RMT Tick RMT. Application of the RMT-test on Real Data: Hash Function and Tick Data of Stock Prices

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1 RMT Tick 1,a) 1 1,b) RMT 2 SHA-1 MD5 RMT SHA TOPIX RMT Application of the RMT-test on Real Data: Hash Function and Tick Data of Stock Prices Xin Yang 1,a) Yuta Mikamori 1 Mieko Tanaka-Yamawaki 1,b) Abstract: The RMT-test, that we have proposed earlier as a tool to measure the randomness of long sequences, is applicable on various data types, integer or real, independent of the length of sequences. We have shown its effectiveness by comparing the degrees of randomness among pseudo-random generators as well as physical random generators. From the practical point of view, however, the advantage of this RMT-test resides in its applicability on real-world data, whose randomness level is far below the reach of the conventional randomness tests, such as NIST or JIS. In this paper, we present our result on applying the RMT-test on two examples: the choice of hash functions and the stock prices, assuming that the high randomness of the sequences indicates the high security level in both cases. In the first example, we compare two popular hash functions, the SHA-1 and the older MD5. The result of the RMT-test shows that the randomness of the output sequences of SHA-1 is indeed higher than the output of MD5. In the second example, we compare the performance of various stock prices in relation to their randomness and show that the stock prices of higher randomness perform better than the stocks of lower randomness, for the stocks in TOPIX500 in the year 2010 and 2011 based on their randomness in the previous three years, Keywords: Random matrix theory, RMT-test, Randomness, Hash value, Safetiness of Stocks 1 Tottori University, Graduate School of Engineering, Department of Information and Electronics a) yx0709@ike.tottori-u.ac.jp b) mieko@ike.tottori-u.ac.jp 1. c 2012 Information Processing Society of Japan 1

2 1 Fig. 1 RMT Example of the qualitative evaluation of the RMT-test (left: passed, right: failed) PLAB ENT NIST-SP DIEHARD [1]-[4] RMT [5] RMT 2. RMT 2.1 RMT Plerou [6][7][8] L N Random Matrix TheoryRMT N Q = L/N P RMT (λ) = Q (λ+ λ)(λ λ ) (1) 2πλ λ ± = ( ) ± (2) Q (1) RMT 1 RMT RMT RMT k m k = 1 N N λ k i (3) i=1 (1) µ k = E(λ k ) = λ+ λ λ k P RMT (λ)dλ (4) error k = m k /µ k 1 (5) N RMT k (trace) 1/N k m k = 1 N N (C k ) i,i = 1 N i=1 N λ k i (6) i=1 [9] k 2 LCG MT 3 5 N = 500 Q = 3 6L = k = 5 k = % 5 [10] N = Q = α = 0.05 N = 200L = 600Q = 36 5% RMT k = 6x = 56 5% c 2012 Information Processing Society of Japan 2

3 1 MD5 SHA-1 RMT 10 () Table 1 Quantitative evaluation of the output sequences from MD5 and SHA-1:The average(s.d.) over 10 samples 2 Fig. 2 Relation between a hash function and its hash value RMT 12 N = 100Q = 36 10% 5% N 200L 600 N L= 12 [11] 3. RMT MD5 SHA-1 RMT SHA-0 RIPEMDMD5 SHA-1 RIPEMD MD5[9] SHA-1[10] MD5 SHA-1 MD5 SHA-1 SHA-1 MD5 MD5 SHA MD SHA1 12 N = 200Q = % 1 MD5 SHA-1 MD5 k = 6 2σ k MD5 SHA (.0030) (.0016) (.0078) (.0039) (.0139) (.0069) (.0211) (.0106) (.0294) (.0154) 5%SHA-1 MD5 SHA % (p 1, p 2,..., p L ) r i = ln p i ln p i 1 (7) (r 1, r 2,..., r L 1 ) 7 LCG N = 500L = [λ, λ + ] 3 3 LCG () () Fig. 3 Evaluation of LCG by means of log-return with overlapping (left) and non-overlapping (right) c 2012 Information Processing Society of Japan 3

4 3 Table 3 TOPIX industrial sectors having high liquidity ,22**,25**,28**, ,3407,40**,41**,42**,44**,46** ,45** **, ,4902,60**,65**,66**,67**,68**,69**,7276,77**, ,61**,62**,63**,64**,6581,6586,70** **,58** ,6201,6902,6995,7003,7012,72**,73** **,84**,8522, **,2730,30**,3382,7453,7532,7649,8028,8184,82**,98**,9983, **,7459,80**,9832,9962, ,4666,88**,89**, ,46**,47**,8056,94**,96**,9766, ** 13 [12] 2 RMT 6 5% R=(r 1, r 2,..., r L 1 ) R 1 =(r 1, r 3,..., r L 1 ) R 2 =(r 2, r 4,..., r L 2 ) (L )R 1 R 2 6 R RMT ( 1 ) TOPIX500 3 TOPIX 33 TOPIX LCG MT ( () ()) Table 2 Errors in RMT-test for the overlapping(left) and nonoverlapping(right) log-return sequences k LCG MT LCG MT TOPIX500 1 RMT N L ( 2 ) RMT % c 2012 Information Processing Society of Japan 4

5 4 Table 4 4 (Q=4) :% Errors in RMT-test for the data length which is the top 4 of each industrial sector () () () (1.9) 4568(1.2) 8053(0.9) 4901(3.3) 4503(1.3) 2768(2.6) 4185(3.6) 4502(1.9) 8058(5.6) 4063(7.0) 4519(3.9) 8031(6.8) Table 5 Purchase all the stock in table 3 on January 4th2010,and calculate the log-return on the end of January 2010 () () () 4901(.032) 4568(1.2) 8053(.052) 6988(.019) 4503(1.3) 8031(-.002) 4185(-.074) 4502(1.9) 2768(-.041) 4063(-.121) 4519(3.9) 8058(-.058) 4 probability Fig time (H) (L) (Q=4) Probability of the profit and loss of stocks which is top 2 in each industrial sector for Q=4 H L 33.3% TOPIX H L ( 3 ) Q=4 RMT N N N 5 LCG (2 ) N = 100 N = 50 RMT N = 50Q = 8 N = 100 Q = 2 5 Q Q = Q = 2 H L Q = 3 H L H L Q = 5 L H Q = 6 L H Q = 4 2 H L 3 6 Q = 3 4 Q = 4 5 Q RMT (Q=2()Q=8()) Fig. 5 Comparison of Q=2 and Q=8 for the same data RMT H L 2 4 (r) r = 1 n (r 1 + r 2 + r r n ) (8) n H L A B 4. RMT (RMT c 2012 Information Processing Society of Japan 5

6 Log return 6 Fig. 6 Yen A time HL Average Log-return of each month: H: highest randomness, L: lowest randomness) A time H L B NIKKEI B [6] Laloux, L.,Cizeaux, P.,Bouchaud, J., Potters, M.: Noise Dressing of Financial Correlation Matrices,Physical Review L, Vol.83, pp (1998). [7] Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L.A.N. and Stanley, H.E.: Universal and Non-Universal Properties of Cross-Correlations in Financial Times Series,Physical Review L, Vol.83, pp (1999). [8] Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L.A.N. and Stanley, H.E.: Random Matrix Approach to Cross Correlation in Fianancial Data,Physical Review E, Vol.65, no (2002). [9] RMT NIST MPS2012-MPS-88(15). [10] Vol.2718pp (2012). [11] Black, J., Cochran, M. and Highland, T.: A Study of the MD5 Attacks: Insights and Improvements (online), jrblack/papers/md5efull.pdf ( ). [12] Locktyukhin, M. and Farrel, K.: Improving the Performance of the Secure Hash Algorithm(SHA-1) (online), ( ). [13] Tanaka-Yamawaki, M., Yang, X. and Itoi, R.: Moment Approach for Quantitative Evaluation of Randomness Based on RMT Formula, Intelligent Decision Technologies Smart Innovation, Systems and Technologies, Vol.16, pp (2012). 7 ( ) Fig. 7 NIKKEI Average chart( ) ) MD5 SHA-1 SHA-1 MD5 TOPIX [1] Hannes,L.: PLAB a System for Testing Random Numbers, Proceedings of the International Workshop Parallel Numerics 94,pp.89-99(1994). [2] ENT: A Pseudorandom Number Sequence Test Program(online), [3] NIST: A Statistical Test Suite (online), csrc.nist.gov/groups/st/toolkit/rng/documentationsoftware.html ( ). [4] Dieharder: A Random Number Test SuiteVersion (online), rgb/general/dieharder.php. [5] RMT Vol.5pp.1-8 (2012). c 2012 Information Processing Society of Japan 6

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