A recipe for an unpredictable random number generator

Transcription

1 Condensed Matter Physics 006, Vol. 9, No (46, A recie for an unredictable random number generator M.A.García-Ñustes, L.Trujillo, J.A.González Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC, A.P 187, Caracas 100 A, Venezuela Received February 7, 006, in final form Aril 19, 006 In this work we resent a model for comuting the random rocesses in digital comuters which solves the roblem of eriodic seuences and hidden errors roduced by correlations. We show that systems with noninvertible non-linearities can roduce unredictable seuences of indeendent random numbers. We illustrate our results with some numerical calculations related to random walks simulations. Key words: random number generator, indeendent random numbers PACS: a, a, Fb, T Many challenging roblems in comutational hysics are associated with reliable realizations of randomness (e.g. Monte Carlo simulations. In a tyical 3-bit format a maximum of 3 floating oint numbers can be reresented. Therefore, a recursive function X n+1 =f(x n, X n 1,...,X n r+1 acting on these numbers generates a seuence X 0, X 1, X,..., X N 1 which should reeat itself. It is known that for any recursive function, a digital comuter can only generate eriodic seuences of numbers 1 7]. These generators are not unredictable. Definition of truly unredictable rocess: The next values are not determined by the revious values. A rocess X n = P(θTZ n is said to be unredictable if for any string of values X 0, X 1, X,...,X m of length m + 1, generated using some θ = θ 1, there are other values of θ for which function X n = P(θTZ n generates exactly the same string of numbers X 0, X 1, X,...,X m, but the next value X m+1 is different, where m is any integer. Note that this kind of rocess cannot be exressed as a ma of tye X n+1 = f(x n, X n 1,...,X n r ]. All the known generators (in some secific hysical calculations give rise to incorrect results because they deviate from randomness,4,5,7]. It is trivial that any eriodic rocess is not unredictable. Suose that m T is the eriod of the seuence generated. Given any string of m T values: X s, X s+1,..., X mt 1; the next value X mt is always known because the rocess is eriodic. On the other hand, for any generator of tye X n+1 = f(x n, X n 1,..., X n r+1, given any string of r values: X s, X s+1,..., X s+r 1 ; the next value X s+r is always determined by the revious r values. Thus it is not unredictable. So the subseuences must be correlated. An examle of this can be found in 5], where the authors have shown that using common seudo random number generators, the roduced random walks resent symmetries, meaning that the generated numbers are not indeendent. On the other hand, the logarithmic lot of the mean distance d versus the number of stes N is not a straight line (as exected theoretically, d N 1/ after N > 10 5 (in fact, it is a raidly decaying function. Here d is defined as the end to end mean-suare distance from the origin of the random walk as a function of the number of stes. Other aers on the effect of the seudorandom number generator on random walk simulations are as follows 11 13]. In the following, we shall show that using non-invertible nonlinear functions, we can create an unredictable random number generator which does not contain visible correlations while simulating a random walk with the length Let us investigate the following function8 10]: X n = P(θ TZ n, (1 c M.A.García-Ñustes, L.Trujillo, J.A.González 367

2 M.A.García-Ñustes, L.Trujillo, J.A.González where P(t is a eriodic function, θ is a real number, T is the eriod of the function P(t, and Z is a noninteger real number. Let Z be a rational number exressed as Z = /, where and are relative rime numbers. Now let us define the following family of seuences X (k,m,s n = P T (θ 0 + m k ( s ( n ], ( where k, m and s are non-negative integers. Parameter k distinguishes different seuences. For all seuences arametrized by k, the strings of m + 1 values X s, X s+1, X s+,...,x s+m are the same. This is because X (k,m,s n = P Tθ 0 ( s ( n ] ( + Tk n s (m n+s = P Tθ 0 s ( n ] for all s n m + s. Note that the number k (n s (m n+s is an integer for s n m + s. So we can have an infinite number of seuences that share the same string of m + 1 values. Nevertheless, the next value X (k,m,s s+1 = P T θ 0 ( s ( (s+1 + T k(m+1 is uncertain. In general X (k,m,s s+1 can take different values. In addition, the value X (k,m,s s 1, ] X (k,m,s s 1 = P T θ 0 ( s ( (s 1 + T k(m+1 is also undetermined from the values of the string X s, X s+1, X s+,..., X s+m. There can be different ossible values. In the case of a generic irrational Z, there are infinite ossibilities for the future and for the ast. From the observation of the string X s, X s+1, X s+,..., X s+m, there is no method for determining the next and the revious values of the seuence. But this is not the only feature of these functions. It can be shown that there are no statistical correlations between X m and X n if m n, and that they are also indeendent in the sense that their robability densities satisfy the relationshi P(X n, X m = P(X n P(X m 14,15]. Moreover, we shall show that, given the function (1, any string of seuences X s, X s+1,...,x s+r constitutes a set of statistically indeendent random variables. Without loss of generality, we assume that P(t has zero mean and can be exressed using the following Fourier reresentation P(t = k= a ke iπkt. We can calculate the r-order correlation functions 14,15]: E(X n1 X nr = dθp(tθz n1 P(TθZ nr = = X k 1= k 1= k r= k r= 1 a k1 a kr dθ ex {iπ(k 1 Z n1 + + k r Z nr Tθ} 0 a k1 a kr δ(k 1 Z n1 + + k r Z nr, 0, (3 where the coefficients k i can be different integers, and δ(n, m = 1 if n = m or δ(n, m = 0 if n m. When all n i are even, the following euation is satisfied E(X n1 s X n s+1 Xnr s+r = E(X n1 s E(X n s+1 E(Xnr s+r. (4 The main roblem in this euation is when one of the numbers n i is odd. In this case, the correlations E(Xs n1 X n s+1 Xnr s+r should be zero. A nonzero correlation in euation (4 exists ], 368

3 Unredictable random number generator only for the sets (n 1, n,..., n r that satisfy the euation k 1 Z n1 + + k r Z nr = 0. For a tyical real number Z, this euation is never satisfied. If we use non-invertible nonlinear functions, tye of (1, we can imlement a Truly Random Number Generator (TRNG. In this case, we roose the following function X n = θ s Z n ] mod 1. (5 Function (5 is an examle of the general case X n = PθTZ n ] studied in this aer. We have shown that the subseuences X s, X s+1,...,x s+r constitute a set of statistically indeendent random variables. The articular case of function (5 is well-known to roduce uniformly distributed numbers 8 10]. Now we shall formulate a central limit theorem. Using theorems roved in revious studies 14 19] and the results obtained from this aer, we obtain the following formula: If Z is a generic real number and X n = (Y n 1/, Y n = θz n ] mod 1, then { lim P α < X 1 + X + X r r r } < β = 1 β π α e ξ dξ. (6 The Gaussian distribution of the sums is correct even for other functions X n = PθZ n ], where P(t is eriodic. This has been shown in numerical simulations 14]. The numbers X n = θz n ] mod 1 are uniformly distributed8 10]. We can simulate different stochastic rocesses (with different distributions using different functions X n = PθTZ n ]. As ρ(x n = 1, ρ(x n+1 = 1, ρ(x n, X n+1 = 1, it is trivial that they are indeendent. It is interesting to check the theoretical redictions using numerical simulations of the behavior of different stochastic rocesses. For instance, let us study the function U n = cosπθz n ]. (7 All the moments and higher-order correlations can be exactly calculated 14,15]: For odd m: E(Un m = 0. (8 If any n i is odd, then Suose now that all n i are even: E(U n0 s Un1 s+1 Unr s+r = E(U n0 s = E(U n0 s U n1 s+1 Unr s+r = 0. (9 ( (n0+n1+ +nr n0 n 0 n0 ( n0 n 0 ( E(U n1 s+1 = n1 n1 n 1 E(U nr s+r = nr ( nr n r Note that the condition for indeendence is satisfied ( n1 n 1 ( nr n r, (10, (11,..., (1. (13 E(Us n0 Un1 s+1 Unr s+r = E(Un0 s E(Un1 s+1 E(Unr s+r, (14 for all integers n 0, n 1,...n r. We have erformed extensive numerical simulations that confirm the values of these moments and the indeendent conditions. An additional checking is as follows. 369

4 M.A.García-Ñustes, L.Trujillo, J.A.González 4500 (a 4500 (b ρ(u n ρ(u n U n U n+1 Figure 1. Probability densities for random variables U n = cosπθz n ] and V n = U n+1. (a ρ(u = (π 1 U 1 ; (b ρ(v = (π 1 V 1. Figure. Probability density ρ(u n, U n+1, when U n = cosπθz n ]. Here ρ(u, V = (π (1 U (1 V 1. The robability density of U n is ρ(u = (π 1 U 1. Define V n = U n+1. The robability density of V n is ρ(v = (π 1 V 1. We have checked both theoretically and numerically that ρ(u, V = (π (1 U (1 V 1, that is ρ(u, V = ρ(uρ(v. This can be observed in figure 1 and figure. In order to avoid comutation roblems, we have used the following rocedure. We change arameter θ after each set of M values of X n, where M is the maximum number for which there are no overflow roblems, such that the next value of X n+1 is obtained with the new θ. Let us define θ s = A(C s + X s + 0.1, (15 where C s is a seuence obtained using the digits of the Chamernowne s number 0] (i.e., : C 0 = , C 1 = , C = , C 3 = , C 4 = , and so on. This seuence is noneriodic. Index s is the order number of θ, such that s = 0 corresonds to the θ used for the first set of M seuence values X 1, X,..., X M ; s = 1 for the second set X M+1, X M+,..., X M, and so on. X 0 reresents the TRNG s seed. Using this method we have generated a very long seuence of random numbers without comutational roblems. To test function (5 as a truly random number generator, we have imlemented a random walk 370

5 Unredictable random number generator simulation rogram in C++. We have made a samling test of a random walk with N = 10 9 stes with 100 realizations with different initial seeds. The mean distance d was being calculated every 1000 stes of the random walk. The Chamernowne seuence of numbers used in the generator was roduced reviously by a short C++ rogram, who created a seuence of a maximum of Chamernowne s numbers. If a larger amount of values to C s is necessary, it can be obtained using a segment code that has 40 thousand values already stored in C s and mixing them, e.g. the algorithm takes the first value of the series C 1, the third C 3 and so on, and adds them at the end of the series, obtaining C s+1 = C 1, C s+ = C 3,...; if more values are necessary, this rocedure or cycle is reeated but now skiing two values C 1, C 3, C 5,...three values C 1, C 4, C 7 and so on. In this way, we can make the C s seuence as large as we wish. We resent a logarithmic lot of the mean distance d versus the number of stes N with N = 10 9 stes with A = and Z = π/ (See figure 3. We can verify that there is no deviation from the theoretical straight line, even for N 10 5 stes, which is a very good test of the reliability of the Random Number Generator used in the random walk simulations (a 10 8 (b log<d> log<d> log(n log (N Figure 3. Logarithmic lot of the mean distance d versus the number of stes N = 10 9 stes. (a for generator (5; (b the same simulation for a generator of tye X n+1 = ax n mod T. We have resented a random number generator based on the roerties of non-invertible transformations of truncated exonential functions. The obtained random rocess is unredictable in the sense that the next values are not determined by the revious values. We have alied this generator to the numerical simulation of statistically indeendent random variables. In the simulation of a random walk with the length 10 9, the random rocess does not contain visible correlations. References 1. Ferrenberg A.M., Landau D.P., Wong Y.J., Phys. Rev. Lett., 199, 69, Grassberger P., Phys. Lett. A, 1993, 181, Vattulainen I., Ala-Nissila T., Kankaala K., Phys. Rev. Lett., 1994, 73, D Souza R.M., Bar-Yam Y., Kardar M., Phys. Rev. E, 1998, 57, Nogués J., Costa-Krämer J.L., Rao K.V., Physica A, 1998, 50, L Ecuyer P., Oer. Res., 1999, 47, Bauke H., Mertens S., J. Stat. Phys., 004, 114, González J.A., Reyes L.I., Suárez J.J., Guerrero L.E., Gutiérrez G., Phys. Lett. A, 00, 95, González J.A., Reyes L.I., Suárez J.J., Guerrero L.E., Gutiérrez G., Physica D, 003, 178, Trujillo L., Suárez J.J., González J.A., Eurohys. Lett., 004, 66, Grassberger P., J. Phys. A: Math. Gen., 1993, 6, Shchur L.N., Heringa J.R., Blöte H.W.J., Physica A, 1997, 41, Shchur L.N., Comut. Phys. Comm., 1999, 11,

6 M.A.García-Ñustes, L.Trujillo, J.A.González 14. González J.A., Trujillo L., Acta Physica Pol. B, 005, 37, González J.A., Trujillo L., J. Phys. Soc. Jaan, 006, 75, Kac M., Stud. Mathematica, 1936, 6, Kac M., Steinhaus H., Stud. Mathematica, 1936, 6, Kac M., Steinhaus H., Stud. Mathematica, 1936, 6, Kac M., Steinhaus H., Stud. Mathematica, 1937, 7, Chamernowne D.G., J. London Math. Soc., 1933, 8, i -,.! i" #, $.. #%& '(*+, i-./., 0 ((1(3414 /.5 I 1*. ** 6 / :713 i :;(4, <7=*. 1 / +. 4 / 6 187, > 6+6 / ?, 0 7 HI AGJG 006 B., K GLAEAGMFGDN K CJ HO P i 19 QKi AF O 006 B. 0 Ri5 +7S7* i T. <+7<7U T 7 T 7:(34 8. <6: / <+7 R (1 i 8 8 V / 7 T < W *(+69, X/ i =U <+7 - S3( T 1<+. V. (. 9 / 7+(3 XRiXT. <(+ i 7:. V. 9 <713 i :7871*( 5 i < <7 T. 37 /. Y. <7 / 6 - U T 7, Z7 1. 1*( T. - ( *.T. (3 i i5 71* XT. T 7;*4 <7+7:;86*. (<(+(:S6V86 i <713 i :7871* i ( - 63(; <6: / V. 1(3. 6= i +( - 34*6*. i 3 W 1*+ W *41 X 7SV. 13( XT., <78 X- 6.T. - 1.T - 3 XRi U W 8. <6: / S3 / 64. \]^_`a i b ]`ac : de f e ghijg klmhnojkl lr e s, f e ths e uf i klmhnojk i lrsh PACS: a, a, Fb, T 37