JOURNAL OF COMPUTERS, VOL. 7, NO. 7, JULY 2012 1671 A Fast New Cryptographic Hash Function Based on Integer Tent Mapping System Jiandong Liu Information Engineering College, Beijing Institute of Petrochemical Technology, Beijing, China [email protected] Xiahui Wang, Kai Yang, Chen Zhao Information Engineering College, Beijing Institute of Petrochemical Technology, Beijing, China {Wnagxiahui, Yangkai0212, zhao_chen}@bipt.edu.cn Abstract—This paper proposes a novel one-way Hash collisions by SHA-1” by Wang and Yu is another function which is based on the Coupled Integer Tent breakthrough in the hash functions history [11]. Mapping System and termed as THA (THA-160, THA-256). Based on the MD iteration structure [3], the The THA-160 compresses a message of arbitrary length into conventional Hash functions (MDx and SHA) have many a fingerprint of 160 bits, well the THA-256 compresses a common design guidelines. The designs of their mixed message of arbitrary length into a fingerprint of 256 bits. The algorithm adopts a piecewise message expansion operations for each round are very similar, and all of scheme. Compared with SHA-1 and SHA-256 message them adopt integer modulo addition and logic function; expansion, the message expansion scheme has enhanced the therefore, many Hash functions have been breached degree of nonlinear diffusion of the message expansion, and successively within a short period of time, which thus increased the computation efficiency. In addition, as indicates the defects in the design of the Hash functions. the major nonlinear component of compression function, the In recent years, in order to obtain more secure hash traditional logic functions are replaced by the integer tent functions, the research on constructing one-way hash map, and so the scheme has ideal properties of diffusion and function has been carried out and it has achieved progress confusion. Furthermore, the parallel iteration structure is by utilizing the sensitivity of the chaotic system to the adopted in the compression functions, which is advantageous to high speed parallel operation of software variation of the variables and parameters [12, 18, 17, 13, and hardware. Preliminary security testing indicates that, 8, 1, 2, 4, 7, 20]. But at present, the Hash function this Hash function has a high degree of security, and it can structure scheme, which is based on the Chaos Theory, is be realized easily with great rapidity. Therefore, it is an not acceptable to the world due to the degradation of ideal substitution for conventional Hash function. dynamic properties of digital chaotic system and the defects of the structure scheme itself [5, 14]. Index Terms—cryptography, Hash function, tent map, To work out the SHA-3, a new cryptographic Hash message expansion, diffusion algorithm standard, the analysis and assessment of the Hash algorithm Competition, held by National Institute of Standards and Technology (NIST), is quite popular all I. INTRODUCTION around the world. Most of the schemes of SHA-3 Hash functions are essentially easy to compute algorithm, accepted by NIST, are improvements of the functions that produce a digital fingerprint of messages or conventional algorithm, which are filled with the data and they are ubiquitous in today’s IT systems and elements of old cryptographic algorithm. However, the have a wide range of applications in security protocols paper combines the research result of the chaotic hash and schemes, such as providing software integrity, digital function with the conventional hash function structure signatures and password protection. Furthermore, hash method, improves the design standard and the message function algorithms have been used for constructing expansion mode of conventional hash function and pseudo-random number generators, key derivation eventually proposes a coupled integer tent mapping algorithms, message authentication codes, as well as system-based cryptographic one-way hashing algorithm stream ciphers and block ciphers. termed as THA(Tent map-based hash algorithm). THA- In recent years, an extensive attention has been given 160 compresses a message of arbitrary length into a to the design and analysis of hash functions within the fingerprint of 160 bits; THA-256 compresses a message field of cryptology. The attacks used to find the of arbitrary length into a fingerprint of 256 bits. The “breakthrough” collisions presented at the rump session algorithm adopts a piece-wise message expansion scheme. of Crypto’04 by Wang et al.[10]. At that conference, Compared with SHA-1 and SHA-256 message expansion, Wang announced the presence of collisions within for the message expansion scheme has enhanced the degree MD4, MD5, HAVAL-128 and RIPEMD. “Finding of nonlinear diffusion of the message expansion, and thus © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.7.1671-1680 1672 JOURNAL OF COMPUTERS, VOL. 7, NO. 7, JULY 2012 increases the computation efficiency. In addition, the P(2k-ε/2-1<x≤2k-1)=P(2k-ε/2≤x≤2k-1)=1-(1-ε/2k+1)； integer tent map replaces the traditional logic functions as k+1 k+1 k the major nonlinear component of compression function, P(xn+1<ε)= ε/2 +1-(1-ε/2 )= ε/2 and so the scheme has ideal properties of diffusion and confusion. Furthermore, the parallel iteration structure is When ε is odd， adopted in the compression functions, which is k+1 P(0≤x <ε/2-1/2)= P(0≤x <(ε-1)/2)=( ε-1)/2 ； advantageous to high speed parallel operation of software and hardware. P(2k-ε/2-1<x≤2k-1)= P(2k-(ε+1)/2≤x≤2k-1) =1-(1-(1-(ε+1)/ 2k+1))； II. INTEGER TENT MAP k+1 k+1 k A. The Standard Uniform Distribution Property of the P(xn+1<ε)= (ε-1)/2 +1-(1-(1+ε)/2 )= ε/2 Integer Tent Map Therefore, F obeys the standard uniform distribution The tent map is 1-D and piecewise-linear map as U[0, l]. Integer tent map has the nonlinear nature of follows [18]: rolled-out and folded-over, and its rolled-out characteristic will finally lead to the exponent separation x i1 of the adjacent points. While its folded-over characteristic , 0 xi1 , (1) keeps generating sequence with boundary, and makes the F : xi 1 x mapping irreversible. i1 , x 1. 1 i1 B. Realization approach of Integer Tent Map Suppose D =231, operator (? :) in C language could be This mapping is chaotic within the real number field 32 and one of its excellent characteristics is the uniform used in GF(2 ) to describe Eq.(2) as follows: distribution function. When the parameter α=0.5, real xn+1=xn<D?(xn<<1)+1:～xn <<1 (3) number operations will be turned into integer operations: Obviously, Eq. (3) could be realized through simple k1 logical judgment, logic reverse and shifting operations. If 2xn 1, xn [0,2 ) F : x n1 k1 (2) it is realized by assembly language or hardware, then the 2(l - xn ), xn [2 ,l] operation could be further simplified: Testing whether the highest digit is 0 or not, if it is 0, then shift one digit to In which, l＝2k-1 and the map F is termed as Integer left and add 1; otherwise, shift one digit left after bitwise Tent Map. The multiplication (division) within the complement operation. Simple as the operation is, it will integer fields in the formula (1) is turned into shifting be translated into jump instruction if we come cross if function in the formula (2). block here. Considering the effect of the jump instruction Definition: if the possible values of the random to the pipeline efficiency of modern Super Pipeline CPU, variable X in the set of integer are 0, 1… l and they are the water flow could be blocked whenever the branch distributed as: forecast fails and thus prolong the instruction period. To 1 avoid if jump, we adopt another equivalent form of Eq. (2) pk P(X k) 32 (described by C language operator): 2 (4) Then, the random variable X will be submitted to the xn+1=(( xn ^(-( xn >>31)))<<1)|(!( xn >>31)) uniform distribution in the integer [0, 1]. It can be seen that, simple instructions are solely used Theorem: The distribution of y=F(x) for integer x∈ in Eq. (4) which completely avoids the jumping operation. [0,l] of randomly chosen is the standard uniform C. Coupled Integer Tent Mapping System distribution U[0, l]． Proof: Considering that x is randomly chosen from [0, Since the Integer Tent Maps are defined within the integer set, the iterative sequence generated from the l], x obeys the standard uniform distribution U[0,l]，i.e. formula must turn to periodical form. In the iteration process, the disturbance is applied to breaking the ∀ ∈(0,l) P(xn ) 2k inherent cycle of the Integer Tent Maps to improve the Also as ε<l, so ε≤2k-2, then: Ergodicity of the Integer Tent Maps. As a result, the iteration series of the systems are randomized. Thus, we P(xn+1<ε)=P(F(x) <ε) build the following coupled mapping system model: = P(F(x) <ε,0≤x<2k-1)+ P(F(x) <ε,2k-1≤x≤l) =P(2x+1<ε,0≤x<2k-1)+ P(2(l-x)<ε,2k-1≤x≤l) ( i+1 ) ( i ) ( i ) x j =F( x j ) + (F ( x j-1 mod p )<<<21) =P(2x+1<ε,0≤2x≤2k-2)+P(x>2k-ε/2-1,2k-1≤x≤2k-1) k k = P(0≤x <ε/2-1/2) +P(2 -ε/2-1<x≤2 -1) ( i ) +(F ( x j+1 mod p )<<<11) (5) When ε is even： The operators in the formula are described in the P(0≤x <ε/2-1/2)= P(0≤x <ε/2)= ε/2k+1； following text.