CryptrecReport.dvi
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- こうだい しもとり
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2 1 NIST NIST Special Publication [1, 2] SP DIEHARD [27] SP Ver DFT Lempel-Ziv [3, 4, 5] SP DFT Lempel-Ziv 1
3 2 SP NIST SP800-22[1, 2] NISTNational Institute of Standards and Technology AESAdvanced Encryption Standard 2.1 SP Frequency (Monobit) Test 2. Frequency Test within a Block 3. Runs Test 4. Test for the Longest Run of Ones in a Block 5.2 Binary Matrix Rank Test 6.DFT Discrete Fourier Transform (Spectral) Test 7. Non-overlapping Template Matching Test 8. Overlapping Template Matching Test 9.Maurer Maurer's Universal Statistical Test 10.Lempel-Ziv Lempel-Ziv Compression Test 11. Linear Complexity Test 12. Serial Test 13. Approximate Entropy Test 2
4 14. Cumulative Sums (Cusum) Test 15. Random Excursions Test 16. Random Excursions Variant Test 2.2 NIST p-value p-value 2 p-value 0.01 (NIST 1000 ) 1.p-value 2.p-value p-value [0, 1) [0, 1) 10 2 p-value m 0.01 p-value 0:99 ± 3 s 0:99 0:01 m (2.1) 3
5 3 SP Generator-Using SHA-1 NIST Generator-Using SHA-1 SP
6 " 0 1 n S n n n Step1 0 1 " =(" 1 ;" 2 ; ;" n ) (3.1) "-1""+1" X =(X 1 ;X 2 ; ;X n ) X i =2" i 1 (1» i» n) (3.1) Step2 S n = X 1 + X X n (3.2) Step3 S n μ =0 ff 2 = n S n (3.3) p-value p-value 0.01 p-value = erfc ψ! jsn j p 2n (3.3) 5
7 3.1.5 De Moivre-Laplace theorem S n 0 n S n 0 1 NIST SP Sn n =1001; 00010; ; 0001; 000; 000 (3.2) S n μ =0 ff 2 = n 3.1 NIST G Using SHA
8 7
9 8
10 3.1: S n 9
11
12 M 1 M/ M: n: ":0 1 " =(" 1 ;" 2 ; " n ) χ 2 (obs):m 1 M/2 χ NIST 100 M 20;M > 0:01n; N < Step1 N = bn=mc Step2 1» i» N i 1 ß i ß i = Step3 χ 2 χ 2 (obs) =4M P M j=1 " (i 1)M +j M NX i=1 (3.4) (ß i 1 2 )2 (3.5) 11
13 Step4 χ 2 (obs) χ 2 χ 2 (obs) χ 2 ( 0.01) P-value P value = igamc( N 2 ; χ2 (obs) ) (3.6) 2 P-value P-value M N 1 ß i (3.5) N χ χ 2 (obs) χ 2 NIST N χ 2 NIST G Using SHA
14 1:n=100 M=20 N=5 2:n=1,000 M=20 N=50 13
15 3:n=10,000 M=200 N=50 4:n=100,000 M=2000 N=50 14
16 5:n=1,000,000 M=20000 N=
17 k k n " "=" 1," 2,," n V n (obs) NIST n 100bit Step1 ß ß np " j j=1 n (3.7) Step2 Runs Test jß 1 j fi Frequency test 2 Runs Test 0 Frequency test ß 1/2 fi = p 2 n Step3 V n (obs) X n 1 V n (obs) =f r(k)g +1 (3.8) " k = " k+1 r(k)=0" k 6= " k+1 r(k)=1 (3.8), 1 V n (obs) k=1 16
18 Step4 V n (obs) V n (obs) 0.01P value P value = erfcf jv n(obs) 2nß(1 ß)j 2 p g (3.9) 2nß(1 ß) P value V n μ2nß(1 ß) ff 2 2 p nß(1 ß) lim P Vn 2nß(1 ß) n!1 2 p» z nß(1 ß) Φ(z) = Z z 1 =Φ(z) (3.10) 1 p 2ß e x2 2 dx (3.11) V n NIST Φ(z) erf c(z) = Z 1 z 2 p ß e x2 dx (3.12) P value = erfc( z p 2 ) (3.13) (3.9) Vn(obs) 100,1000,10 4,10 5,10 6 bit V n (obs) NIST G-Using SHA
19 u 3.2: n=100bit u 3.3: n=1000bit 18
20 u 3.4: n=10 4 bit u 3.5: n=10 5 bit 19
21 u 3.6: n=10 6 bit
22 M n " "=" 1," 2,," n M 1 N χ 2 (obs) n M 128» n » n » n M n NIST n 128bit Step1 n M N = b n M c Step2 2 ν i ν i M=8 M=128 M=10 4 ν 0»1»4»10 ν ν ν ν ν ν
23 2 M ν i Step3 χ 2 (obs) χ 2 (obs) = KX i=0 (ν i Nß i ) 2 Nß i (3.14) K, ß i 3 M K ß i M K ß i 8 3 ν 0 ß 0 = ν 0 ß 0 = ν 1 ß 1 = ν 1 ß 1 = ν 2 ß 2 = ν 2 ß 2 = ν 3 ß 3 = ν 3 ß 3 = ν 0 ß 0 = ν 4 ß 4 = ν 1 ß 1 = ν 5 ß 5 = ν 2 ß 2 = ν 6 ß 6 = ν 3 ß 3 = ν 4 ß 4 = ν 5 ß 5 = M K, ß i Step4 χ 2 (obs) χ 2 (obs) 0.01P value P value = igamcf K 2 ; χ2 (obs) g (3.15) 2 P value ν K+1 ν 0 ν K M r M r (ν» ) U = min M r +1; h M j(m+1) M r P (ν» mjr) = 1 ψ M r i! U X j=0 ψ!ψ M r +1 M j(m +1) ( 1) j j M r 22!
24 P (ν» m) = MX r=0 ψ M r! P (ν» mjr) 1 2 M (3.16) (3.16) χ 2 ν ß (3.14) K P value ψ K igamc 2 ; χ2 (obs) 2! = 1 ( K 2 )2 K 2 Z 1 χ 2 (obs) e u 2 u K 2 1 du (3.17) ν ß 3 ν i N ν 0 = ν i n N 10 6 bit NIST G-Using SHA-1 23
25 Š Šš wš 3.7: n=128bit Š Šš wš 3.8: n=8192bit 24
26 Š Šš wš 3.9: n=10 6 bit χ 2 (obs) n 128, 8192, 10 6 bit χ 2 (obs) NIST G-Using SHA
27 3.10: n=100bit «3.11: n=8192bit 26
28 «3.12: n=10 6 bit
29 n: ":0 1 " =(" 1 ;" 2 ; " n ) M: M=32 Q: Q=32 χ 2 (obs): χ NIST 38912bit Step1 j k n N = M Q MQ Q M Q Step2 R l (1» l» N) Step3 F M : R l = M F M 1:R l = M 1 N F M F M 1 : 28
30 Step4 χ 2 χ 2 (obs) = (F M 0:2888N) 2 0:2888N + (F M 1 0:5776N) 2 + (N F M F M 1 0:1336N) 2 0:5776N 0:1336N (3.18) Step5 χ 2 (obs) χ 2 χ 2 (obs) χ 2 ( 0.01) P-value P value = e χ2 (obs)=2 (3.19) P-value Kovalenko [9], Marsaglia Tsay [10] M Q 2 r =0; 1; 2; ;m r 1 Y p r =2 r(q+m r) MQ i=1 (1 2 i Q )(1 2 i M ) 1 2 i r (3.20) M N 32 p M ß 1Y j=1 [1 1 ]=0:2888 ::: (3.21) 2j p M 1 ß 2p m ß 0:5776 ::: (3.22) p M 2 ß 4p M 9 ß 0:1284 ::: (3.23) j k M 10 (» 0:05) n N = N M MQ M-1 M-2 3 F M ;F M 1 ;N F M FM 1 (3.18) χ 2 (obs) 2 χ 2 χ 2 29
31 3.5.6 χ 2 (obs) χ 2 NIST 2 χ 2 NIST G Using SHA
32 1:38912bit 2:100,000bit 31
33 3:1,000,000bit
34 3.6 DFT DFT 0 1 ±1 Discrete Fourier Transform " :0 1 n : d : n NIST 1000bit Step1 0 1 n " = " 1 ;" 2 ;:::;" n (" i 20; 1;i = 1; 2; ;n) 1 1 X = x 1 ;x 2 ; ;xx n i =2" i 1 Step2 Step1 X f j = nx k=1 x k exp ψ i 2ß(k 1)j n! i p 1 (3.24) Discrete Fourier Transform f j j = 0; 1; ;n 1f j = μ f n j j =0; n 2 f 0 ; ;f b n 1 2 c Step3 f j jf j j 95 T = p 3n T N 0 =0:95n=2 Step4 jf j j T N 1 N 1 N 0 d = q n(0:95)(0:05)=2 33 (3.25)
35 d Step5 d d 0.01P value P value = erfc ψ! jdj p 2 (3.26) P value DFT [3]Kim [4] [5] d n bit d NIST G-Using SHA
36 3.13: n =10 3 bit 3.14: n =10 4 bit 35
37 3.15: n =10 5 bit 3.16: n =10 6 bit 36
38 3.6.7 NIST d 2 [3, 4] 1 T f j jf j j F q (x) =1 exp( x2 ) n p T p = (ln 0:05)n = 2: n T NIST T = 3n d 1 d n = : dn =10 6 bit 3.17 d d 0 (0:7) 2 N 1 50 [5] 37
39 N m N χ " 0 1 n m B M N N =2 3 =8 χ 2 (obs) χ m 9 10 n 2 20 =1; 048; Step1 " M N Step2 m B 1 m j(1» j» N) B W j Step3 W j (3.27) μ(3.28) ff 2 χ 2 χ 2 (obs) (3.29) μ = M m +1 2 m (3.27) 38
40 1 ff 2 = M 2 2m 1 m 2 2m χ 2 (obs) = NX j=1 (W j μ) 2 (3.28) ff 2 (3.29) Step4 χ 2 (obs) N χ 2 χ 2 (obs) (3.30) p-value p-value 0.01 ψ N p-value = igamc 2 ; χ 2 (obs) 2! (3.30) Step2 m W j B =(" 0 1;" 0 2; ;" 0 m) B 6= n j; 1» j» m 1;" 0 j+k = " 0 k;k =1; 2; ;m j o (3.31) M W j (3.27)(3.28) μ ff 2 (3.29) χ 2 (obs) N χ n 2 20 =1; 048; 576 m 9 10 B W j (3.29) χ 2 (obs) m =9 B " " m =10" " NIST G Using SHA W j W j (3.27)(3.28) μ ff
41 χ 2 (obs) χ 2 (obs) N =8 χ
42 3.18: W j 41
43 3.19: χ 2 (obs) χ 2 42
44
45 N m K +1=6 χ " 0 1 n m B m 1 K K =5 M M =1032 N χ 2 (obs) χ m 9 10 n 1,000, Step1 " M N Step2 m B 1 j(1» j» N) W j Step3 0 x 0 1 x 1 2 x 2 3 x 3 4 x 4 5 x 5 Step2 W j x i (0» i» 5) ν i 44
46 Step4 x i ß i (3.32) ß 0 = e 9 ß 1 = 2 e ß 2 = e ( +2) 8 ψ! ß 3 = e 2 8 ψ ! ß 4 = e ß 5 =1 (ß 0 + ß 1 + ß 2 + ß 3 + ß 4 ) >= >; (3.32) = 2 (3.33) = M m +1 2 m (3.34) Step5 χ 2 χ 2 (obs) (3.35) χ 2 (obs) = 5X i=0 (ν i Nß i ) 2 Nß i (3.35) Step6 χ 2 (obs) K =5χ 2 χ 2 (obs) (3.36) p-valuep-value 0.01 ψ 5 p-value = igamc 2 ; χ 2 (obs) 2! (3.36) W j [11] U P (U = u) P (U = u) = e 2 u ux l=1 ψ u 1 l 1! l l! (3.37) 45
47 P (U =0),P (U = 1),,P (U =4) P (U =0)=e 9 P (U =1)= 2 e P (U =2)= e ( +2) 8 ψ! P (U =3)= e 2 8 ψ ! P (U =4)= e >= >; (3.38) ß 0 ;ß 1 ; ;ß 5 (3.32) W j K +1=6 (3.35) χ 2 (obs) 6 1=5 χ χ 2 (obs) n 1; 000; 000 m 9 10 ß i (3.32) (3.35) χ 2 χ 2 (obs) K =5 χ NIST G Using SHA
48 3.20: ß i 47
49 3.21: χ 2 (obs) 5 χ 2 48
50
51 3.9 Maurer Maurer L " 0 1 n L Q K f n L 6» L» 16 Q Q =10 2 L n 6» L» 15 (10 2 L L )L» n<(10 2 L L+1 )(L +1) L =16 (10 2 L L )L» n Step1 " L Q K Q + K = bn=lc Step2 T j 0 i L 2 j(0» j» 2 L 1) T j = i (1» i» Q) 50
52 3.1: L n Q =10 2 L 6 387,840904, ,9602,068, ,068,4804,654, ,654,08010,342, ,342,40022,753, ,753,28049,643, ,643,520107,560, ,560,960231,669, ,669,760496,435, ,435,2001,059,061, ,059,061, T j j 2 L Step3 sum 0 i L 2 j sum = sum + log 2 (i T j ) (3.39) T j = i (Q +1» i» Q + K) (3.39) sum Q =0 (3.39) 0 sum i = sum i 1 +log 2 (i T j ) (Q +1» i» Q + K) (3:39) 0 Step4 f n = sum K (3.40) Step5 f n 3.2 μ(l)(3.41),(3.42) ff f n (3.43) p-value p-value
53 3.2: L μ(l) V (L) L μ(l) V (L) ff = c s V (L) K c =0:7 0:8 4+ L + 32 L p-value = erfc K 3=L 15 ψfi fifififi fi f n μ(l) fifififi! p 2ff (3.41) (3.42) (3.43) f n μ(l) [12] 1X μ(l) =2 L (1 2 L ) i 1 log 2 i (3.44) i=1 f n ff (3.41) Coron Naccache c [13] c =0:7 0:8 L + 1:6+ 12:8 L K 4=L (3.45) (3.45) (3.42) 52
54 3.9.6 fn L 6 7 (3.40) f n 3.2 μ(l)(3.41),(3.42) ff 3.22 L 6 f n (3.45) (3.42) ff (3.45) ff NIST G Using SHA-1 n L =6 500,000L =7 1,000, : ff L =6 (3.42) (3.45) ff
55 3.22: f n 54
56 3.23: (3.45) (L=6) 55
57 3.9.7 (3.42) (3.45) 56
58 3.10 Lempel-Ziv Lempel-Ziv 0 1 Lempel-Ziv " :01 n : W (n) : n NIST 10 6 bit Step1 0 1 n " = " 1 ;" 2 ;:::;" n (" i 20; 1;i = 1; 2; ;n) W (n) W (n) E[W (n)] Step2 W (n) ff[w (n)] 0.01P value P value = 1 ψ! E[W (n)] W (n) 2 erfc p 2ff 2 (3.46) P value 0.01 W (n) E[W (n)] ff 2 [W (n)] NIST n E[W (n)] = n!1 lim log 2 n ff 2 [W (n)] ß n[c + ffi(log 2 n)] C =0:26600jffi()j < 10 6 log 3 2 n (3.47) (3.48) 57
59 n =10 6 bit W (n) E[W (n)] ff 2 [W (n)] (3.47)(3.48) E[W (n)] = 50171:66594ff 2 [W (n)] = 33:59365 [6]NIST G-Using SHA-1 Blum-Blum-Shub E[W (n)] = 69588: ff 2 [W (n)] = 73: W (n) n 10 6 bit W (n) NIST G-Using SHA NIST (3.47) (3.48) 58
60 3.24: n =10 6 bit W(n) NIST 3.25: n =10 6 bit NIST 59
61 W (n) NIST NIST n =10 6 bit NIST 60
62 M n : " : "=" 1," 2,," n M : K : K 6 χ 2 (obs) : NIST n 10 6 bit, M 500»M» Step1 n M N = b n M c Step2 Berlekamp-Massey N L i L i i LFSR L i L i +1 Step3 μ μ = M +1 M 9+( 1)M (3.49) M Step4 1» i» N T i T i =( 1) M (L i μ)+ 2 9 (3.50) Step5 T i 1 ν 0 ν 6 61
63 ν i T i ν 0 T i» 2:5 ν 1 2:5 <T i» 1:5 ν 2 1:5 <T i» 0:5 ν 3 0:5 <T i» 0:5 ν 4 0:5 <T i» 1:5 ν 5 1:5 <T i» 2:5 ν 6 T i > 2:5 T i ν i Step6 χ 2 (obs) χ 2 (obs) = KX i=0 (ν i Nß i ) 2 Nß i (3.51) ß i ß 0 = , ß 1 = , ß 2 =0.125, ß 3 =0.5, ß 4 =0.25, ß 5 =0.0625, ß 6 = Step7 χ 2 (obs) χ 2 (obs) 0.01P value P value = igamcf K 2 ; χ2 (obs) g (3.52) 2 P value n 2 s n L(s n )=L n n EL(s n ) [18] (L n μ n )=ffi n n 2 1 n T n =( 1) n [L n ο n ]+ 2 9 ο = n r n 18 (3.53) (3.54) 62
64 T (3.56) P (T =0)= 1 2 (3.55) P (T = k) = 1 (k =1; 2; ) 22k (3.56) P (T = k) = 1 (k = 1; 2; ) 2jkj+1 2 (3.57) P (T k>0) = k» 0 (3.57) P (T» k) = k 2 (3.58) jkj 1 (3.59) T obs P-value k [jt obs j]+1p-value k = 1 (3.60) 2k 2 2 2k 1 P-value n MN n M N (3.53) T M M K +1N T M K +1 ν 0 ;ν 1 ; ;ν K ν 0 + ν ν K = N ß 0 ;ß 1 ; ;ß k (3.56) (3.57) (3.56) (3.57) M M » M» 5000 M χ 2 K P-value 1 ( K 2 )2 K 2 Z 1 χ 2 (obs) ψ! e u K K 2 u 2 1 du = igamc 2 ; χ2 (obs) 2 χ 2 (3.61) Nminß i 5 (i =0; 1; ;K) (3.62) 63
65 M N (K 6) (T» 2:5)( 2:5 <T» 1:5)( 1:5 <T» 0:5)( 0:5 <T» 0:5) (0:5 <T» 1:5)(1:5 <T» 2:5) (T >2:5) ß 0 =0:0147ß 1 =0:03124ß 2 =0:12500ß 3 = 0:50000ß 5 =0:6250ß 6 =0: χ 2 (obs) n 10 6 bit,m 500,1000 χ 2 (obs) NIST G-Using SHA
66 «3.26: M=500 «3.27: M=
67
68 m m m: n: ":0 1 " =(" 1 ;" 2 ; " n ) 5ψ 2 m 52 ψ 2 m: m NIST m n m<blog 2 nc Step1 " (m-1) " 0 Step2 m i 1 ;i 2 ; i m ν i1 i m (m-1) (m-2) ν i1 i m 1,ν i1 i m 2 Step3 ψ 2 m = 2m n X i 1 ;i 2 ; i m (ν i1 i m n 2 m )2 = 2m n X i 1 ;i 2 ; i m ν 2 i 1 i m n (3.63) 67
69 ψ 2 m 1 = 2m 1 n ψ 2 m 2 = 2m 2 n X i 1 ;i 2 ; i m 1 X i 1 ;i 2 ; i m 2 Step4 (ν i1 i m 1 n 2 m 1 )2 = 2m 1 n (ν i1 i m 2 n 2 m 2 )2 = 2m 2 n X i 1 ;i 2 ; i m 1 X i 1 ;i 2 ; i m 2 ν 2 i 1 i m 1 n (3.64) ν 2 i 1 i m 2 n (3.65) 5ψ 2 m = ψ 2 m ψ 2 m 1 (3.66) 5 2 ψ 2 m = ψ 2 m 2ψ 2 m 1 + ψ 2 m 2 (3.67) Step5 5ψ 2 m 52 ψ 2 m χ2 5ψ 2 m,52 ψ 2 m χ 2 ( 0.01) P-value P value1=igamc(2 m 2 ; 5ψ 2 m=2) (3.68) P value2=igamc(2 m 3 ; 5 2 ψ 2 m=2) (3.69) P-value Serial Test " 0 =(" 1 ;" 2 ; " n ;" 1 ;" 2 ; ;" m 1 ) i 1 ;i 2 ; ;i m m (i 1 ;i 2 ; ;i m ) ν i1 ;i 2 ; ;i m (3.63) ψm 2 χ2 ν i1 ;i 2 ; ;i m χ 2 χ 2 (3.66),(3.67) 5ψm 2 2m 1 χ ψm 2 2m 2 χ 2 [16] [17] [18] 5ψm 2 χ2 Good [19] ψ 2 m,52 ψ 2 m χ2 NIST G-Using SHA-1 1,000, NIST 68
70 5ψ 2 m 2m ψ 2 m 2m 2 χ 2 m =
71 m=2 1:5ψ 2 m 2:5 2 ψ 2 m 70
72 m=3 3:5ψ 2 m 4:5 2 ψ 2 m 71
73 m=16 k k>30 p 2X p 2k 1 [22] X 5:5ψm 2 6:5 2 ψ 2 m 72
74
75 m m m: n: ":0 1 " =(" 1 ;" 2 ; " n ) χ 2 (obs): ApEn(m) χ NIST m,n m<blog 2 nc Step1 " (m-1) n m Step2 m ]i i m 2 Step3 i Ci m = ]i n Step4 ffi (m) = 2X m 1 i=0 74 C m i log C m i (3.70)
76 Step5 m=m+1 Step1 Step4 Step6 χ 2 =2n[log 2 ApEn(m)] (3.71) ApEn(m) =ffi (m) ffi (m+1) (3.72) Step7 χ 2 χ 2 χ 2 χ 2 ( 0.01) P-value P value = igamc(2 m 1 ; χ2 2 ) (3.73) P-value (1996) Pincus Singer [20]Yi(m) =(" i ; ;" i m 1) C m i = 1 n +1 m ]fj :1» j<n m; Y j(m) =Y i (m)g = ß` (3.74) Φ (m) = X n+1 m 1 n +1 m i=1 log C m i (3.75) Ci m Y i (m) Φ (m) m 2m Φ (m) = 2 m X `=1 ß` log ß` (3.76) ß` ` =(i 1 ;:::;i m ) m(m 1) ApEn ApEn(m) =Φ (m) Φ (m+1) (3.77) 75
77 ApEn(0) Φ (1) ApEn(m) 1 m ApEn(m) [20] Pincus and Kalman(1997) ApEn(m) m-irregular(m-random) eß p 2 p ApEn(m)m =0; 1; 2 p 3 ß m () ApEn(m) log 2 n[log 2 ApEn(m)] 2m 2 Rukhin(2000) χ 2 (obs) =n[log2 ApEn(m)] P-value igamc(2 m 1 χ2 (obs) (3.78) 2 (" 1 ;:::;" n " 1 ;:::;" m 1 ) ν i1 :::i m (i 1 ;:::;i m ) ~Φ (m) = X i 1 :::i m ν ii :::i m log ν ii :::i m (3.79) Ap ~ En(m) = ~ Φ (m) ~ Φ (m+1) (3.80) Jensen s m log s Ap ~ En(m) log s<apen(m) (s =2) log s n = S m m n ApEn(m) m n Φ (m) ~Φ (m) Pincus χ 2 χ 2 NIST G-Using SHA-1 1,000, NIST 2 m χ 2 76
78 1:m=2 2:m=3 77
79 k k>30 p 2X p 2k 1 [22] X 2 3:m=10 4:m=11 78
80 5:m=12 6:m=13 79
81 6:m= n =100 m>12 m 2 m n 1 n 2 m m n=100,m=14 NIST m<blog 2 nc 2 m> blog 2 nc 5 n=100 m<12 m<blog 2 nc 7 80
82 n " 0 1 "=" 1," 2,...," n z -1, n = 100bit Step1 " 0,1 X i =2" i X i Step2 X S i mode=0 X 1 mode=1 X n Step3 z=max 15k5njS k j max 15k5njS k j S k Step4 z (3.84) z ( 0.01) P value P value =1 ( n z X 1)=4 k=( n z +1)=4 Φ( X ( n z 1)=4 k=( n z 3)=4 Φ( (4k +1)z p n ) Φ( (4k 1)z p )+ n (4k +3)z (4k +1)z p ) Φ( p ) n n (3.81) P value
83 Φ(z) = 1 p 2ß Z z 1 expf u2 gdu (3.82) ± S 0 k = X n + :::+ X n k+1 max 1»k»n js k j ψ max15k5njs lim P k j p n!1 n! 5 z = p 1 Z z 2ß z 1X = 4 ß j=0 ( ( 1) j 2j +1 exp ) 1X ( 1) k (u 2kz)2 exp ( du 2 ) (2j +1)2 ß 2 = H(z) z >0 8z 2 k= 1 (3.83) max 1»k»n js k j(obs)= p n z P-value (3.84)(3.87) G(z) 1 H(max 1»k»n js k j(obs)= p n)=1 G(max 1»k»n js k j(obs)= p n) (3.83) H(z) z H(z) G(z) G(z) = p 1 Z z ) 1X ( 1) k (u 2kz)2 exp ( du 2ß z 2 = 1X k= 1 =Φ(z) Φ( z)+2 =Φ(z) Φ( z)+2 1X k=1 k= 1 ( 1) k Φ((2k +1)z) Φ((2k 1)z) (3.84) 1X k=1 ( 1) k Φ((2k +1)z) Φ((2k 1)z) 2Φ((4k 1)z) Φ((4k +1)z) Φ((4k 3)z) (3.85) 82
84 ß Φ(z) Φ( z) 22Φ(3z) Φ(5z) Φ(z) (3.86) ß 1 p 4 expf z2 g z!1 (3.87) 2ßz 2 (z) (3.84) k H(z) = 1X k= 1 Φ((4k +1)z) Φ((4k 1)z) 1X k= 1 Φ((4k +3)z) Φ((4k +1)z) (3.88) Revesz(1990)p [24] P max 15k5njS k j = z =1 1X k= 1 + P ((4k 1)z <S n < (4k +1)z)) 1X k= 1 P ((4k +1)z<S n < (4k +3)z)) (3.89) (3.81) P-value 1 H(z) (3.89)(3.88) z n 1001,00010,000100,0001,000,000bit mode=0mode=1 z (3.84) k =1 (3.85) (3.84) k =3 NIST G-Using SHA
85 w 3.28: 100bit(mode=0) w 3.29: 100bit(mode=1) 84
86 w 3.30: 1000bit(mode=0) w 3.31: 1000bit(mode=1) 85
87 w 3.32: 10000bit(mode=0) w 3.33: 10000bit(mode=1) 86
88 w 3.34: bit(mode=0) w 3.35: bit(mode=1) 87
89 w 3.36: bit(mode=0) w 3.37: bit(mode=1) 88
90
91 (-4-114) x n " 0 1 "=" 1," 2,...," n χ 2 (obs) χ n = 10 6 bit Step1 " 0,1 X i =2" i X i Step2 X X 1 S i S =S i Step3 S 0 S 0 S 0 = 0;s 1 ;s 2 ;:::;s n ; 0 Step4 J J S 0 0 J 0 J<500 Step5 8 x( 4 5 x 5 1; 1 5 x 5 4) x Step6 8 x ν k (x) ν k (x) x k (0 5 k 5 5) ν 5 (x) x 5 Step7 8 x χ 2 (obs) 90
92 χ 2 (obs) = 5X k=0 (ν k (x) Jß k (x)) 2 Jß k (x) (3.90) ß k (x) x k ß k (x) 1 1 jxj ß 0 (x) ß 1 (x) ß 2 (x) ß 3 (x) ß 4 (x) ß 5 (x) Step8 χ 2 (obs) χ 2 χ 2 (obs) χ 2 ( 0.01) P value ψ! 5 P value = igamc 2 ; χ2 (obs) (3.91) 2 P value J lim n!1 P ψ Jpn <z! = s 2 Z z e u2 2 du; z>0 (3.92) ß 0 J P value J J<500 J<500 91
93 x k ß k (x) ß 0 (x) =1 1 2jxj ß k (x) = 1 ψ 1 1! k 1 k =1; 2; 3; 4 (3.93) 4x 2 2jxj ß 5 (x) = 1 ψ 1 1! 4 2jxj 2jxj ß k (x) χ 2 (obs) (3.90) χ 2 (obs) 5 χ 2 P value 1 P (3.91) ψ! 5 2 ; χ2 (obs) 2 (3.94) χ 2 (obs) χ 2 n 10 6 bit χ 2 (obs) χ 2 NIST G-Using SHA
94 3.38: x= : x=-3 93
95 3.40: x= : x=-1 94
96 3.42: x=1 3.43: x=2 95
97 3.44: x=3 3.45: x=4 96
98
99 (-9-1,19) x n " 0 1 "=" 1," 2,...," n ο(x) n 10 6 bit Step1 " 0,1 X i =2" i X i Step2 X X 1 S i S =S i Step3 S 0 S 0 S 0 = 0;s 1 ;s 2 ;:::;s n ; 0 S 0 J Step4 ο(x) ο(x) 9» x» 1; 1» x» 9 18 x x S 0 Step5 ο(x) ο(x) ( 0.01) P value P value = erfc q 1 jο(x) Jj A (3.95) 2J(4jxj 2) P value
100 J J<500 J<500 ο(x) J J(4jxj 2) (3.96) 0 1 lim ο J(x) J J!1 qj(4jxj 2) <z A =Φ(z) (3.96) P value (3.95) ο(x) n 10 6 bit ο(x) NIST G-Using SHA
101 u š 3.46: x=-9 u š 3.47: x=-8 100
102 u š 3.48: x=-7 u š 3.49: x=-6 101
103 u š 3.50: x=-5 u š 3.51: x=-4 102
104 u š 3.52: x=-3 u š 3.53: x=-2 103
105 u š 3.54: x=-1 u š 3.55: x=1 104
106 u š 3.56: x=2 u š 3.57: x=3 105
107 u š 3.58: x=4 u š 3.59: x=5 106
108 u š 3.60: x=6 u š 3.61: x=7 107
109 u š 3.62: x=8 u š 3.63: x=9 108
110 x z ο(x) J< x ο(x) x x x ο(x) x ο(x)
111 4 SP Generator-Using SHA-1 SP ffl Generator-Using SHA-1 ffl DFT Lempel-Ziv DFT Lempel-Ziv 110
112 DFT Maurer 10.Lempel-Ziv
113 [1] NISTSpecial Publication A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications [2] NISTSpecial Publication NIST Statistical Test Suite [3] Randomness Test using Discrete Fourier Transform [4] NIST Vol.103 No.499 ISEC [5] NIST SP DFT Vol.104 No.200 ISEC pp [6] Lempel-Ziv rep ID0206.pdf ID0206.pdf [7] /rep ID0207.pdf ID0207.pdf [8] rundum/ 112
114 rundum inve.pdf [9] I.N.Kovalenko(1972), "Distribution of the linear rank of a random matrix", Theory of Probability and its Applications.17,pp [10] G.Marsaglia and L.H.Tsay(1985), "Matrices and the structure of random number sequences", Linear Algebra and its Applications.Vol.67,pp [11] O.Chrysaphinou and S.Papastavridis, A Limit Theorem on the Number of Overlapping Appearances of a Pattern in a Sequence of Independent Trials.",Probability Theory and Related Fields,Vol.79(1988),pp [12] Ueli M.Maurer, A Universal Statistical Test for Random Bit Generators",Journal of Cryptology.Vol.5,No.2,1992,pp [13] J-S Coron and D.Naccache, An Accurate Evaluation of Maurer's Universal Test",Proceedings of SAC'98(Lecture Notes in Computer Science),Berlin:Springer-Verlag,1998 [14] H.Gustafson, E.Dawson, L.Nielsen, and W.Caelli (1994), "A computer package for measuring the strength of encryption algorithms," Computers and Security.13, pp [15] R.A. Rueppel, Analysis and Design of Stream Ciphers. New York: Springer, [16] M.Kimberley(1987), "Comparison of two Statistical tests for keystream sequences", Electronics Letters.23,pp [17] D.E.Knuth, "The Art of Computer Programming.Vol.2,3rd ed.reading", Addison-Wesley,Inc.,pp [18] A.J.Menezes,P.C.van Oorschot,and S.A.Vanstone(1997), "Handbook of Applied Cryptography.Boca Raton, FL", CRC Press,p.181 [19] I.J.Good(1953), "The serial test for sampling numbers and other tests for randomness", Proc.Cambridge Philos.Soc.. 47,pp [20] S.Pincus and B.H.Singer,"Randomness and degrees of irregularity",proc.natl.acad.sci.usa.vol93,march 1996,pp [21] A.Rukhin(2000),"Approximate entropy for testing randomness",journal of Applied Probably.Vol.37,2000 [22] 113
115 [23] Frank Spitzer,"Principles of Random Walk"Princceton, Van Nostrand, 1964(especially p.269) [24] Pal Revesz, "Random Walk in Random And Non-Random Environments" Singapore, World Scientific, 1990 [25] NIST Randomness Testing of the Advanced Encryption Standard Candidate Algorithms [26] NIST Randomness Testing of the Advanced Encryption Standard Finalist Candidates [27] G.Marsaglia DIEHARD ( [28] D.E.Knuth The Art of Computer Programming Vol Seminumerical Algorithms Third Edition Addison Wesley [29] [30] NISTFIPS PUB Digital Signature StandardDSS
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NIST 及び DIEHARD テストによる RPG100 乱数評価 FDK( 株 )RPG 推進室 2003/12/16 導入 RPG100 から生成される乱数を 2 つの有名なテストを用いて評価します 一方は米国機関 NIST により公開されている資料に基づくテストで 他方は Marsaglia 博士により提供されているテストです 乱数を一度テストしただけでは それが常にテストを満足する乱数性を持っていることを確認できないことから
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