Device and method for outputting random vector string, and information recording medium

Description: TECHNICAL FIELD [Detailed description of the invention]

【0001】

[Technical field to which the invention belongs]

The present invention relates to a random vector sequence output device, an output method, and an information recording medium.

【0002】

In particular, when there are two vector sequence generation methods in which the density function of the output random vector sequence distribution is a known analytical function, combine them to create a higher dimensional random vector sequence. The present invention relates to an output device for a random vector sequence that outputs what the density function of the distribution is obtained as an analytical function, an output method, and an information recording medium that records a program that realizes these.

【0003】

[Conventional technology]

Conventionally, many methods for generating random numbers using recurrence formulas have been known. In the Monte Carlo method, which conducts simulated experiments such as physics and engineering, the random numbers generated in this way are used.

【0004】

In addition, in the CDMA (Code Division Multiple Access) method used in mobile phones, each user uses a PN (Pseudo Noise) code obtained from random numbers in order to enable a large number of users to effectively use the limited radio band. Assigned to.

【0005】

In addition, with the development of communication technologies such as the Internet, the need for confidentiality of communications is increasing, and it is becoming common to use a technique called public key cryptography to maintain confidentiality. Also in this method, random numbers are used to generate the public key.

【0006】

In order to obtain such a random number, a method using a recurrence formula has been widely used. The period of random numbers has been a problem in the recurrence formula by multiplication that has been known for a long time. However, in recent years, due to the development of chaos theory, random numbers obtained by using rational maps derived from the addition theorem of elliptic functions (including trigonometric functions) in recurrence formulas have the following advantageous properties. It is known and its importance is increasing.

【0007】

(1) Since the output random number sequence has no period, the same sequence will not be output repeatedly. (2) If a rational number is given as the seed of random numbers (initial value given to the recurrence formula), all the numbers included in the obtained random number sequence become rational numbers. (3) The density function that represents the distribution of random numbers is a known analytic function.

【0008】

Known rational maps include the Ulam-von Neumann map [Equation 1], the cubic map [Equation 2], and the quintic map [Equation 3].

【0009】

[Number 1]

<img file="JP2000276331A_D0001.tif" />【0010】

[Number 2]

<img file="JP2000276331A_D0002.tif" />【0011】

[Number 3]

<img file="JP2000276331A_D0003.tif" />【0012】

Regardless of which of these rational maps is selected, if an appropriate initial value ξ (0 &lt;ξ &lt;1) is given and a random number sequence x [i] is obtained by the following recurrence formula, this random number sequence The density function that represents the distribution of x [i] is represented by [Equation 4].

[0] = ξ x [i + 1] = f (x [i]) (i 0) [0013]

[Number 4]

<img file="JP2000276331A_D0004.tif" />【0014】

A rational map with parameters can also be used as a recurrence formula, and such rational maps include Katsura-Fukuda map, generalized Uram-von Neumann map [Equation 5], generalized cubic map, and generalized Chebyshev map. and so on.

【0015】

[Number 5]

<img file="JP2000276331A_D0005.tif" />【0016】

For example, when a random number sequence is obtained by the above recurrence formula using the generalized Uram-von Neumann map [Equation 5], its distribution is also expressed by a density function [Equation 6] having the same parameters.

【0017】

[Number 6]

<img file="JP2000276331A_D0006.tif" />【0018】

The Katsura-Fukuda map is set to m = 0 in the generalized Ulam-von Neumann map [Equation 5].

【0019】

A method for generating random numbers using these rational maps is disclosed in Japanese Patent Application Laid-Open No. 10-283344 according to the application by the inventors of the present application. The theoretical background is disclosed in the following documents. SM Ulam and J. von Neumann, Bull. Math. Soc. 53 (1947) 1120. RL Adler and TJ Rivlin, Proc. Am. Math. Soc. 15 (1964) 794. K. Umeno, Method of constructing exactly solvable chaos, Phys. Rev. E (1997) Vol.55: 5280-5284. [0020]

Conventionally, in such a random number generation method, a random number sequence (random one-dimensional vector sequence) can be obtained by giving a scalar value (one-dimensional vector) as a seed of a random number.

【0021】

[Problems to be Solved by the Invention]

However, the conventional random number generation method has the following problems.

【0022】

That is, the Monte Carlo method in a space of two or more dimensions requires a random sequence of vectors of two or more dimensions. However, in the conventional random number generation method, the obtained random number sequence is a sequence of scalar values (one-dimensional vector sequence). If the required number of values are selected one by one, there is a problem that the random number distribution is biased and the convergence is deteriorated.

【0023】

In addition, when generating public key cryptography, it is necessary to obtain a random number consisting of a pair of two integers, but in the conventional random number generation method, this pair cannot be generated at the same time, which is malicious. There was a problem that the defense against the decryptor was insufficient.

【0024】

In this way, a set of a plurality of random numbers is generated as one vector at the same time, and this can be used as a column to output a random vector sequence, and the density function of the distribution of these vector sequences can be analytically obtained. The demand for various output devices and output methods is extremely large.

【0025】

The present invention has been made to solve the above problems, and there are two methods for generating a vector sequence such that the density function of the distribution of the output random vector sequence is a known analytical function. At one point, a random vector sequence output device, output method, and output device, and output method, that combine these to output a higher-dimensional random vector sequence whose distribution density function is obtained as an analytical function. , An object of the present invention is to provide an information recording medium in which a program for realizing these is recorded.

【0026】

[Means for solving problems]

In order to achieve the above object, the following invention will be disclosed in accordance with the principle of the present invention.

【0027】

As shown in FIG. 1, the random vector sequence output device 100 of the present invention includes a first storage means 101, a first calculation means 102, a second storage means 103, and a second calculation means 104. The output means 105, the first update means 106, and the second update means 107 are provided. (A) The first storage means 101 stores a vector x having one or more dimensions, and (b) the first The calculation means 102 of 1 calculates the vector x'= f (x) as a result of applying the first rational vector mapping f to the vector x stored in the first storage means 101. [0028]

(c) The second storage means 103 stores a vector y of one or more dimensions, and (d) the second calculation means 104 is the result vector x'calculated by the first calculation means 101 and the second. The vector y'= g (x', y) as a result of applying the second rational vector mapping g to the one-dimensional or higher vector y stored in the storage means 103 of 2 is calculated. [0029]

(e) The output means 105 outputs a vector z'combining the result vector x'calculated by the first calculation means 102 and the result vector y'calculated by the second calculation means 104. (f) The first update means 106 stores and updates the vector x'of the result calculated by the first calculation means 102 in the first storage means 101, and (g) the second update means 107 , The vector y'of the result calculated by the second calculation means 104 is stored in the second storage means 103 and updated.

【0030】

Here, as the rational vector maps f and g, in addition to the maps based on the chaos theory described later, arbitrary maps used in the recurrence formula for generating random numbers can be used. For example, it is possible to use a mapping that multiplies a huge prime number to find the remainder.

【0031】

Further, in the random vector sequence output device of the present invention, the vector sequence obtained by applying the first rational vector mapping f to the vector x of one dimension or more 0 times or more. x, f (x), f (f (x)), f (f (f (x))), ... The density function of the limit distribution of is an analytical function, [0032]

A vector sequence obtained by applying a map g (λ, ·) obtained by giving a vector λ of one dimension or more as a parameter to the second rational vector map g to a vector y of one dimension or more 0 times or more. y, g (λ, y), g (λ, g (λ, y)), g (λ, g (λ, g (λ, y))), ... The density function of the limit distribution of is configured to be an analytical function with the parameter λ.

【0033】

Further, the first rational vector map f of the output device of the random vector sequence of the present invention is a rational map derived from the addition theorem of the elliptical function, particularly the Uram-von Neumann map, the cubic map, the quintic map, or , Katsura-Fukuda map, generalized Uram-von Neumann map, generalized cubic map, or generalized Chevisch map with predetermined parameters.

【0034】

Further, the second rational vector map g of the present invention is a rational map derived from the addition theorem of the elliptic function, particularly Katsura-Fukuda map, generalized Uram-von Neumann map, generalized cubic map, generalized Chevischev map. Can be either.

【0035】

When the first rational vector map f and the rational map derived from the addition theorem of the elliptical function as described above are selected as the second rational vector map g, the density function of the distribution of the sequence of vectors output sequentially by the output means 105. Can be obtained from the density function of the random sequence obtained from these maps.

【0036】

As shown in FIG. 2, the random vector sequence output device 200 of the present invention includes a generation means 201, a first output means 202, a second output means 203, and a third output means 204. , (a) The generation means 201 receives a vector ζ of two or more dimensions, and from this, generates a vector ξ of one dimension or more and a vector η of one dimension or more. [0037]

(b) The first output means 202 accepts the vector ξ generated by the generation means 201, and a recurrence formula using the first rational vector map f. x [0] = ζ x [i + 1] = f (x [i]) (where i 0) Outputs the vector sequence x [i] obtained by [0038]

(c) The second output means 203 receives the vector η generated by the generation means 201 and the vector sequence x [i] output by the first output means 202, and receives the second rational vector mapping g. Recurrence formula using y [0] = η y [i + 1] = g (x [i + 1], y [i]) (where i 0) Outputs the vector sequence y [i] obtained by [0039]

(d) The third output means 204 combines the vector sequence x [i] output by the first output means 202 and the vector sequence y [i] output by the second output means 203. The resulting vector sequence z [i] is output as a result.

【0040】

Further, in the random vector sequence output device of the present invention, the vector sequence obtained by applying the first rational vector mapping f to the vector x of one dimension or more 0 times or more. x, f (x), f (f (x)), f (f (f (x))), ... The density function of the limit distribution of is an analytical function, [0041]

A vector sequence obtained by applying a map g (λ, ·) obtained by giving a vector λ of one dimension or more as a parameter to the second rational vector map g to a vector y of one dimension or more 0 times or more. y, g (λ, y), g (λ, g (λ, y)), g (λ, g (λ, g (λ, y))), ... The density function of the limit distribution of is configured to be an analytical function with the parameter λ.

【0042】

Also, the first rational vector map f of the random vector sequence output device of the present invention is a rational map derived from the addition theorem of the elliptical function, particularly the Uram-von Neumann map, the cubic map, the quintic map, or , Katsura-Fukuda map, generalized Uram-von Neumann map, generalized cubic map or generalized Chevisch map with predetermined parameters.

【0043】

Further, the second rational vector map g of the output device of the random vector sequence of the present invention is a rational map derived from the addition theorem of the elliptic function, particularly Katsura-Fukuda map, generalized Uram-von Neumann map, general. It can be either a cubic cubic map or a generalized Chebyshev map.

【0044】

In this case as well, the density function of the distribution of the output vector sequence can be obtained analytically from the rational vector maps f and g.

【0045】

Further, the first output means of the random vector string output device of the present invention can also be the random vector string output device of the present invention. That is, (1) First, a random vector sequence output device X is constructed from a certain rational vector map f and a rational vector map g having parameters.

【0046】

(2) Next, the output device is made to correspond to a certain rational vector map f', and a new random vector sequence output device Y is similarly constructed from this and the rational vector map g'having a parameter. The vector dimension of the vector sequence output by Y is larger than the vector dimension of the vector sequence output by X.

【0047】

(3) By repeating this, it is possible to construct an output device of a random vector sequence of any dimension.

【0048】

The method for outputting a random vector sequence of the present invention includes the following steps. (a) The first acquisition step of acquiring a vector x of one dimension or more stored in the first storage means, and (b) the first rational vector mapping f to the vector x acquired in the first acquisition step. The first calculation step to calculate the vector x'= f (x) resulting from the application of [0049]

(c) A second acquisition step of acquiring a one-dimensional or higher vector y stored in the second storage means, (d) a vector x'calculated in the first calculation step, and a second acquisition step. The second calculation step to calculate the vector y'= g (x', y) resulting from applying the second rational vector mapping g to the vector y obtained in [0050]

(e) An output step that outputs a vector z'that combines the result vector x'calculated in the first calculation step and the result vector y'calculated in the second calculation step. [0051]

(f) The result vector x'calculated in the first calculation step is stored in the first storage means and updated in the first update step, and (g) the result calculated in the second calculation step. A second update step in which the vector y'is stored and updated in the second storage means.

【0052】

Further, in the random vector sequence output method of the present invention, a vector sequence obtained by applying the first rational vector map f to a vector x having one dimension or more 0 times or more. x, f (x), f (f (x)), f (f (f (x))), ... The density function of the limit distribution of is an analytical function, [0053]

A vector sequence obtained by applying a map g (λ, ·) obtained by giving a vector λ of one dimension or more as a parameter to the second rational vector map g to a vector y of one dimension or more 0 times or more. y, g (λ, y), g (λ, g (λ, y)), g (λ, g (λ, g (λ, y))), ... The density function of the limit distribution of is configured to be an analytical function with the parameter λ.

【0054】

Further, in the method of outputting a random vector sequence of the present invention, the first rational vector map f is a rational map derived from the addition theorem of the elliptical function, particularly the Uram-von Neumann map, the cubic map, and the quintic map. Alternatively, it can be either the Katsura-Fukuda map, the generalized Uram-von Neumann map, the generalized cubic map, or the generalized Chevisch map with predetermined parameters.

【0055】

Further, in the method of outputting a random vector sequence of the present invention, the second rational vector map g is one of the Katsura-Fukuda map, the generalized Uram-von Neumann map, the generalized cubic map, and the generalized Chebyshev map. be able to.

【0056】

In this case as well, the density function of the distribution of the output vector sequence can be obtained analytically from the rational vector maps f and g.

【0057】

A program that realizes an output device that outputs a random vector string of the present invention and an output method is recorded on an information recording medium such as a compact disk, floppy disk, hard disk, magneto-optical disk, digital video disk, magnetic tape, or semiconductor memory. can do.

【0058】

The program recorded on the information recording medium of the present invention is executed by an information processing device such as a general-purpose computer, a game device, a mobile information terminal, or a mobile telephone in a storage device, a calculation device, an output device, or the like. It is possible to realize an output device that outputs various vector sequences and an output method.

【0059】

Further, the information recording medium on which the program of the present invention is recorded can be distributed and sold independently of the information processing apparatus.

【0060】

BEST MODE FOR CARRYING OUT THE INVENTION

An embodiment of the present invention will be described below. It should be noted that the embodiments described below are for illustration purposes only and do not limit the scope of the present invention. Therefore, those skilled in the art can adopt embodiments in which each or all of these elements are replaced with equivalent ones, but these embodiments are also included in the scope of the present invention.

【0061】

(First Example) FIG. 3 is a block configuration diagram of the information processing device of the embodiment in which the output device of the random vector sequence of the present invention is realized in an information processing device such as a general-purpose computer.

【0062】

The information processing device 301 is controlled by a CPU (Central Processing Unit) 302, and temporarily stores data and the like in a main storage device 303 such as RAM (Random Access Memory), and is a hard disk, a floppy disk, or a CD-ROM (Compact). Disk Read Only Memory), magnetic tape, magneto-optical disk, and other external storage devices 304 store programs executed by the CPU 302.

【0063】

When the power is turned on to the information processing device 301, the CPU 302 first executes a program called an initial program loader stored in the ROM (Read Only Memory) 308, and then the operating system from the external storage device 304 or the like. The program of the above and the program of the application are loaded into the main memory 303 and executed.

【0064】

The executed result can be stored as a file in the external storage device 304 or displayed on a display device 305 such as a CRT (Cathode Ray Tube) or a liquid crystal display. The user of the information processing device gives an instruction to the information processing device using an input device 306 such as a mouse or a keyboard.

【0065】

Here, when the information processing device 301 functions as the output device 100 of the random vector sequence shown in FIG. 1, the main storage device 303 functions as the first storage means 101 and the second storage means 103, and the CPU 302. Functions as the first calculation means 102, the second calculation means 104, the first update means 106, and the second update means 107, and the external storage device 304 outputs the result as a file, the output means 105. The display device 305 functions as an output means 105 when displaying and outputting the result, and the main storage device 303 functions as an output means 105 when the result is used in another program.

【0066】

When the information processing device 301 functions as the output device 200 for the random vector sequence shown in FIG. 2, the CPU 302 is used together with the main storage device 303 and, if necessary, the external storage device 304 and the display device 305. It works and functions as a generation means 201, a first output means 202, a second output means 203, and a third output means 204.

【0067】

Further, the main storage device 303 and the external storage device 304 function as the information recording medium of the present invention. The ROM 308 can also function as the information recording medium of the present invention.

【0068】

Hereinafter, the processing of the random vector string output device of the present invention will be described with reference to FIG. FIG. 4 is a flowchart showing the flow of processing of the present invention.

【0069】

In the following, for convenience of explanation, the Uram-von Neumann map [Equation 1] is used as the rational map f, and the Katsura-Fukuda map is used as the rational map g. It is easy for those skilled in the art, and these embodiments are also included in the scope of the present invention.

【0070】

First, the CPU 302 acquires the seed of the random number from the current time and the like (step S401). In this case, the rational mapping f is applied to the scalar value (one-dimensional vector), and the rational mapping g is applied to the scalar value (one-dimensional vector) along with the parameters of the scalar value (one-dimensional vector). So we need two scalar value seeds. From a different point of view, we get a two-dimensional vector as a seed of random numbers.

【0071】

The seed of the random number can also be input by the user from the input device 306, and this may be combined with a numerical value such as a time. These are useful when generating public keys.

【0072】

Next, the CPU 302 stores the acquired seeds in the first storage means 101 and the second storage means 103 in the main storage device 303, respectively (step S402). As a result, the initial value for generating a random number is set.

【0073】

The process from step S401 to step S402 corresponds to the process executed by the generation means 201 of the device 200 that outputs the random vector sequence shown in FIG.

【0074】

Further, the CPU 302 acquires the value x stored in the first storage means 101 (step S403), calculates the value x'= f (x) using this (step S404), and calculates the calculated value. The x'is stored in the first storage means 101 and updated (step S405).

【0075】

That is, in step S404, the CPU 302 functions as the first calculation means.

【0076】

The CPU 302 then obtains the updated value x'stored in the first storage means 101 and the value y stored in the second storage means 103 (step S406), and uses these to obtain the value. y'= g (x', y) is calculated (step S407), and the calculated value y'is stored in the second storage means 103 and updated (step S408).

【0077】

That is, in step S407, the CPU 302 functions as a second calculation means.

【0078】

Finally, the CPU 302 combines the updated value x'stored in the first storage means 101 with the updated value y'stored in the second storage means 103 to provide an external storage device. Output to 304 etc. (step S409) and return to step S403.

【0079】

The result of combining one n-dimensional vector and another m-dimensional vector is a (n + m) -dimensional vector, and the elements are the n-dimensional vector elements first, and then the m-dimensional vector. It is an arrangement of the elements of. The processing of the generation means 201 of the output device 200 of the random vector sequence shown in FIG. 2 can be realized by performing the inverse operation of the vector combination.

【0080】

By repeating this process, a random sequence of two-dimensional vectors is output to the external storage device 304.

【0081】

The process executed by the first output means 202 of the random vector sequence output device 200 shown in FIG. 2 is in steps S403 to S405, and the process executed by the second output means 203 is in steps S406 to S408. , The processing executed by the third output means 204 is realized by step S406, respectively.

【0082】

In this embodiment, as described above, the generalized Ulam-von Neumann map [Equation 1] is adopted as the rational map f, but the density function of the distribution of random numbers obtained when this is used in the recurrence formula. Is shown in Fig. 5. As shown in Figure 5, this is a non-uniform density function defined in the range 0 x 1, which is infinite at x = 0 and x = 1 and down in the range 0 &lt;x &lt;1. It is a convex function.

【0083】

Further, in this embodiment, as described above, the Katsura-Fukuda function [Equation 7] is adopted as the rational map g, but the distribution of random numbers obtained when the parameter x'is fixed and used in the recurrence formula is also , [Equation 8] can be obtained analytically.

【0084】

[Number 7]

<img file="JP2000276331A_D0007.tif" />【0085】

[Number 8]

<img file="JP2000276331A_D0008.tif" />【0086】

FIG. 6 shows a sequential plot of the two-dimensional vector output in step S409 as coordinate values, and FIG. 7 shows the two-dimensional vector output in step S409 as coordinate values, which is used as a histogram. Show things. Further, FIG. 8 shows a state when the histogram shown in FIG. 7 is cut at a certain cross section.

【0087】

The inventor describes the recurrence formula of the present invention as "generally, a rational map f having a density function ρ (x) and a rational map g (x, ·) having a parameter x and having a density function ν (x, y). When combined by the method using, the density function of the distribution of the output vector sequence is ν (x, y) ρ (x). " Therefore, when the density function of the distribution is a known analytical function, when these are combined, the density function of the distribution can be obtained analytically as the product of the density functions of the original distribution.

【0088】

It can be seen that this conclusion is correct from the fact that the shape of the graph shown in FIG. 8 is almost the same as the shape of the graph shown in FIG. In addition, it is known that the bias of random numbers, which has been a problem in the conventional method, is small.

【0089】

It should be noted that, by appropriately changing the order of each step or separately executing the steps that perform the same processing, it is possible to realize the same processing as the control flow in the above embodiment, but such an embodiment is also present. Included in the scope of the invention.

【0090】

(Second Example) In the second embodiment of the present invention, an information processing device such as a general-purpose computer does not constitute an output device for a random vector sequence, but an electronic circuit.

【0091】

That is, both the first storage means 101 and the second storage means 103 of the random vector sequence output device 100 shown in FIG. 1 can be configured by a storage circuit based on a flip-flop or the like.

【0092】

Both the first calculation means 102 and the second calculation means 104 can be configured by a combination of an addition circuit and a multiplication circuit.

【0093】

The output means 105 can be configured by the output lines of the circuits constituting the second calculation means 104.

【0094】

The first update means 106 and the second update means 107 are the first storage means for the output lines of the circuits constituting the first calculation means 102 and the second calculation means 104, respectively, with a constant clock delay. It can be configured by returning to 101 and the second storage means 103 for storage.

【0095】

As described above, by configuring the output device of the random vector string of the present invention by the dedicated electronic circuit, a simple and space-saving configuration with low power consumption is required, for example, a personal digital assistant or a mobile telephone. The present invention can be applied to a device.

【0096】

(Third Example) In the above embodiment, a rational map derived from the addition theorem of the elliptic function is used as the rational map, but it can also be an elliptic integral, a super-elliptic integral, or a map derived from a variant of these. , Some have similar properties, and such mappings may be utilized. It is also possible to use a mapping that represents the recurrence formula of the random number generation method that has been used conventionally.

【0097】

(Fourth Example) Since the density function of the random vector sequence obtained in the present invention can be obtained analytically, the random vector sequence of arbitrary dimensions obtained by the present invention can be obtained by the inverse function method of von Neumann. It is possible to generate a random vector sequence with a uniform distribution.

【0098】

(Fifth Example) The present invention can be realized by stream processing which is frequently used in an operating system such as UNIX, a logical language such as prolog and GHC, and a functional language such as Lisp and Haskell. That is, for the rational vector maps f and g as described above, the process of generating the data stream of the following vector sequence x [i] (represented by predicates, functions, procedures, etc. in programming languages) and x [0], x [1], x [2], x [3], ... x [0] = ζ x [i + 1] = f (x [i]) (where i 0) By receiving the data stream x [i] output by this process A in order and preparing the process B to generate the data stream of the following vector string y [i], y [0], y [1], y [2], y [3], ... y [0] = η y [i + 1] = g (x [i + 1], y [i]) (where i 0) The present invention can be realized.

【0099】

Communication between process A and process B can be described by the so-called manufacturer-consumer model, and known techniques such as demand-driven generation and data stream buffering can be used.

【0100】

[Effect of the invention]

As described above, according to the present invention, the following effects are obtained.

【0101】

When there are two vector sequence generation methods in which the density function of the output random vector sequence distribution is a known analytical function, these are combined to form a higher dimensional random vector sequence. Therefore, it is possible to provide an output device and an output method of a random vector sequence that outputs what the density function of the distribution is obtained as an analytical function.

【0102】

In the present invention, since the product of the distribution density functions of the known vector sequence generation method becomes the density function of the distribution of a higher-dimensional random vector sequence, the characteristics of the distribution can be easily obtained, and various applications can be obtained. Can contribute to. In particular, when a rational mapping is adopted as the generation method, it has a feature that all the elements of the obtained vector sequence are also rational numbers, and has an advantage that the calculation accuracy on the computer can be strictly preserved.

【0103】

Such a random vector sequence can be used in the Monte Carlo method, CDMA in mobile communication and optical communication, public key cryptography in communication such as the Internet, and the like.

【0104】

Further, the information recording medium on which the program is recorded can be easily distributed or sold as a software product independently of the hardware of the information processing device. If the program recorded on the information recording medium of the present invention is executed by an information processing device such as a general-purpose computer, the random vector string output device and output method according to the above invention can be realized.

[Simple explanation of drawings]

[Figure 1]

It is a block block diagram which shows the structure of the output device of the random vector sequence of this invention.

[Figure 2]

It is a block block diagram which shows the structure of the output device of the random vector sequence of this invention.

[Fig. 3]

It is a block block diagram of the information processing apparatus of the Example which realizes the output apparatus of a random vector string of this invention in an information processing apparatus such as a general-purpose computer.

[Fig. 4]

It is a flowchart which shows the flow of the process of this invention.

[Fig. 5]

It is a graph which shows the density function of the distribution of random numbers obtained when the generalized Ulam-von Neumann map is used in the recurrence formula.

[Fig. 6]

It is explanatory drawing which sequentially plotted with the 2D vector output in step S409 of the flowchart shown in FIG. 4 as a coordinate value.

[Fig. 7]

This is a histogram using the two-dimensional vector output in step S409 of the flowchart shown in FIG. 4 as coordinate values.

[Fig. 8]

It is a graph when the histogram shown in FIG. 7 is cut at a certain cross section.

[Explanation of symbols]

100 Random vector sequence output device 101 First means of storage 102 First calculation method 103 Second storage means 104 Second calculation method 105 Output means 106 The first of the additional new means 107 Second update method 200 Random vector sequence output device 201 Generating means 202 First output means 203 Second output means 204 Third output means 301 Information processing equipment 302 CPU 303 Main storage 304 External storage 305 Display device 306 Input device 308 ROM