JP2011145464A - Device and program for generation of pseudo-random number - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To extend a cycle and secure maximum cycle length and uniformity. <P>SOLUTION: The pseudo-random number generating device includes: a bijective mapping preparing means 12 which prepares at least one original bijective mapping and a bijective mapping for cycle extension obtained by mapping calculation of a reverse mapping of the original bijective mapping and an ergodic bijective mapping of the original bijective mapping using a mapping function set by an initial value setter 11; a mapping value calculator 13 which repeats processing to obtain the mapping value using the bijective mapping for cycle extension further after obtaining the mapping value using the original bijective mapping only once, prepared by the bijective mapping preparing means 12; a bit string extractor 15 which extracts a bit string from the mapping value obtained by the mapping value calculator 13; and a pseudo-random number output means 15 which outputs the bit string extracted by the bit string extractor 14 until reaching the bit length set by the initial value setter. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a pseudo-random number generation device and a pseudo-random number generation program that generate pseudo-random numbers using bijective mapping.

  A conventional pseudo-random number generator is known that employs a pseudo-random number generation method using a non-linear (chaos) mapping in a finite operation implementation (see Patent Document 1). In this conventional example, a mapping that does not guarantee bijection is used. In addition, even if it is guaranteed to be bijection, a long period cannot be obtained by falling into the short period band, so a mechanism for monitoring the period is necessary.

  As a pseudo-random number generator that guarantees the maximum period and uniformity, a linear congruential method and an M-sequence random number (GFSR: Generalized Feedbacked Shift Register) are known. However, the cycle is limited, and the generation pattern is also a method of outputting the one determined in one direction, and the generation pattern is limited. Further, in these generation methods, each value appears only once in a single cycle. Therefore, if a certain value is observed, then that value does not appear in that cycle.

  Twisted GFSR and Mersenne Twister exist as an improvement of the GFSR, but the parameter band for extending the period is a fixed value, and there is an effort to search for it.

  Further, as a pseudo-random number generation device using mapping change or mapping conversion, a device using a mapping such as a two-dimensional cat map is known (see Patent Document 2). However, this two-dimensional cat map is said to be inappropriate for encryption and decryption, and it is not possible to extend the period.

JP 2003-152706 A JP 2007-264147 A

  The present invention has been made in view of the present situation in the generation of pseudo-random numbers as described above, and its purpose is to extend the period, ensure the maximum period length and uniformity, and generate a large number of random numbers. Another object of the present invention is to provide a pseudo random number generation device and a pseudo random number generation program using bijective mapping.

  The pseudorandom number generator according to the present invention uses a mapping function necessary for generating a pseudorandom number, an initial value setting means for setting an initial value of the bit length of the pseudorandom number, and a mapping function set by the initial value setting means. A period extending bijective map obtained by mapping at least one bijective map, a reverse map of the original bijective map, and an ergodic bijective map of the original bijective map. And a bijection mapping preparation means for preparing the periodical extension bijection after obtaining a mapping value by using the original bijection mapping prepared once by the bijection mapping preparation means. A mapping value calculating means for repeatedly obtaining a mapping value using mapping; a bit string extracting means for extracting a bit string from the mapping value obtained by the mapping value calculating means; and a bit string extracted by the bit string extracting means as the initial value. Value setting Characterized by comprising a pseudo-random number output means for outputting to the bit length set by the unit.

  In the pseudo-random number generation device according to the present invention, the bijection mapping preparation means includes a mapping function based on a linear congruence method, a mapping based on an M-sequence random number, a mapping based on a modified tent function, or a mapping method other than the above. One of the projection maps is used.

  In the pseudorandom number generator according to the present invention, the bijection map preparation means includes an ergodic complete unit for each period determined by multiplication of the number of prepared bijection maps and the number of sets included in the bijection map. It is characterized by preparing a new bijective map for period extension with a modified projective map.

  In the pseudorandom number generation device according to the present invention, the bijection map preparation means prepares a bijection map for period extension when a bijection map of the prepared number of original bijective maps is made. To do.

  In the pseudorandom number generation device according to the present invention, the bijection map preparation means prepares a period extension bijection map in advance before bijection mapping of the number of bijection maps is made.

  The program for generating pseudo-random numbers according to the present invention includes a computer that generates pseudo-random numbers, an initial value setting unit that sets an initial value of a mapping function necessary for generating random numbers and a bit length of pseudo-random numbers, and the initial value setting unit. Using a set mapping function, it is obtained by calculating a map of at least one original bijective map, an inverse map of the original bijective map, and an ergodic bijective map for the original bijective map. A bijection mapping preparation means for preparing a period extension bijection map, a bijection map preparation means prepared by the bijection map preparation means, and after obtaining a map value by using the original bijection map once each further Map value calculation means for repeatedly obtaining a map value using a bijection map for period extension, bit string extraction means for extracting a bit string from the map value obtained by the map value calculation means, and extraction by the bit string extraction means The bit string, characterized in that to function as a pseudo random number output means for outputting to the bit length set by the initial value setting means.

  The program for generating pseudo-random numbers according to the present invention uses a computer that generates pseudo-random numbers as a bijection mapping preparation means, mapping by linear congruential method, mapping by M-sequence random numbers, mapping by modified tent functions, or other methods than the above It is made to function so that any one of the bijection maps created by (1) may be used as a mapping function.

  The pseudo random number generation program according to the present invention uses a computer that generates pseudo random numbers as a bijection mapping preparation unit, and is determined by multiplying the number of prepared bijection maps by the number of sets included in the bijection map. It is characterized in that it functions so as to prepare a new bijection map for period extension in which the ergodic bijection map is changed for each cycle.

  The pseudo-random number generation program according to the present invention uses a computer that generates pseudo-random numbers as a bijection mapping preparation means, and when a bijection map of the prepared original bijection map number is made, It is characterized by functioning to prepare a projective map.

  The program for generating pseudo-random numbers according to the present invention uses a computer that generates pseudo-random numbers as a bijection mapping preparation unit, and a bijection map for period extension is used in advance before bijection mapping of the number of bijection maps is made. Characterized by functioning to prepare

  According to the present invention, at least one original bijective map, an inverse map of the original bijective map, and a period extension obtained by a map calculation of an ergodic bijective map with respect to the original bijective map. The bijection map is prepared, the prepared bijective map is used once to obtain a map value, and then the process of obtaining the map value using the period extending bijection map is repeated. Thus, the map value is calculated, so that the period extension is performed by the bijection map for period extension, and the maximum period length and uniformity can be ensured.

  In the present invention, any bijective map can be used without limitation, and the order of mapping can be changed and reused, so abundant generation patterns can be provided.

The figure which shows the map value cyclic state for demonstrating the period extension method of the map value by a bijection group in the pseudorandom number generator of this invention. The figure for demonstrating the production | generation of the period extension bijection map of the map value by the bijection group in the pseudorandom number generation apparatus of this invention. The block diagram which shows the structure of the pseudorandom number generator which concerns on this invention. The block diagram of the computer system which comprises the pseudorandom number generation apparatus which concerns on this invention. The flowchart for demonstrating operation | movement of the 1st Example of the pseudorandom number generator which concerns on this invention. The flowchart for demonstrating operation | movement of the 2nd Example of the pseudorandom number generator which concerns on this invention. The figure which shows the row | line | column of each initial value using the linear congruential method as a mapping by the bijection group in the pseudorandom number generation apparatus of this invention. The figure which shows the row | line | column of each initial value using M series (GFSR) as a mapping by the bijection group in the pseudorandom number generator of this invention. The figure which shows the row | line | column of each initial value using M series (TGFSR) as a mapping by the bijection group in the pseudorandom number generator of this invention. The figure which shows the map of a deformation | transformation tent map used as a map by the bijection group in the pseudorandom number generator of this invention. The figure which shows the map transition using a deformation | transformation tent map as a map by the bijection group in the pseudorandom number generator of this invention.

A pseudorandom number generation device and a pseudorandom number generation program according to the present invention will be described below with reference to the accompanying drawings. The principle is as follows. As an example, the map shown in Fig. 1 when three maps "f ₀ , f ₁ , f ₂ " of the all-injection group of the finite set x (the number of sets is 4) shown in the following (Formula 1) is prepared. For example,

  These recurrence formulas are expressed by the following (formula 2).

In the above description, the order y _i of the mapping functions f _{yi to be} selected is not limited, but y _i = (i + 1) mod 3 (i = 0, 1,...) Is used in the example of FIG.

The numerical sequence obtained by repeatedly performing mapping by the above mapping has an initial value x ₀ = 0,
0, 2, 1, 3, 1, 2, 2, 0, 3
A periodic band having a cycle length of 9 is obtained (the pattern of the number sequence as the mapping result is the same even with the initial values x ₀ = 2 and x ₀ = 3). When the initial value x ₀ = 1,
1, 3, 0
A period band with a period length of 3 is obtained. Thus, when the example mapping is used as it is, the generation pattern exists in two periodic bands.

  Thus, simply using a bijection map, the periodic bands are divided into several, there are short periods, and the frequency of occurrence of each value is not uniform. As such a system, there is a random number sequence obtained by using a non-linear mapping under a finite operation, which becomes a problem when trying to obtain a pseudo-random number sequence in which long periodicity and uniformity are guaranteed.

Here, by changing “f ₂ ” to the ideal map “Q”, the maximum period length 12 when three types of maps are used is realized, and the appearance frequency of each value is uniformly 3. Allows you to get a sequence of numbers. That is, an ideal bijection map “Q” for obtaining a pseudo-random number sequence in which long periodicity and uniformity are guaranteed is created as follows.

First, an inverse map “f ₀ ⁻¹ , f ₁ ⁻¹ ” of the bijective map “f ₀ , f ₁ ” used in FIG. 1 is generated. In addition, an ergodic bijective map “α” is prepared. Then, using these inverse maps “f ₀ ⁻¹ , f ₁ ⁻¹ ” and the ergodic bijective map “α”, a calculation for creating a map “Q” as shown in FIG. 2 (Q = αf ₀ ^{− 1} f ₁ ^-1 ).

Thus, by changing the bijection map “f ₂ ” used in FIG. 1 to “Q”, the initial value x ₀ = 1 is obtained.
0, 2, 1, 1, 3, 0, 2, 0, 3, 3, 1, 2
It is possible to obtain a numerical sequence having a maximum cycle length of 12 repeating the above, and a numerical sequence in which the appearance frequency of each value is a uniform value (three times).

  That is, when the number of bijective maps used in FIG. 2 is “3”, the number of sets is “4”, and three types of bijective maps are used, the maximum period length is It is obtained from 3 × 4 = 12. And as above-mentioned, it turns out that the pseudorandom number sequence in which the appearance frequency of each value is 3 times is obtained.

In the above, the mapping function “f _yi ” (y _i = imodK: i is the number of mappings, K is the number of prepared bijective maps) may be any bijection, and the mapping function “f _yi ” is Change every mapping.

  The map “α” used in the period extension calculation used in FIG. 2 may be a bijective map having a maximum period that is divided into several period bands and does not fall into a short period (in the present invention, the period extension calculation is performed). The map “α” used for the above is called an “ergodic bijective map”). For example, the set elements of the mapping can be simply shifted (f (x) = (x + 1) mod N where N is the number of sets). It is important to calculate and create a map “Q” that extends the period using the inverse of the selected bijective map and an ergodic bijective map. As a result, it was found that a random number sequence having the maximum period length and uniformity can be generated.

The bijection map that can be selected, such as the linear congruence method known for the pseudorandom number generation method with the maximum period, M series (GSFR), and the improved version with extended GSFR period (Twisted GFSR, Mersenne Twister, etc.) The completed one can also be used, but there is no particular limitation as long as it is a bijection map even if it includes a short period band. In these systems, the random number sequence pattern is determined, but it is possible to create more random number sequence patterns by changing the order of the bijective map “f _yi ” or changing the map “Q”. is there.

  The pseudorandom number generator according to the embodiment of the present invention is configured as shown in FIG. That is, an initial value setting unit 11, a bijection mapping preparation unit 12, a mapping value calculation unit 13, a bit string extraction unit 14, and a pseudo random number output unit 15 are provided. The initial value setting means 11 sets an initial value of the mapping function and the bit length of the pseudo random number necessary for generating the pseudo random number. The initial value can be input from input means such as a keyboard of a computer system.

  The bijective mapping preparation means 12 uses the mapping function set by the initial value setting means 11 to at least one original bijective map, an inverse of the original bijective map, and the original bijective. A period extending bijective map obtained by mapping calculation with an ergodic bijective map is prepared. The map value calculation means 13 obtains a map value by using the original bijection map prepared by the bijection map preparation means 12 once, and further maps by the bijection map for period extension. The process of obtaining a value is repeated.

  The bit string extraction unit 14 extracts a bit string from the mapping value obtained by the mapping value calculation unit 13. The pseudo-random number output means 15 outputs the bit string extracted by the bit string extraction means 14 until the bit length set by the initial value setting means 11 is reached.

  The pseudorandom number generator is realized by a computer system 100 shown in FIG. The computer system 100 includes a CPU 101 as a computer that generates pseudo-random numbers, and a memory 102 in which a pseudo-random number generation program used by the CPU 101 is stored, and calculation results and the like performed by the CPU 101 are stored.

  The CPU 101 includes an input unit 103 for inputting data such as a keyboard and a mouse, and an output unit 104 for outputting the generated pseudo-random number to a display output, a print output, or another system.

  The CPU 101 functions as each unit in FIG. 3 by executing a pseudo random number generation program corresponding to the flowchart shown in FIG. The operation of the pseudorandom number generator will be described below based on this flowchart. At the start, initialization processing is performed (S11). In the initialization process S11, for example, a seed Seed is given, and when there are a plurality of mapping functions prepared in advance based on the seed Seed, one mapping function is selected. Mapping functions include linear congruential mapping, M-sequence random number mapping, GFSR (Generalized Feedback Shift Register) mapping, Twisted GFSR (GFSR improvement) mapping, Mersenne Twister (TGFSR improvement) mapping, modified tent mapping Any other bijective map satisfying the bijection (such as the map shown in FIG. 2) can be used. In the case of having one mapping function, selection based on seed Seed is not performed. The seed Seed is given by, for example, generating an appropriate numerical value every time the program starts, and performs function selection using a table that associates the seed Seed with a mapping function.

Further, the initialization step S11 smell, determines the mapping the initial value x ₀ which is based on seeds Seed (determination technique similar to the mapping function), initialization of the mapping count i to 0, setting the order of the mapping random pseudorandom The bit length n is set (input by the user).

Following step S11, it generates the K total injective mapping f ₀ ~f _K-1 from the mapping function selected above and to prepare for mapping (S12). The bijection maps f _{0 to} f _K−1 prepared here are called original bijection maps in the sense that they are generated from the above mapping function. Next to step S12, using the original bijective map prepared in step S12, mapping calculation is performed based on the mapping order set in the initialization process of step S11 to obtain one mapping value. . (S13).

In the mapping calculation (S13), when the number of the original bijection maps f _y1 prepared in step S12 is K, the calculation is performed by the following recurrence formula (Formula 3). Further, the order of mapping is selected by the function h.

When the mapping value x _{i + 1} is obtained in step S13, i is counted up to i + 1 (S14), and a predetermined bit at the position set in the initialization process in step S11 is extracted and stored in the buffer ( S15). For example, the extraction formula for extracting the least significant bit is x _{i + 1} & 1.

  Following step S15, it is detected whether it is time to output the accumulated bit string (S16). If YES, the accumulated bit string is output (S17). As the output timing, (1) output when the buffer is full, (2) output when requested by the system, (3) output when requested by the user, or the like is initialized. If NO in step S16, the process proceeds to step S18.

  When the process branches to NO in step S16 or when the process in step S17 ends, it is determined whether or not to prepare a bijection map for period extension (S18). That is, it is detected whether or not the mapping by the K-time original bijection mapping has been completed. Here, if “YES” is determined in the step S18, a period extending bijective map is prepared in a step S19.

Specifically, as already mentioned, an original bijective mapping _{"f 0, f 1 ···,} f K-1" reverse mapping ^{_{"f 0 -1, f 1 -1}} ···, f K- ₁ ^-1 ”is generated, and an ergodic bijection map“ α ”is prepared. Then, the inverse map “f ₀ ⁻¹ , f ₁ ⁻¹ ..., F _K-1 ⁻¹ ” and the ergodic bijective map “α” are multiplied to create the map “Q”. . Here, the relationship after the following (Formula 4) is established.

Following this step S19, the process proceeds to step S13, and a map calculation is performed using the period extending bijective map x _{i + 1} prepared in step S19 to obtain one map value (S13). On the other hand, if NO in step S18, the process proceeds to step S20, where it is detected whether the random bit length of the pseudo-random number generated so far has become the initially set length n (S20). Returning to step S12, the process is continued, and when YES is determined in step S20, the process ends and the process is ended.

  In this way, the bijective map preparation means 12 changed the ergodic bijective map for each period determined by the multiplication of the number of prepared bijective maps and the number of sets included in the bijective map. A new bijective map for period extension is prepared. In the flowchart shown in FIG. 5, the bijection map preparation means 12 prepares a bijection map for period extension when the bijection map of the prepared original bijection map number is made (step S19). However, regardless of this, as shown in the flowchart of FIG. 6, the bijection map for period extension is prepared in advance before the bijection map of the bijection map number is made (step S30). good.

In the flowchart of FIG. 6, in step S30, K bijective maps f _{0 to} f _K-1 are generated from the map function selected in step S11 to prepare the map, and the original bijective map “ f ₀ , f ₁ ..., f _K-1 ”and the inverse map“ f ₀ ⁻¹ , f ₁ ⁻¹ ..., f _K-1 ⁻¹ ”is generated, and ergodic bijection Prepare the map “α”. Then, the inverse map “f ₀ ⁻¹ , f ₁ ⁻¹ ..., F _K-1 ⁻¹ ” and the ergodic bijective map “α” are multiplied to create the map “Q”. .

  The created map “Q” is stored in a buffer in the memory 102 in preparation for processing, and used in step S13 in a necessary cycle. The ergodic bijective map “α” can be obtained by simply shifting the set elements of the map (f (x) = (x + 1) mod N, where N is the number of sets). In this case, a different α may be created for each period by changing the shift amount for each period, and a bijection map for period extension may be prepared using this α. Creating a different α for each period may be performed in step S19 in the flowchart of FIG.

  In the flowchart of FIG. 6, except for step S30, the process of step S18 for determining whether or not to prepare a period extension bijection map and the step S19 of preparing a period extension bijection map in the flowchart of FIG. Is deleted. In step S20, if branched to NO, the process returns to step S13 to continue the processing. Except for this, the same processing as that described in FIG. 5 is performed.

  Next, an example of pseudorandom number generation processing performed when the mapping function is a linear congruential mapping will be described. A commonly used linear congruential equation is given by the following (Equation 5).

As an example, if M = 7, and a = 2 and c = 0, the bijection map “f ₀ ”

The column according to (Equation 6) above has an initial value X ₀ = 1,
A period band of period length 3 that repeats 1, 2, 4;
When the initial value X ₀ = 3, there are a total of two period bands, including a period band with a period length of 3 that repeats 3, 6, and 5.

When a = 5 and c = 0, if the bijection map is “f ₁ ”

From the above (Equation 7), a row that circulates with a maximum period length 6 of 1, 5, 4, 6, 2, 3 is obtained.
A map sequence is created using two bijective maps “f ₀ , f ₁ ” and a map “Q” that realizes the maximum period and uniformity. "F _0, f _1" reverse mapping ^{_{"f 0 -1, f 1 -1}} " and, here ergodic bijective mapping alpha, is as follows.

By preparing such a bijection map “α” and the inverse map “f ₀ ⁻¹ , f ₁ ⁻¹ ”, a map “Q” that realizes the maximum period and uniformity is Q = αf ₀ ^−1. Calculating with f ₁ ^-1

FIG. 7 shows a column obtained from the initial values of “f ₀ → f ₁ → Q”. When the initial value is 1
1, 2, 3, 2, 4, 6, 3, 6, 2, 4, 1, 5, 5, 3, 1, 6, 5, 4
Thus, a sequence satisfying a uniform appearance frequency with a maximum period length of 3 × 6 = 18 is obtained.

  Next, an example of pseudo-random number generation processing performed when the mapping function is a mapping of a GFSR (Generalized Feedback Shift Register) known by M-sequence random numbers will be described. The recurrence formula of GFSR is as (Equation 8). Here, when n = 3 and m = 1, (Equation 9) is obtained.

As a result, a circulating numerical sequence having a cycle length of 7 (2 ⁿ −1) can be generated. Assuming that the bijection map of the GFSR is “f ₀ ”, f ₀ is expressed by the following (Equation 10).

Next, a map sequence is created using “f ₀ ” and a map “Q” that realizes the maximum period and uniformity. Incidentally, the degree of freedom in setting the GFSR is only the initial value x _0, the pattern of the sequence is only one. "F _0" and the inverse mapping "f ₀ ^-1", here ergodic bijective mapping alpha, is as follows.

When such a bijection map “α” and the above inverse map “f ₀ ⁻¹ ” are prepared and a map “Q” that realizes the maximum period and uniformity is calculated by Q = αf ₀ ⁻¹ , the following is obtained. Q is obtained.

The mapping of “f ₀ → Q” is repeated using f ₀ and Q, and the numerical sequence obtained from each initial value is as shown in FIG. When the initial value is 1
1, 4, 2, 5, 3, 1, 4, 2, 5, 6, 6, 7, 7, 3, 1
Thus, the maximum cycle length is 2 × 7 = 14, and a numerical sequence satisfying a uniform appearance frequency is obtained.

  Next, an example of pseudo-random number generation processing performed when the mapping function is a mapping including TGFSR (Twisted Generalized Feedback Shift Shift Register), which is an improved version of GFSR known by M-sequence random numbers, will be described. The recurrence formula of TGFSR is expressed by the following (formula 11), and A is the following matrix.

If the vector a in the bottom row of the matrix is selected to achieve the maximum period, the period will be 2 ^nω-1 . Here, when n = 2, m = 1, and ω = 2, (Equation 11) becomes the recurrence formula of the following (Equation 12), and this recurrence formula is calculated.

When the bijection map is calculated with the matrix of A as follows, the following “f ₀ ” is obtained.

The map of “f ₀ ” above is
1, 8, 10, 14, 11, 6 with a period length of 6, 2, 4, 5, 13, 7, 9 with a period length of 6, and 3, 12, 15 with a period length of 3, It has a periodic band consisting of three short periods. For this reason, “f ₀ ” alone does not provide the maximum period and does not result in TGFSR.

When the bijection map is calculated with the matrix of A as follows, “f ₁ ” is obtained.

上記“ f₁”の写像では、
1, 12, 15, 7, 13, 3, 8, 10, 14, 11, 2, 4, 5, 9, 6
の最大周期を実現する周期長１５の数列が得られる。“f₁”はエルゴード的であり、TGFSRであるといえる。“f₀”の逆写像“f₀ ^-1 ”と、“f₁”の逆写像“f₁ ^-1 ”と、更に、ここでエルゴード的な全単射写像αとを次の通りに用意する。

In the map of “f ₁ ” above,
1, 12, 15, 7, 13, 3, 8, 10, 14, 11, 2, 4, 5, 9, 6
A numerical sequence with a period length of 15 that realizes the maximum period is obtained. “F ₁ ” is ergodic and can be said to be TGFSR. Reverse mapping ^{_{"f 0""f 0 -1}} ", the inverse mapping "f ₁ ^-1" of "f _1", further, wherein preparing the ergodic bijective mapping α as follows .

When the map “Q” realizing the maximum period and uniformity is calculated by Q = αf ₀ ⁻¹ f ₁ ⁻¹ , the following Q is obtained.

The columns obtained from the initial values of “f ₀ → f ₁ → Q” are as shown in FIG. If the initial value is 1,
1, 8, 10, 2, 4, 5, 3, 12, 15, 4, 5, 9, 5, 13, 3, 6, 1, 12, 7, 9, 6, 8, 10, 14, 9, 2, 4, 10, 14, 11, 11, 6, 1, 12, 15, 7, 13, 7, 13, 14, 11, 2, 15, 3, 8
And the maximum cycle length is 3 × 15 = 45, and a number sequence satisfying a uniform appearance frequency is obtained.

In this design, there are generally 2 ^ω−1 +1 bijection maps at arbitrary n and ω. Therefore, the period of the bijection map can be at least 2 ^ω−1 +1. At this time, the prime period of the pseudo random number sequence is (2 ^ω−1 +1) (2 ^nω −1), and the appearance frequency of each value is uniformly 2 ^ω−1 +1. For example, when ω = 32 and n = 624, the prime period is (2 ³¹ +1) (2 ¹⁹⁹⁶⁸ −1), the same calculation accuracy is 32 bits, and the period 2 of the Mersenne Twister 19997 whose state dimension is 32 × 624 ^A much longer periodic train is obtained than ^19937-1 .

Next, an example of pseudorandom number generation processing performed when the mapping function is a modified tent mapping that is a bijection group using a one-dimensional mapping will be described. When the calculation accuracy is L bits, the modified tent mapping formula expressed by the calculation width N = 2 ^L −1 is as follows, and a map of the modified tent mapping is shown in FIG.

When the number of bijective maps that can be prepared from the modified tent map equation is K, bijective maps of each map “f _{M j} ” can be prepared as follows. Here, the maximum number of bijective maps that can be prepared is K ≦ N. The inverse map “f _{M j} ⁻¹ ” is as follows.

  Further, an ergodic bijective map “α” is prepared, and a calculation for creating a map “Q” is executed as in the following equation.

As described above, after K maps are calculated using the prepared K maps “f _{M j} ”, the map “Q” is applied to perform cyclic mapping as shown in FIG. In this case, since the number of elements in the set of each bijection map “f _{M j} ” is N + 1 of the calculation accuracy, it is a prime period (N + 1) (K + 1), and each value appears K + 1 times. The prime cycle length is (N + 1) ² when K = N, and a pseudo-random number sequence with guaranteed uniformity can be obtained.

11 Initial value setting means 12 Bijection mapping preparation means 13 Mapping value calculation means 14 Bit string extraction means 15 Pseudorandom number output means 100 Computer system 102 Memory 103 Input section 104 Output section

Claims (10)

An initial value setting means for setting a mapping function necessary for generating a pseudo-random number and an initial value of the bit length of the pseudo-random number;
Using the mapping function set by the initial value setting means, at least one original bijective map, an inverse map of the original bijective map, and an ergodic bijective map for the original bijective map, A bijection map preparation means for preparing a bijection map for period extension obtained by the map calculation of
A map value prepared by the bijective map preparation means, wherein a map value is obtained by using the original bijective map once, and then a process of obtaining a map value by using the period extending bijective map is repeated. Calculation means;
A bit string extracting means for extracting a bit string from the mapped value obtained by the mapped value calculating means;
And a pseudo-random number output unit that outputs the bit string extracted by the bit string extraction unit until the bit length set by the initial value setting unit is reached.   The bijection mapping preparation means uses any one of a mapping by a linear congruential method, a mapping by an M-sequence random number, a mapping by a modified tent function, or a bijection map created by a method other than the above as a mapping function. The pseudorandom number generation device according to claim 1.   The bijective mapping preparation means a new cycle extension that changes the ergodic bijective map for each cycle determined by multiplying the number of prepared bijective maps by the number of sets included in the bijective map. 3. The pseudorandom number generator according to claim 1 or 2, wherein a bijection map is prepared.   The bijection map preparation means prepares a bijection map for period extension when a bijection map having the prepared number of original bijective maps is made. The pseudo-random number generator according to item.   The bijection map preparation means prepares a bijection map for period extension in advance before bijection mapping of the number of bijection maps is made. The pseudorandom number generator described. A computer that generates pseudo-random numbers,
An initial value setting means for setting an initial value of a mapping function necessary for generating a pseudorandom number and a bit length of the pseudorandom number;
Using the mapping function set by the initial value setting means, at least one original bijective map, an inverse map of the original bijective map, and an ergodic bijective map for the original bijective map, A bijection map preparation means for preparing a bijection map for period extension obtained by the map calculation of
A map value prepared by the bijective map preparation means, wherein a map value is obtained by using the original bijective map once, and then a process of obtaining a map value by using the period extending bijective map is repeated. Calculation means,
A bit string extracting means for extracting a bit string from the mapped value obtained by the mapped value calculating means;
A pseudorandom number generation program that functions as a pseudorandom number output unit that outputs a bit string extracted by the bit string extraction unit until a bit length set by the initial value setting unit is reached. A computer that generates pseudo-random numbers,
As a bijection mapping preparation means, a function using any one of a mapping by a linear congruential method, a mapping by an M-sequence random number, a mapping by a modified tent function, or a bijection mapping created by a method other than the above is used as a mapping function. The pseudo-random number generation program according to claim 6. A computer that generates pseudo-random numbers,
As a bijection mapping preparation means, for a new period extension by changing the ergodic bijection map for each period determined by multiplying the number of prepared bijection maps and the number of sets included in the bijection map The pseudo-random number generation device according to claim 6 or 7, wherein the pseudo-random number generation device functions to prepare a bijection map. A computer that generates pseudo-random numbers,
9. A bijective map preparation means that functions so as to prepare a bijective map for period extension when a bijective map of the prepared number of original bijective maps is made. The pseudorandom number generation program according to any one of the above. A computer that generates pseudo-random numbers,
9. The bijection map preparation means is made to function so as to prepare a bijection map for period extension in advance before bijection maps of the number of bijective maps are made. Or the pseudo random number generation program according to item 1.

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