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4 NIST Special Publicatio Maurer Lempel-Ziv NIST SP NIST SP liear trucated cogruetial geerator Freize, Håstad, Kaa, Lagarias, Shamir

5 5.4 Kuth A B

6 1 [6] NIST Special Publicatio [1] DIEHARD [10] 2 3 NIST SP NIST SP liear trucated cogruetial geerator 3

7 2 [6] 2.1 {0, 1} 2 NIST Special Publicatio (NIST SP800-22) [1] DIEHARD [10] NIST SP DIEHARD OPSO OQSO 4

8 1. 1 (frequecy (moobit) test) 2. (frequecy test withi a block) 3. (rus test) 4. (test for the logest ru of oes i a block) 5. 2 (biary matrix rak test) 6. (discrete Fourier trasform (spectral) test) 7. (o-overlappig template matchig test) 8. (overlappig template matchig test) 9. Maurer (Maurer s uiversal statistical test) 10. Lempel-Ziv (Lempel-Ziv compressio test) 11. (liear complexity test) 12. (serial test) 13. (approximate etropy test) 14. (cumulative sums (cusum) test) 15. (radom excursios test) 16. (radom excursios variat test) 2.1: NIST SP

9 1. (birthday spacigs test) 2. OPERM5 (overlappig 5-permutatio test) (biary rak test for matrices) (biary rak test for matrices) (biary rak test for 6 8 matrices) 6. (bitstream test) 7. OPSO (overlappig-pairs-space-occupacy test ) 8. OQSO (overlappig-quadruples-space-occupacy test ) 9. DNA (DNA test) (cout-the-1 s test o a stream of bytes) (cout-the-1 s test for specific bytes) 12. (parkig lot test) 13. (miimum distace test) 14. 3DSPHERES (3d-spheres test) 15. (squeeze test) 16. (overlappig sums test) 17. (rus test) 18. (craps test) 2.2: DIEHARD 6

10 1. G Usig SHA-1 2. Liear Cogruetial 3. Blum-Blum-Shub 4. Micali-Schorr 5. Modular Expoetiatio 6. Quadratic Cogruetial I 7. Quadratic Cogruetial II 8. Cubic Cogruetial 9. XOR 10. ANSI X9.17 (3-DES) 11. G Usig DES 2.3: NIST SP NIST SP DIEHARD NIST SP DIEHARD NIST SP NIST SP NIST SP Lempel-Ziv NIST SP

11 1. A multiply-with-carry (MWC) geerator 2. A MWC geerator o pairs of 16 bits 3. The Mother of all radom umber geerators 4. The KISS geerator 5. The simple but very good geerator COMBO 6. The lagged Fiboacci-MWC combiatio ULTRA 7. A combiatio MWC/subtract-with-borrow (SWB) geerator 8. A exteded cogruetial geerator 9. The super-duper geerator 10. A subtract-with-borrow geerator 11. Ay specified cogruetial geerator 12. The 31-bit geerator ra2 from Numerical Recipes 13. Ay specified shift-register geerator 14. The system geerator i Microsoft Fortra 15. Ay lagged-fiboacci geerator 16. A iverse cogruetial geerator 2.4: DIEHARD 8

12 3 NIST Special Publicatio NIST SP NIST SP {0, 1} 2 χ 2 p-value p-value 0.01 p-value < 0.01 p-value p-value erfc erfc(z) = z 2 π e x2 dx χ 2 p-value igamc igamc(a, z) = 1 Γ(a) Γ(a) = 0 e t t a 1 dt z e t t a 1 dt x p-value x = (x 1, x 2,..., x ) p-value x = (x 1, x 2,..., x ) 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) 9

13 S = X 1 + X X 2. s obs = S / 3. p-value = erfc(s obs / 2) x M p-value x = (x 1, x 2,..., x ) χ 2 p-value 1. x N = /M M 2. 1 i N i 1 π i π i = M j=1 x (i 1)M+j M N 3. χ 2 (obs) = 4M (π i 1/2) 2 i=1 4. p-value = igamc(n/2, χ 2 (obs)/2) x p-value x = (x 1, x 2,..., x ) p-value i=1 1. x 1 π = x i 2. π 1/2 2/ p-value = V (obs) = r(k) + 1 k=1 r(k) = { 0 x k = x k+1 1 x k x k+1 10

14 ( ) V (obs) 2π(1 π) 4. p-value = erfc 2 2π(1 π) x p-value x = (x 1, x 2,..., x ) χ 2 p-value 1. x N = /M M M < 6272 M = < M M = 8, 128, i v i χ 2 (obs) = K = N = K (v i Nπ i ) 2 i=0 Nπ i K N π i 3 M = 8 5 M = M = M = 8 49 M = M = p-value = igamc(k/2, χ 2 (obs)/2) i M

15 x M Q p-value x = (x 1, x 2,..., x ) M = 32 Q = 32 χ 2 p-value 1. x N = /(MQ) MQ M Q 2 Q 2. 1 l N R(l) 3. F M F M 1 F M = R(l) = M F M 1 = R(l) = M 1 4. χ 2 (obs) = (F M N) N + (F M N) N + (N F M F M N) N 5. p-value = e χ2 (obs)/ x p-value x = (x 1, x 2,..., x ) p-value 1. x 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) 2. X S = DF T (X) 3. S S /2 M = modulus(s ) = S modulus 4. 95% T = 3 12

16 5. N 0 = 0.95/2 N 0 X T 6. T N 1 7. d = N 1 N 0 (0.95)(0.05)/2 8. p-value = erfc( d / 2) x m B M N p-value x = (x 1, x 2,..., x ) m x M = 2 17 N = 2 3 χ 2 p-value 1. x M N 2. 1 j N j B W j m B 1 m 3. µ σ 2 µ = M m + 1 ( 2 m 1 σ 2 = M 2 m 2m 1 ) 2 2m N 4. χ 2 (W j µ) 2 (obs) = j=1 σ 2 5. p-value = igamc(n/2, χ 2 (obs)/2) 13

17 3.1.8 x m B K M N p-value x = (x 1, x 2,..., x ) m K = 5 x M = 1032 N = 968 χ 2 p-value 1. x M N 2. B m B 1 B 6 0 i 4 i i i = 5 i 5 i v i 3. π i = v i /N λ = M m m η = λ/2 π 0 = e η π 1 = η 2 e η π 2 = ηe η 8 π 3 = ηe η 8 π 4 = ηe η 16 (η + 2) ( ) η η + 1 ( η η η ) 4. χ 2 (obs) = 5 (v i Nπ i ) 2 i=0 Nπ i 5. p-value = igamc(5/2, χ 2 (obs)/2) 14

18 3.1.9 Maurer x L Q p-value x = (x 1, x 2,..., x ) p-value 1. x (Q + K) L Q K Q + K = /L 2. 2 L T j (j = 0, 1,..., 2 L 1) (a) T j 0 (b) i = 1 i = Q i L 2 j T j = i T j j 2 L 3. (a) sum 0 (b) i = Q + 1 i = Q + K i L 2 j sum = sum + log(i T j ) T j = i Q f = sum K ( ) f 5. p-value = erfc µ(l) 2σ(L) µ(l) σ 2 (L) 15

19 L µ(l) σ 2 (L) Lempel-Ziv x p-value x = (x 1, x 2,..., x ) p-value 1. x W obs ( ) 2. p-value = 1 2 erfc µ Wobs 2σ = 10 6 µ = σ 2 = G usig SHA x M K p-value x = (x 1, x 2,..., x ) K = 6 χ 2 p-value 1. x M N = MN 2. Berlekamp-Massey L i (1 i N) 16

20 3. µ µ = M ( 1)M+1 36 (M/3 + 2/9) 2 M 4. 1 i N T i = ( 1) M (L i µ) ν 0, ν 1,..., ν 6 ν 0 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 T i < T i < T i < T i < T i < T i 2.5 T i > 2.5 K 6. χ 2 (ν i Nπ i ) 2 (obs) = π 0 = π 1 = π 2 = Nπ i=0 i π 3 = 0.5 π 4 = 0.25 π 5 = π 6 = p-value = igamc(k/2, χ 2 (obs)/2) x m p-value x = (x 1, x 2,..., x ) χ 2 p-value 2 1. x m 1 x + m 1 x 2. x m i 1 i 2 i m ν i1 i 2 i m m 1 i 1 i 2 i m 1 ν i1 i 2 i m 1 m 2 i 1 i 2 i m 2 ν i1 i 2 i m 2 17

21 3. Ψ m 2 Ψ m 1 2 Ψ m 2 2 = 2m = 2m 1 = 2m 2 i 1 i 2 i m i 1 i 2 i m 1 i 1 i 2 i m 2 ( ν i1 i 2 i m 2 m ) 2 ( ν i1 i 2 i m 1 ) 2 2 m 1 ( ν i1 i 2 i m 2 ) 2 2 m 2 4. Ψ m 2 2 Ψ m 2 = Ψ m 2 Ψ m 1 2 = Ψ m 2 2Ψ m Ψ m p-value p-value1 = igamc(2 m 2, Ψ m 2 ) p-value2 = igamc(2 m 3, 2 Ψ m 2 ) x m p-value x = (x 1, x 2,..., x ) χ 2 p-value 1. x m 1 x + m 1 x 2. x m i #i i m 2 3. C m i = #i 4. ϕ (m) = 2 m 1 i=0 C m i log C m i 5. m = m χ 2 (obs) = 2(log 2 (ϕ (m) ϕ (m+1) )) 7. p-value = igamc(2 m 1, χ 2 (obs)/2) 18

22 x mode p-value x = (x 1, x 2,..., x ) (mode = 0) (mode = 1) p-value 1. x 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) 2. 1 k S k = k i=1 k i=1 X i X i+1 mode = 0 mode = 1 3. z = max 1 k S k 4. p-value Φ p-value = 1 /z 1 4 k= /z+1 4 ( ( ) ( )) (4k + 1)z (4k 1)z Φ Φ + /z 1 4 k= /z 3 4 ( ( ) ( )) (4k + 3)z (4k + 1)z Φ Φ x p-value x = (x 1, x 2,..., x ) χ 2 p-value 8 1. x 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) 19

23 k 2. 1 k S k = X i i=1 3. S = (0, S 1, S 2,..., S, 0) 4. S 0 0 J J 0 J < x 1 1 x 4 8 x x 6. 8 x k = 0, 1, 2, 3, 4 x k ν k (x) x 5 ν 5 (x) x 5 ν k (x)j k= x χ 2 (obs) = 5 (ν k (x) Jπ k (x)) 2 k=0 Jπ k (x) π k (x) x π 0 (x) π 1 (x) π 2 (x) π 3 (x) π 4 (x) π 5 (x) ± ± ± ± x p-value = igamc(5/2, χ 2 (obs)/2) x p-value x = (x 1, x 2,..., x ) p-value x 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) k 2. 1 k S k = X i i=1 3. S = (0, S 1, S 2,..., S, 0) S J 20

24 4. 9 x 1 1 x 9 18 x x S ξ(x) ( ) ξ(x) J x p-value = erfc 2J(4 x 2) p-value p-value 1 m p-value ± 3 m 2 [0, 1) 10 p-value χ 2 1 i 10 F i [(i 1)/10, i/10) p-value χ 2 = 10 i=1 (F i m/10) 2 m/10 p-value = igamc(9/2, χ 2 /2) p-value NIST SP NIST SP x

25 Maurer ( ) 7(1280)

26 3.1: 10 bit 500 (U P ) 1. G Usig SHA-1 2. Liear Cogruetial 3. Blum-Blum-Shub 4. Micali-Schorr 5. Modular Expoetiatio 6. Quadratic Cogruetial I 7. Quadratic Cogruetial II 8. Cubic Cogruetial 9. XOR 10. ANSI X9.17 (3-DES) 11. G Usig DES U P U P U P U P U P U P x x x 1 x x x 2 Maurer Lempel-Ziv U P U P U P U P U P x x 1 x x x 2 x x Maurer x x Lempel-Ziv x x 23

27 3.2: 10 bit 1000 (U P ) 1. G Usig SHA-1 2. Liear Cogruetial 3. Blum-Blum-Shub 4. Micali-Schorr 5. Modular Expoetiatio 6. Quadratic Cogruetial I 7. Quadratic Cogruetial II 8. Cubic Cogruetial 9. XOR 10. ANSI X9.17 (3-DES) 11. G Usig DES U P U P U P U P U P U P x x x x x x x 1 x x x x x 2 x x Maurer Lempel-Ziv x x x x U P U P U P U P U P x x x x x 1 x x x x 2 x x Maurer x x x x Lempel-Ziv x 24

28 3.3: 100 bit 500 (U P ) 1. G Usig SHA-1 2. Liear Cogruetial 3. Blum-Blum-Shub 4. Micali-Schorr 5. Modular Expoetiatio 6. Quadratic Cogruetial I 7. Quadratic Cogruetial II 8. Cubic Cogruetial 9. XOR 10. ANSI X9.17 (3-DES) 11. G Usig DES U P U P U P U P U P U P x x x x x x x x 1 x x x x 2 Maurer Lempel-Ziv x x U P U P U P U P U P x x x x x x 1 x x x x 2 x x x x x x x x x x Maurer x x x x x Lempel-Ziv x x x 25

29 4 NIST SP NIST SP NIST SP x = (x 1, x 2,..., x ) x = (x 1, x 2,..., x ) 1 1 X = (X 1, X 2,..., X ) X i = 2x i 1 (1 i ) 2. X S = (S 1, S 2,..., S ) 3. S S /2 M = modulus(s ) = S S = (S 1, S 2,..., S /2 ) i = 1, 2,..., /2 S i 4. 95% T = 3 5. N 0 = 0.95/2 N 0 X S 1, S 2,..., S /2 T S i 6. S 1, S 2,..., S /2 T S i N 1 7. d = N 1 N 0 (0.95)(0.05)/2 8. X d p-value = erfc( d / 2) 4.2 NIST SP [7] NIST SP S = (S 1, S 2,..., S ) S j = X k cos k=1 2π(k 1)j + i X k si k=1 2π(k 1)j 26

30 i = 1 3 S S = (S 1, S 2,..., S /2 ) j = 1, 2,..., /2 1 S j S j S j = S j c j = 2π(k 1)j X k cos k=1 s j = 2π(k 1)j X k si k=1 S j 2 = c j 2 + s j 2 X c j s j µ = 0 σ 2 = /2 N(0, /2) c j s j 1 2 c j (X) = 1 2 X X k=1 X X k cos k=1 ( 1 2 (c j (X)) 2 = 1 2 X k cos X X k=1 = 1 ( 2 2π(k 1)j 2 X k cos = 1 2 = = = 2 k=1 k=1 X k=1 ( cos ( cos 2π(k 1)j ( cos 2 2π(k 1)j ) 2 4π(k 1)j ) ) 2 2π(k 1)j = 0 ) 2 2π(k 1)j ) 2 + X l X l2 cos 2π(l 1 1)j l 1 l 2 cos 2π(l 2 1)j y def = ( cj σ ) 2 ( sj ) 2 S j 2 + = σ σ 2 y 2 χ 2 - { 1 y 2 e 2 y > 0 T 2 (y) = 0 y 0 27

31 1 [9] N(µ, σ 2 ) z 1, z 2,..., z y = 1 σ 2 (z i µ) 2 i=1 χ 2 4 T T S j 5% y = S j 2 T 2 /σ 2 1 T 2 y 2 e 2 dy = e 2σ 2 = 0.05 T 2 = 2σ 2 l 0.05 = ( l 0.05) l T N 1 N = /2 p = 0 95 Np q [7] (N/2)pq 4.1 Mathematica % Np q c j, s j X 4.1 c j s j (N/2)pq Np q σ NIST SP NIST SP T = N 1 N 0 7 d = (0.95)(0.05)/

32 4.1: 1 (a) 10 bit 500 p-value G Usig SHA Liear Cogruetial Blum-Blum-Shub Micali-Schorr Modular Expoetiatio Quadratic Cogruetial I Quadratic Cogruetial II Cubic Cogruetial XOR ANSI X9.17 (3-DES) G Usig DES (b) 10 bit 1000 p-value G Usig SHA x Liear Cogruetial Blum-Blum-Shub Micali-Schorr Modular Expoetiatio Quadratic Cogruetial I Quadratic Cogruetial II Cubic Cogruetial XOR ANSI X9.17 (3-DES) G Usig DES x 29

33 1s = Table[ Cout[ Positive[ Take[32 Abs[Fourier[Table[2 Radom[Iteger] - 1, {1024}]]], 512] - Sqrt[ ] ], False ], {1000} ]; 0 = ; d1 = (1s - 0) / Sqrt[ / 2]; d2 = (1s - 0) / Sqrt[ / 4]; {Cout[Positive[d1-1.96], True], Cout[Positive[d2-1.96], True]} 4.1: N 1 Mathematica 4.2: bit 500 p-value G Usig SHA Liear Cogruetial Blum-Blum-Shub Micali-Schorr Modular Expoetiatio Quadratic Cogruetial I Quadratic Cogruetial II x x Cubic Cogruetial x x XOR ANSI X9.17 (3-DES) G Usig DES

34 5 5.1 NIST SP NIST SP Liear Cogruetial Geerator (LCG) s 0 LCG x 0, x 1, x 2, s 1, s 2,... s i+1 = a s i mod x 0, x 1, x 2,... { 0 s i+1 /(2 31 1) < 0.5 x i = 1 LCG s 0 = a = LCG s 1, s 2, s 3,... LCG liear trucated cogruetial geerator liear trucated cogruetial geerator 5.2 liear trucated cogruetial geerator (liear cogruetial geerator) s 0, s 1, s 2,... s i = a s i 1 + b mod m s i t x i x 0, x 1, x 2,... liear trucated cogruetial geerator s i x i si x i = 2 t. 5.3 Freize, Håstad, Kaa, Lagarias, Shamir Freize, Håstad, Kaa, Lagarias, Shamir [3, 4, 5] s i = a s i 1 + b mod m 31

35 b = 0 a, m x 0, x 1, x 2,... s 0, s 1, s 2,... b = 0 ˆx i = x i x i 1 ŝ i = s i s i 1 ŝ i = a ŝ i 1 mod m x i s i t s i β t = (1 β) s i = x i 2 β + y i y i s i β Freize L k b 1, b 2,..., b k b i = (b i,1, b i,2,..., b i,k ) { m j = 1 b 1,j = 0 i = 2, 3,..., k a i 1 j = 1 b i,j = 1 j = i 0 L v l v l,1 s 1 + v l,2 s v l,k s k 0 (mod m) liear trucated cogruetial geerator Freize 1. v 1, v 2,..., v k i = 1, 2,..., k k v i,j s j = j=1 k v i,j x j 2 β + j=1 k v i,j y j 0 (mod m) j=1 2. i = 1, 2,..., k k v i,j y j < m 2 j=1 k k v i,j x j 2 β v i,j s j j=1 j=1 s 1, s 2,..., s k 32

36 2 m ε > 0 k C(ε, k) m > C(ε, k) (1 β) > (k 1 + ε) + c(k) 1 O(m ε ) a k v i,j y j < m 2 j=1 c(k) = k 2 + (k 1) log log k Freize + k 2 x i ( 1 k +ε) + k 2 +(k 1) log log k + 2 NIST SP LCG x i 1 Freize LCG 2 k v i,j y j < m 2 j=1 x j 1 y j m 2 Freize x i Freize 5.4 Kuth Kuth [8] s i = a s i 1 + c mod m m = 2 k a mod 4 = 1, c mod 2 = 1 k x 0, x 1, x 2,... a, c, s 0 [8] k = h + l A s i /2 l (0 i < 2 l ) a c s 0 33

37 B s i /2 l (0 i < N) a c s 0 A l > 1 O(2 l ) B h 2 O(k 2 2 2l /N 2 ) A s i l s (l) i = s i /2 l 1 mod 2 s (l) i 1 A s i /2 l (0 i 2 l ) s i /2 l 1 (0 i 2 l 1 ) s i /2 l (0 i 2 l ) s (l) i = s i /2 l 1 mod 2 (0 i 2 l 1 ) 1 0 t < k y (t) +1 = s +2 t s mod 2 k y (t) +1 = a y(t) mod 2 k y (t) 2 t 2 1 l < k 1 b l {0, 1} s (l) = s (l+1) +2 l 1 s (l+1) + b l mod 2 2 b l s i /2 l (0 i < 2 l ) i = s i /2 l 1 mod 2 (0 i < 2 l 1 ) 1 s +2 l 1 y (l 1) + s mod 2 k l 1 0 y (l 1) s (l) = 1 s (l) +2 l 1 b l s i /2 l (0 i < 2 l ) s i /2 l 1 (0 i 2 l 1 ) s i /2 l (0 i 2 l ) l 2 b l s 0 /2 l 2 s 2 l 1/2 l + s 2 l/2 l 2s (l) (mod 4) s (l) 0 = s (l+1) s (l+1) 2 l b l mod 2 s 2 l/2 l b l 34 s (l)

38 h = 1 b l 0 1 h 2 b l = 0 b l = 1 A 1. h 2 b l = 0 b l = 1 h = 1 b l = < 2 l 1 s (l) = s (l+1) +2 l 1 s (l+1) + b l mod 2 s (l) x (l) = 1 x (l) 2 l 1 0 l 1 3. l 2 s 0 /2 l 2 s 2 l 1/2 l + s 2 l/2 l 2s (l) (mod 4) s (l) 0 = s (l+1) s (l+1) 2 l b l mod 2 b l 2 4. (x (1) 0, x(1) 1, x(1) 2 ) 2 (0, 1, 0) (1, 0, 1) a = x 2 x 1 mod 2 k x 1 x 0 c = x 1 a x 0 mod 2 k l = 1 [8] B B N s = s + b mod 2 k s +1 = as + (c (a 1)b) mod 2 k s mi = mi {x mod 2 l } s max = max {x mod 2 l } 0 <N 0 <N s mi b < 2 l s max 0 < N s /2 l = s /2 l b s 0 = s 0 + b mod 2 k s 0 y 0 = s 1 s 0 mod 2 k 35

39 1 s s0 a 1 2 l = 2 l + a 1 y0 2 l + ɛ mod 2 h ɛ 0 1 B 1. (a, y 0 ) 2. (a, y 0 ) s 0 (a, y 0 ) 2 t + 1 < N t 0 = s 2 t s 0 mod 2 k y (t) = (1 + a 2t 1 )y (t 1) mod 2 k t 1 y 0 = y (0) 0 t y (t) 2 l + κ (t) s+2 t s ( ) = 2 l 2 l mod 2 h κ (t) 0 1 y (t) 0 h = 1 h 2 y (t) Kuth NIST SP LCG 2 s i B s i liear trucated cogruetial geerator Freize, Håstad, Kaa, Lagarias, Shamir Kuth NIST SP LCG liear trucated cogruetial geerator NIST SP LCG 36

40 [1] NIST, Special Publicatio , A statistical test suite for radom ad pseudoradom umber geerators for cryptographic applicatios, [2] R. N. Bracewell, The fourier trasform ad its applicatios, McGraw-Hill, [3] A. M. Frieze, J. Hastad, R. Kaa, J. C. Lagarias, ad A. Shamir, Recostructig trucated iteger variables satisfyig liear cogrueces, SIAM J. Comput., 17(2): , [4] A. M. Frieze, R. Kaa, ad J. C. Lagarias, Liear cogruetial geerators do ot produce radom sequeces, I 25th IEEE Symposium o Foudatios of Computer Sciece (FOCS), pp , [5] J. Hastad ad A. Shamir, The cryptographic security of trucated liearly related variables, I 17th ACM Symposium o Theory of Computig (STOC), pp , [6],, [7],,, NIST,, ISEC , pp , [8] D. E. Kuth, Decipherig a Liear Cogruetial Ecryptio, IEEE Tras. Iformatio Theory, vol. IT-31, o. 1, pp , [9] [ ],,, [10] G. Marsaglia, DIEHARD, geo/diehard.html, 37

CryptrecReport.dvi

CryptrecReport.dvi 16 12 1 NIST NIST Special Publication 800-22[1, 2] SP 800-22 2 DIEHARD [27] SP 800-22Ver.1.516 189 DFT Lempel-Ziv [3, 4, 5] SP 800-22 16 DFT Lempel-Ziv 1 2 SP 800-22 NIST SP800-22[1, 2] NISTNational Institute