enciphering and deciphering the hash function are used for authentication and data integrity [1]. UNIVERSI Journal (.universi.mk) Page 112

2 Hash Function Definition In general the hash function is H: {0,1}^* {0,1}^n for some n. In order for H to be a hash function it is needed some basic properties. It can applied to any block size of data, produce a fixed length output and [3] Preimage resistant, for given y {0,1}^n it is hard to find x {0,1}^n such that H(x)=y. Second preimage resistant, for given x {0,1}^* it is hard to find x^' {0,1}^*, x x^' such that H(x)=H(x^'). Collision resistant, it is hard to find x,x^' {0,1}^*, such x x^' and H(x)=H(x^'). One-ay hash function is very important cryptographic primitive it is used for data integrity and authentication. The output length of value of hash function is fixed and input is variable length. Most usually cryptographic hash functions used today are SHA-2, SHA-3 here MD5 is broken [4] and it as used to break SSL. Most of the hash functions have to give an output greater than 160 bit, this is because of the birthday attack, hich says that to find a collision for a cryptographic hash function ith n bit output ith probability 50% e expect n/2 input values [5]. Noadays ith the computing resources capability all algorithms that have more than 2^80 input cases for brute force attack are considered secure, because of this the output of hash function must be greater than 160 bit to be secure (it means 2^(160/2)=2^80). Also NIST in its Secure Hash Standard uses 160 bit hash value, this makes even harder for the birthday attack, it requires 2^80 random text to find to hash codes ith same value [4]. Permutation Matrices A square matrix is called permutation matrix if each ro and column of the matrix has exactly one 1 and all other entries are 0 [6]. By definition π: 1,2,, m {1,2,, m} or π = 1 2 π[1] π(2) m π(m) here π 1, π(2),, π(m) {1,2,, m}. The permutation above can be ritten as matrix (permutation matrix): P ij = 1, if i = π(j) 0, otherise The number of permutation matrices of size n is n! [6]. Permutation matrices are a class of invertible matrices in GF(2). In GF(2) each element is either 0 or 1, addition is the binary exclusive-or operator (denoted ) and multiplication is the binary and operator. The arithmetic of ro and column in permutation matrix is performed over the commutative ring Z/2Z [16]. Permutation matrices have properties that determinant is 1 or -1 and the inverse is the transposed of the matrix. Furthermore the product of to permutation matrices is a permutation matrix. 1) det P = 1 or 1 2) PP T = P T P = I, here I is identity matrix UNIVERSI Journal (.universi.mk) Page 113

3 When a permutation matrix P is multiplied ith a matrix M from the left it ill permute the ros of M (the elements of column vector), hen P is multiplied ith M from the right it ill permute the columns of M (the elements of a ro vector) [7]. As e can see the permutation matrices are invertible (the property above), but the folloing lemma ill give a result for some property of these matrices [8]. When e refer to a matrix P, that means that P, is a square matrix in GF(2) ith ros and columns. Lemma 1: If some matrix P, has precisely ones, then P is invertible iff it is a permutation matrix [8]. Lemma 2: Let P 1 and P 2 be permutation matrices. The sum P 1 +P 2 is not invertible [8]. Proof: Let P be the sum of to permutation matrices P=P 1 +P 2. We have to cases:i: Suposse there exists i and j such that P 1 (i, j)= P 2 i, j = 1, then the ith ro of P contains all zeros, so P is not invertible. II: Assume that there are no such i and j as in first case, then P ill have precisely to ones each ro and column. By induction it is proven that such matrices are not invertible [8]. The permutation matrices ill be generated by usin Fisher Yates algorithm, the basic of this algorithm is generating a random permutation of numbers from 1 to m, and it goes as follos [9] 1. Write don the numbers from 1 to m 2. Pick a random number k beteen one and the number of unstruck numbers remaining (inclusive) 3. Counting from the lo end, strike out the kth number not yet struck out, and rite it don elsehere. 4. Repeat from step 2 until all numbers have been struck out. 5. The sequence of numbers ritten don in step 3 is no a random permutation of the original numbers. The code implemented in Java [10]. Proposed Parallel Algorithm In [1] authors proposed a one ay hash function and gain results against SHA-1, here e are propose a parallel version of this sequential one ay hash function. The algorithm from [1] as based on CBC (Cipher Block Chaining), hich ill not allo to do the computation in parallel, so e propose a CTR (Counter mode) hich can be parallelized easily. Belo is shon the schema (Figure 2.) of this proposed parallel algorithm. in UNIVERSI Journal (.universi.mk) Page 114

4 B 1 IV CTR 1 H B 2 IV CTR 2 H B N IV CTR N H Figure 2. Schema of proposed parallel version of one ay hash algorithm The IV is initial value and CTR i is a counter value hich increases in every step, first e concatenate IV ith CTR i than e use H function (permutation matrix), and afterards the result is XOR-ed ith first block of message. The output ill be used in second round of hashing, by XOR-ing ith next block (result) from second block of message. This is easily implemented in Java because e stop if the number of blocks is one. The final value is the hash value of input message. Comparative Analysis H N Comparative analyses are done against our proposed parallel algorithm and sequential version proposed by [1]. We implemented our proposed parallel algorithm in Java. For our experiment e used a test bed ith 4 core ith 3.4 GHz. The effect of changing file size for calculating hash value as chosen. Belo is table ith our gained results after the experiment (Table 1.), as e can see the proposed parallel algorithm is faster for the file size greater than 8 KB, and for large amount of data it is 14 times faster than sequential model. From 1 KB to 4 KB the proposed parallel algorithm is sloer than sequential algorithm, it is because of tree structure of our parallel algorithm, it can not take advantage of its parallel compuiting against sequential computing. Where for data amount larger than 4 KB it is faster and reaches the speedup to 14. UNIVERSI Journal (.universi.mk) Page 115

5 Table 1. Time (ms) for calculating hash value ith proposed parallel model and sequential model for different file size File size (bit) Calculating hash Calculating hash value (ms)- sequential model value (ms)- parallel model Speedup N/A N/A N/A N/A N/A N/A Hashing sequential (ms) Hashing paralel (ms) Figure 3. Graphical vie of performance of proposed parallel algorithm against sequential algorithm UNIVERSI Journal (.universi.mk) Page 116

6 Conclusion and Future Work In this paper e proposed a parallel model of earlier proposed sequential model. For a small amount of data (1 KB 4 KB) this proposed parallel model is not faster than sequential model, but for larger amount of data (greater than 4 KB) it is faster. For larger amount of data the speedup increases exponentially. In future e need to see for non invertible matrices in GF(p m ) and to speed up the calculating process of hash value in Graphical Processing Unit. References 1. A Class of Non Invertible Matrices in GF (2) for Practical One Way Hash Algorithm. Artan Berisha, Behar Baxhaku, Artan Alidema. 18, Ne York : International Journal of Computer Applications, 2012, Vol Bauer, Craig. Secret History - The story of Cryptology. s.l. : CRC Press, Stallings, William. Cryptography and Netork Security, Principles and Practice. s.l.:prentoice Hall, Schneir, Bruce. Applied Cryptography. s.l. : Wiley Computer Publishing, Christof Paar, Jan Pelzl. Understanding Cryptography. s.l. : Springer, Zhang, Fuzhen. Matrix Theory. s.l. : Springer, Thoma H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction To Algorithms : MIT Press. 8. James S. Plank, Adam L. Buchsbaum. Some Class of Invertible Matrices in GF(2). s.l. : University of Tennessee, Statistical tables for biological, agricultural and medical research. Fisher, R.A, Yates, F. London : Oliver&Boyd Java, Shuffle an array or a list - Algorithm in. [Online] UNIVERSI Journal (.universi.mk) Page 117