外文原文1-一个基于修改后的Logistics映射的图片加密方案

2011 Fourth International Workshop on Chaos-Fractals Theories and ApplicationsAn Image Encryption Scheme Based on Modified Logistic MapYue SunSchool of Electronics InformationHangzhou Dianzi UniversityHangzhou, Chinae-mail: AbstractIn order to improve the security of chaoticencryption algorithm, a modified chaotic map, which is basedon the Logistic map, is proposed in this paper. Compared withoriginal logistic map，the proposed map makes it always bechaotic, and expands the iteration range from original (0, 1) to(0, 4) (0.25). We designed an encryption scheme based onthe proposed map for implementing image encryption. Somesimulation results show that the modified Logistic mappossesses bigger key space, faster sequence generation rate andfaster encryption speed compared with the original logistic map.Keywords-chaos; modified logistic map; encryptionGuangyi WangSchool of Electronics InformationHangzhou Dianzi UniversityHangzhou, Chinae-mail: I. INTRODUCTIONIn recent years, digital information has been widely usedin many fields. Any of us has a lot of private multimediasources and these sources should be protected from any kindof unauthorized manipulation.Chaos has nonlinear dynamic behaviou and chaoticsequence is paseudorandom and sensitive to the initialconditions 1. Because of the nature of chaotic sequence, itis widely used in cryptology, control and communications2-7. Discrete chaotic system has high efficiency but lowsecurity because the range of its key space is smal， whilecontinuous chaotic system has higher security, but lowefficiency, because its calculation is complex.This paper proposes a modified Logistic map, whichcontains two parameters and can expand its onto mappingfrom original 0 - 1 to 0 - 40.25. This is important forexpanding key space of chaotic sequence and enhancing rateof change of chaotic signal. An encryption algorithm isdesigned based on this chaotic map and some analysis ispresented to show its good efficiency.II. CHAOTIC SEQUENCE GENERATIONThe one-dimensional chaotic maps have been widely usedrecently in various information encryptions because of itshigh-level simplicity and efficiency. But they have manydefects, for example small key space and weak security. Wemade some modifications on logistic map to overcome theseweaknesses.A. A new modified Logistic MapLogistic Map is defined asWhere is a constant parameter, x n ( 0 ,1) and (0, 4) .When 3 . 6 , 4 , the Logistic map has a chaotic attractor,Where is a constant parameter and xn 0, 4） .（The coefficient of the first term in equation (2) guarantee thatthe modified Logistic map is always chaotic, while thecoefficient , of the second term make its map range to be4 .Fig. 1 and Fig. 2 show the bifurcation diagrams forLogistic Map and the modified Logistic map respectively,from which we can see the differences between the two maps.In logistic map only when (3.6,4 it is chaotic, but in themodified Logistic map, under the parameter is set to beand when 4 the map is an onto mapping. Separatingright side of Logistic map into two parts, we find that theparameter of first term determines the properties of the mapand second parameter determines the range of iterationvalues of this map. Hence, taking the firstparameter 4 and the second parameter 1/ , weobtain a modified Logistic map, and it is given byx n 1 4 x n x n2 , n 1 , 2 , 3 .(2)106 , it always keeps chaotic when (0,106 . From Fig.s1 and 2, we can also see that the value range of iteration for6the modified map is 4 10 .Figure 1. The bifurcation diagram of the logistic mapx n 1 x n (1 x n )， n 1, 2 ,3 （ 1） Figure 2. The bifurcation diagram of modified systemB. Chaotic Sequence Generating MethodEven if the range of iteration for modified Logistic maphas been extended, but there are some drawbacks on the179978-0-7695-4560-8/11 $26.00 2011 IEEEDOI 10.1109/IWCFTA.2011.56sequence generation for the map. First, the calculation offloating point numbers is complex and some optimizationneed to be done to fix this problem. Second, the range of thesequence can not be large enough when the CPU is low end,such as 16bit or 8bit. So we need to try to get a larger rangethan the CPUs bit width.In equation (2), the multiplication and the divisionbecome into bit-computation when 2m and all thecomputations becomes integer computations. As we knowthe speed of bit-computation is much faster thanmultiplication or division, and the speed of integercomputation is much faster than float computation.We use four formulas to generate chaotic sequences, eachof them has different parameter from others. Theprocedure of sequence generation is shown in Fig. 3.1). An array, whose elements values are equal to theirindex numbers, is needed. For example the array is a 256 ,the values of its elements are: 2). An array is filled with the low bytes of chaoticsequence and the length is 256. For example the array isb256, the values of its elements are:a i i , i 0 ,1, 2 , , 255 （ 3）key. Replace ai with ak. The procedure can be describedin Fig. 4:b i i | 0 i 255 ;(4)3). A variable k bi is needed to generate encryptionFigure 3. Procedure of the sequence generationThe result of first formula recursion is set as initial valuesof the next formula. We push the low 8 bit of every result ofthe recursions into the chaotic sequence. It is like that therange of chaotic sequence is extended indirectly. And moreformulas can be united into a large system to generate thesequence with bigger cycle.III. ALGORITHM ANALYSISThere are many cryptography methods in history andsome of them are still popular with us today. The principle ofthese cryptographies includes a lot of complex calculationsto improve the security. DES 8 and AES are the examplesof them. But if the target to deal with is very large, it willtake a very long time for these cryptographies. We proposedan encryption algorithm with good efficiency based on themodified chaotic system.A. Algorithm principlesAs we know any kind of information can be counted inbyte and any operation on the information is done by dealingwith bytes. The range of one byte is 0,255 and there are256 different values. The value of key should be able toindicate different byte with different key value. So the keyshould has 256 different values. The steps to get encryptionkeys are as below:Figure 4. Procedure of the encryption keyThe values of encryption keys distribute in 0， and255the length of this key is 256. Any element of this key array isdifferent from others. For example the generated encryptionkey is as Table:TABLE I.1245.103.ENCRYPTION KEY20071.250129.The decryption keys are generated from the encryptingkey. We can get the decrypting key as Table II if given theencrypting key as Table I.TABLE II.71200.90100.DECRETION KEY129250.2451.The procedure of generating decryption key can bedescribed as Fig. 5.180IV. APPLICATION IN IMAGE ENCRYPTIONNow we encrypt the image using both the chaoticsequences. The Fig. 6 gives the original image and itsgradation histogram. Fig. 7 gives encrypted image by logisticchaotic sequence and its gradation histogram. Fig. 8 showsthe encrypted image by modified chaotic sequence and itsgradation histogram. There is a little difference between Fig.7 and Fig. 8. The modified chaotic sequence is generated byinteger computation and the logistic chaotic sequence isgenerated by float computation. The key space of the formeris much larger than the later, and the speed of formersequence is faster than the later sequence too.Figure 5. Procedure of the decryption keyNow that, encryption is just replacing each byte of thetarget data with the corresponding value in the encryptingkey array. So the encrypted information should be as TableIII and the decrypted information should be as Table IV:TABLE III.71 90TABLE IV.200 100ENCRYPTED DATA245 129 3 Figure 6. The original imageDECRYPTED DATA1 250 10A variable can be set to control the frequency of the keyschanging. For example, if the variable is set to 200, it meansthat the keys change every 200 bytes.B. Performance analysisThe Tableshows the speed of generating chaoticsequences from both the maps, which the value of parameter14takes 2 in modified logistic map and the value ofparameter of Logistic map takes 3.6 . From Table V wecan see that the speed of key generation from the former ismuch faster than that from the later.TABLE V.Length(byte)Logistic(us)Modified(us)10000142.187.2SPEED OF SEQUENCE GENERATING1000001410.6890.5100000014384.28859.610000000144068.892003.4Figure 8. Encrypted by modified sequenceFigure 7. Encrypted by logistic sequenceTable shows the tested results of encryption speed forthe DES and the proposed algorithm in this paper. Theconfiguration of the testing computer is 32bit/1.0G/2.99Ghz.It can be seen from Table that the speed of the proposedalgorithm is much faster than the speed of DES.TABLE VI.Length(byte)Logistic(ms)Modified(ms)102411.5200.068V. CONCLUSIONSPEED OF SEQUENCE GENERATING1024*2562950.02317.4021024*5125790.24035.0101024*102411800.30069.810A modified logistic map is proposed in this paper forgenerating good chaotic sequences for image encryption.The proposed map makes it always to be chaotic, andexpands the iteration range of values from original (0, 1) to(0, 4).A scheme for image encryption is designed based on theproposed map. Some simulation results show that the181modified chaotic system has advanced characters comparedwith original logistic system, such as bigger key space, fastersequence generation rate, faster encryption speed and highersafety. Hence, the proposed modified Logistic map has thevery good application prospect as the pseudorandomsequences for secret communications and other informationencryption fields.ACKNOWLEDGMENTThis work was supported by the National Nature ScienceFoundation of China. (Grant No 60971046).REFERENCES1 Xuefeng Zhang, Jiulun Fan “Extended logistic chaotic sequence andits performance analysis,” Tsinghua science and technology , volume12, July 2007,pp156-161.2345678