JP7079546B2 - Pseudo-random number generator and pseudo-random number generator - Google Patents

Description

The present invention relates to a pseudo-random number generator and a pseudo-random number generator.

Conventionally, several methods for generating pseudo-random numbers by one-dimensional mapping (tent mapping or logistic mapping) have been proposed. These acquire values as random numbers each time they are mapped by one-dimensional mapping. However, in the implementation of finite arithmetic precision, there is a problem that the random number performance deteriorates due to entering a fairly short periodic band for the arithmetic accuracy. For this reason, we have adopted a mechanism to change the sequence to extend the period and improve the random number performance.

In order to extend the period in the conventional random number generation by one-dimensional mapping, the random number generation sequence is self-feedback or disturbed by an uncertain sequence from the outside. Therefore, with such a random number generation method, it is not possible to obtain a guarantee of uniformity that the same value once output theoretically is not output, and the random number property deteriorates when the numerical value is biased. It contained concerns. In addition, there is a problem that the maximum cycle length cannot be definitively estimated.

Patent Document 1 discloses a device for generating a chaotic random number sequence in consideration of periodicity. In Patent Document 1, a means for skillfully extending a period in a random number generation method using a one-dimensional map is introduced, but the aspect of guaranteeing uniformity is not mentioned, and there is a limit to the extension of the period length.

Further, "linear congruential method", "linear feedback shift register (LFSR)", "Mersenne Twister" and the like are known as those that can obtain a uniform random number in one cycle. However, in these cases, it is necessary to empirically search in advance for the optimum parameters such that random numbers are generated with a uniform value at the maximum cycle.

Patent Document 2 discloses a periodic extension method in which a prime number is used when selecting a sequence of random numbers from a sequence of random numbers generated by a random number generation module. However, the one described in Patent Document 2 does not guarantee the uniformity of the period.

Japanese Unexamined Patent Publication No. 9-292978 Japanese Unexamined Patent Publication No. 2009-3925

The present invention has been made in view of the problems in the conventional random number generation as described above, and is a pseudo-random number generator and a pseudo-random number capable of generating a pseudo-random number sequence having a substantially unlimited period length. The purpose is to provide a generator.

An embodiment of the pseudo-random number generator according to the present invention is a tent mapping calculation means that performs a mapping calculation by a tent mapping, and when the maximum section of the tent mapping is multiplied by an integer in order to enable the tent mapping by an integer calculation. Detects whether the equation (P-1) / 2 using the prime number P of is a prime number, and if (P-1) / 2 is not a prime number, factor (P-1) / 2 into a prime number to obtain the prime factor. Based on whether (P-1) / 2 is a prime number or the prime factor satisfies a predetermined conditional expression, the tent mapping parameter selection means for selecting the prime number P and giving it to the tent mapping calculation means, and the tent. It is characterized by comprising a random number generation means for generating a prime number sequence having a predetermined length based on the calculation result by the mapping calculation means.

The block diagram which shows the structure of embodiment of the pseudo-random number generation apparatus which concerns on this invention. The flowchart which shows the operation of the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the map of the tent map used in the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the time-series change of the tent map used in the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the map of the logistic map which has the relationship of topological conjugation with the tent map used in the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the time-series change of the logistic map which has a phase conjugation relationship with the tent map used in the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the value fluctuation of the time series for showing the relationship of the phase conjugation of the tent map and the logistic map used in the embodiment of the pseudo-random number generator which concerns on this invention. The figure which shows the value when the method of the multiplier of 2 is adopted by the prime number P. The figure which shows the time-series change numerically when the tent mapping operation is performed by the equation (8) using some prime numbers P. The flowchart which shows the operation by 1st Embodiment of the tent mapping parameter selection means of the pseudo-random number generation apparatus which concerns on this invention. Information on whether or not "(P-1) / 2" is a prime number and "(P-1) / 2" when the prime number P takes a value in the row direction (when it is a prime number, "(P-1) / 2" -1) A diagram showing a table listing the result values obtained by factoring "(P-1) / 2" into "(P-1) / 2" with "○" in the "2 is a prime number" column. The figure which shows the tent mapping operation value of 2 series and the value of a random number bit realized by an example of the random number extraction method of the pseudo-random number generation apparatus which concerns on this invention in time series in a column direction. The flowchart which shows the processing of the 2nd random number generation method which is an example of the random number extraction method embodiment of the pseudo-random number generation apparatus which concerns on this invention. The figure which shows the tent mapping map of the tent mapping operation of the previous stage in the method of FIG. As an example of the random number extraction method embodiment of the pseudo-random number generator according to the present invention, for the random number generator, the result of the tent mapping calculation and all or part of the value of the prime number P are calculated, and a predetermined value is obtained from this calculation result. The figure which shows the numerical example when it is a long random number sequence. The figure which shows the flow of calculation when 7 is adopted as P by the processing of the 3rd random number generation method which is an example of the random number extraction method embodiment of the pseudo-random number generation apparatus which concerns on this invention. It is a figure which shows the flow of calculation when 43 is adopted as P by the processing of the 3rd random number generation method which is an example of the embodiment of the random number extraction method of the pseudo-random number generation apparatus which concerns on this invention. The flowchart of the embodiment of the P selection process including the "determination excluding an impossible short cycle" adopted by the embodiment of the pseudo-random number generator according to the present invention. The figure which shows the nature of the multiplication of the remainder by the joint arithmetic.

In the present embodiment, it is realized that the value that appears once in the random number sequence generated by the tent map never appears again. As a result, it is possible to generate a random number sequence guaranteed within that one cycle, and it is possible to extend the cycle endlessly.

The tent map has a topological conjugation relationship with the logistic map, and the tent map corresponds to the phase of the logistic map. Focusing on the number of maps whose phase returns to the original, the value P obtained by multiplying the maximum interval [0,1] by P in order to enable tent mapping by integer operation corresponds to the above "number of maps whose phase returns to the original". However, the period length of the map can be known from Fermat's little theorem.

Therefore, the maximum period is realized by using a prime number (a natural number that is indivisible and has a remainder other than 1 or its own number) for this value P. That is, a prime number is adopted as the above value P. Then, the maximum period and the guarantee of uniformity are determined by using Fermat's little theorem or the formula shown later in this embodiment, and the selected prime number is used to perform mapping by tent mapping for one cycle to obtain a random number. do.

When the iteration of the tent map goes around one cycle, the next prime number candidate is prepared, it is judged again whether the maximum period and uniformity can be guaranteed, and the method of selecting the prime number is adopted (parameter transition control shown later). Means 40). By performing a tent map that can be realized with a prime number for one cycle of the maximum cycle, then selecting a different prime number and repeating the mapping of the tent map one cycle at a time, a system that can expand the cycle virtually endlessly is generated. And make it possible to obtain a random number sequence that guarantees uniformity.

The merit of using the tent map is that it can be configured by a simple operation of subtraction and shift operation as compared with the conventional pseudo-random number generation method. Therefore, a random number sequence having an ergodic property that has a small calculation load and guarantees the maximum period and uniformity can be acquired in an endlessly long period.

Further, in the present embodiment, since the random number performance is poor if the numerical value for each iteration of the tent mapping with the slope 2 is used as it is, a method of extracting a random number sequence from the numerical value X _i is also described.

<Embodiment 1> A prime number is adopted as a value P that P times the maximum interval [0,1] of the tent map. Hereinafter, an embodiment of the pseudo-random number device according to the present invention will be described with reference to the accompanying drawings. In the present embodiment, in the case of performing random number generation using a tent map, a random number value generated by using a prime number as a value P obtained by multiplying the maximum interval [0,1] by P in order to enable the tent map by an integer operation. Make the cycle length uniform and obtain a sufficiently long cycle.

As shown in FIG. 1A, an embodiment of the pseudo-random number generation device according to the present invention includes a tent map calculation means 10, a tent map parameter selection means 20, and a random number generation means 30. The tent mapping calculation means 10 performs a mapping calculation by tent mapping. The tent map parameter selection means 20 is a prime number and a maximum cycle length condition based on (P-1) / 2 as a value P when multiplying the maximum interval by an integer in order to enable the tent map by an integer operation. Is selected and given to the tent mapping calculation means 10. The random number generation means 30 generates a sequence of predetermined lengths based on the calculation result by the tent mapping calculation means 10.

In this embodiment, the parameter transition control means 40 is provided. The parameter transition control means 40 receives the notification every time the tent map calculation means 10 performs a calculation for one cycle of the tent map, and controls the tent map parameter selection means 20 to give a new P. The parameter transition control means 40 can be omitted in the pseudo-random number generation device having a configuration in which the generation of the pseudo-random numbers is completed within one cycle of the tent map.

(Map)
○ About the tent map The tent map is defined by the following equation (1).

A map of the tent map is shown in FIG. 2, and a time series in which i of the equation (1) is taken on the horizontal axis and the vertical axis is X _i is shown in FIG.

In Japanese Unexamined Patent Publication No. 2016-039418, a tent mapping type A / D conversion circuit is used to acquire bit 0 when “X _i <0.5” and when “0.5 ≦ X _i ”. , It is shown that when bit 1 is acquired and 4 bits are continuously obtained, 16 bit strings can be obtained by selecting from each interval in which the interval [0, 1] is evenly divided into 16 with respect to the initial value X ₀ . Therefore, since any pattern can be potentially generated by the tent map, it can be seen that if a bit string that can pass the random number test is generated by the method of taking the initial value X ₀ , it can be used as a random number generator having high random number performance.

Further, since any binary value can be taken depending on the setting of the initial value X ₀ of the tent map, a series having periodic behavior can be given as long as possible. Therefore, if the cycle length can be known in advance from the initial value X ₀ of the tent map, the tent map for that cycle length is performed and output as a random number sequence, and then the mapping of the cycle band having a different cycle length is performed. Thus, it is considered that an unlimited number of pseudo-random numbers with different cycles can be obtained by repeating mappings with different repetition cycles. That is, the initial value X ₀ is used to map the tent for one cycle and output as a random number sequence, then the initial value X ₀ is changed and the tent map for the next cycle is performed and output as a random number sequence. Therefore, it is considered that an unlimited number of pseudo-random numbers with different cycles can be obtained.

Considering that the initial value X ₀ of the tent map is set in this way, a circular decimal number (rational number [a number represented by a fraction]) may be set. Further, if the cycle length is known in advance from the initial value X ₀ , it is advantageous for the design of random numbers. Here, it can be utilized that the tent map is known to be in topological conjugation with the logistic map. In other words, since the phase of the logistic map corresponds to the tent map, it is possible to obtain a uniform random number generation sequence by deriving the determination of the maximum period at which the phase returns to the original state using Fermat's little theorem explained below. ..

○ Topological conjugation between the tent map and the logistic map The logistic map is defined by the following equation (2).

A map of the logistic map is shown in FIG. 4, and a time series in which i of the equation (2) is the horizontal axis and the vertical axis is X _i is shown in FIG.
In the time series of FIG. 5, the initial value X ₀ = 0.3 is set, and the mapping transition of Eq. (2) is
X ₁ = 4x0.3 (1-0.3) = 0.84
Is.

The result X ₁ = 0.84 is repeatedly input and
X ₂ = 4x0.84 (1-0.84) = 0.5376
Is obtained. In this way, the output value is repeated as an input value, and the vertical axis X _i is transitioned in the open interval (0, 1).

The tent map is defined by Eq. (1), and the iterated interval of the map is an open interval (0,1), where the interval (0, π) multiplied by the pi (3.14159 ...) is used as X _i . Can be transformed into the following equation (3) by replacing with the equation with θ _i .

Here, θ of the equation (3) is substituted as θ ₁ of the equation (1), and at this time, the logistic map is set to X ₁ and the variable transformation of the following equation (4) is performed.

By capturing the mapping of the tent mapping equation (3) as the phase of the equation (4), it can be seen that the logistic mapping always has a phase-conjugated relationship that synchronizes with the equation (4).

From FIG. 6, it can be seen that the tent map system and the logistic map system are synchronized, and it can be visually read that there is a topological conjugation relationship.
Substituting the initial value X ₀ of the logistic map equation (2) for the trigonometric function sin ² θ,

Focusing on the phase "2 ⁿ θ" of the trigonometric function of the equation (5), which is a modification of the logistic mapping equation (2) by selecting the initial value X ₀ from FIG. 6, simply adding π to the initial value θ is an arbitrary integer (here). Then, when the value divided by (the prime number) is entered, it can be seen that the iteration of the logistic map transitions in a short cycle.

For example, when the value “θ = π / 11 (0.2855599332 ...)” obtained by dividing π by 11 is set, when X ₁₁ is calculated from the equation (5), X ₀ ⁼ sin ² 20 (π / 11) = sin. ² (π / 11)
X ₁ = sin ² 2 ¹ (π / 11) = sin ² (2π / 11)
X ₂ = sin ² 2 ² (π / 11) = sin ² (4π / 11)
X ₃ = sin ² 2 ³ (π / 11) = sin ² (8π / 11)
X ₄ = sin ² 2 ⁴ (π / 11) = sin ² (16π / 11)
= Sin ² (π + 5π / 11) = (-sin (5π / 11)) ²
= Sin ² (5π / 11)
X ₅ = sin ^{2 25} (π / 11) = sin ² (32π / 11)
= Sin ² (2π + 10π / 11) = sin ² (10π / 11)
= Sin ² (π-10π / 11) = sin ² (π / 11)
X ₆ = sin ² 2 ⁶ (π / 11) = sin ² (64π / 11)
= Sin ² (4π + π + 9π / 11) = (-sin (9π / 11)) ²
= Sin ² (9π / 11) = sin ² (π-9π / 11)
= Sin ² (2π / 11)
X ₇ = sin ² 2 ⁷ (π / 11) = sin ² (128π / 11)
= Sin ² (10π + π + 7π / 11) = (-sin (7π / 11)) ²
= Sin ² (7π / 11) = sin ² (π-7π / 11)
= Sin ² (4π / 11)
X ₈ = sin ² 2 ⁸ (π / 11) = sin ² (256π / 11)
= Sin ² (22π + π + 3π / 11) = (-sin (3π / 11)) ²
= Sin ² (3π / 11) = sin ² (π-3π / 11) = sin ² (8π / 11)
X ₉ = sin ² 2 ⁹ (π / 11) = sin ² (512π / 11)
= Sin ² (46π + 6π / 11) = sin ² (6π / 11)
= Sin ² (π-6π / 11) = sin ² (5π / 11)
X ₁₀ = sin ² 2 ¹⁰ (π / 11) = sin ² (1024π / 11)
= Sin ² (92π + π + π / 11) = (-sin (π / 11)) ²
= Sin ² (π / 11)
X ₁₁ = sin ² 2 ¹¹ (π / 11) = sin ² (2048π / 11)
= Sin ² (186π + 2π / 11) = sin ² (2π / 11)

When focusing on "X ₀ " and "X ₅ ", the phase θ returns to "π / 11", and the phase θ of "X ₁₀ " is also "π / 11", so the logistic map When "initial value X ₀ = sin ² (π / 11)" is set, it can be read that the map has a period of 5 and the mapping changes. Further, in the tent map equation (3), when “initial value X ₀ = π / 11”, it can be seen that the map repeats semipermanently with a period length of 5 from the relationship with the topological conjugation with the logistic map.
In the tent mapping equation (1), the maximum interval of the equation (3) is [0, π], so if "initial value X ₀ = 1/11" divided by π is set, the mapping repeats with a period length of 5. It turns out.

Focusing on the phase "2 ⁿ θ", the initial value "θ = π / 11" is multiplied by "2 ⁿ " to obtain the phase, and the phase returns to the original value at 2π (360 °). Looking at "n = 11", the phase obtained by raising 2 to the 11th power is
2 ¹¹ (π / 11) = 2048π / 11 = 186π + 2π / 11 = 2π / 11
This is 2 ¹¹ = 2048, and when divided by "11", the remainder is 2 (quotient is 186). Therefore, in phase, "2048π / 11" and "2π / 11" are synonymous. Focusing on the value of this remainder, the value 2 of the numerator whose "modulo" of the joint arithmetic indicating the remainder obtained by dividing 2 to the 11th power by 11 corresponds to "2π / 11".

Modular arithmetic is expressed by the following formula.

The above modulo formula shows that when 2 ¹¹ = 2048 is divided by 11, the remainder is 2, and when the modulo 11 method is adopted, the remainder is 2.

In Fermat's little theorem, "p" is a prime number in congruence arithmetic, and "a" is an integer that is not a multiple of "p"("a" and "p" are relatively prime [greatest common divisor is 1], that is, "p" is It is known that the following equation (6) holds when the number is prime).

In the above example, assuming that the prime number "p" is 11 and "a" is 2, the relationship of the equation (6) is established. It can also be seen that the following equation (7) that the remainder of dividing "a" to the "p-1" power by "p" is 1 holds (meaning that both sides of equation (6) are divided by "a"). ..

FIG. 7 shows the calculation result of the equation (6) when “p” is set to “a = 2” from the prime numbers 7 to 47.
The numerical value 1 is shaded, but when "p = 11", that is, when the multiplier is "p-1 = 10", the value adopted by the method is 1, and when "p = 19", the multiplier is ". When p-1 = 18 ”, the value obtained by the method is also 1, and it can be confirmed that the equation (7) holds.

From the above, when the prime number "p" is given as the denominator as the initial value of the phase of the equation (5) of the logistic mapping and the product is multiplied by "2 ^p ", the number that is not divisible from the equation (6) is the improper fraction (2 ^p ). Since the true fraction of the mixed fraction (186) from π / p) is "2π / p", 2 of the element of the true fraction is shown as the remainder of the division. From this, it can be seen from Fermat's little theorem that there is a logistic map when the denominator "p" is a prime number and a period in which the value returns when the tent map whose phase is "p-1".
That is, in the tent mapping equation (1), if "initial value X ₀ = 1 / p" is given, X _p-1 returns to "1 / p" and the initial value X ₀ , and the period of "p-1". Is expected to have.

Here, the following equation (8) is obtained by expanding the maximum interval from [0,1] to P times so that the equation (1) can be calculated with an integer.

Since the interval is multiplied by P in the above equation (8), "initial value X ₀ = 1 / p" is given in equation (1), but "initial value X ₀ = 1" is given in equation (8). ..

FIG. 8 shows each X _i obtained when P is given a “initial value X ₀ = 1” in a tent mapping using prime numbers 7 to 47 and the mapping is repeated. Here, the value of X _i should be doubled from the tent map equation (8) and take an even number, but if an odd number is taken for the initial value X ₀ , the value that the initial value X ₀ returns to the original value can be confirmed. , When the equation (8) is “P / 2 ≦ X _i ”, the value of “(P—X _i )” is shown as X _i .
In 2 ^p-1 of FIG. 7, the value obtained by the method (modulo) by P is 1, but similarly, in the tent map of FIG. 8, the value of the “P-1” th mapping is returned to 1. Can be read.

Here, paying attention to the column of prime numbers 11, the result of taking the multiplier of 2 on the left side is that the value obtained by the method is 1 when "n" is 10, and the period is 10, but "n" is in the tent map. The value at 5 is also 1, and the cycle is 5.
As you can see from the example of tracing by giving the phase "θ = π / 11" of the initial value X ₀ of the logistic map equation (5), it is the same value as X ₀ at X ₅ , but it is trigonometric. Because of the function
X ₅ = sin ^{2 25} (π / 11) = sin ² (32π / 11)
= Sin ² (2π + 10π / 11) = sin ² (10π / 11)
However, in the trigonometric complement angle formula, "sin (10π / 11)"
= Sin (π-10π / 11) ”, so
= Sin ² (π-10π / 11) = sin ² (π / 11)
Is obtained and returns to the initial value X ₀ .

That is, as shown in FIG. 7, the value obtained by taking the method of 11 for the multiplier of 2 is taken once for each {1,2,3,4,5,6,7,8,9,10}, and the cycle length is 10. Again, when the phase of the sine function is "2 ⁿ θ", the phases are {6π / 11, 7π / 11, 8π / 11, 9π / 11, 10π / 11} and {5π / 11, 4π / 11}. , 3π / 11 / 2π / 11, 1π / 11}, since the sine function (sinθ) takes the same value, it is found that the period length of the logistic map and the tent map is 5, and it is found that the period length of the logistic map and the tent map is 5. The value to be taken is {1,2,3,4,5} once each.

According to the rule of Fermat's little theorem in FIG. 7, when n = ⁵ , the value obtained by adopting the modulo of 25 is 10, but when the values from n = 5 to n = 9 are listed, {10 , 9, 7, 3, 6}, and it can be seen that the number obtained by subtracting each value from 11 is {1, 2, 4, 8, 5}, and the phase of the logistic map equation (5). By applying to, it is confirmed that the logistic map repeats with a period length of 5.

In the above example, the case where the prime number 11 is used as the method is shown, but when the prime number is represented by the variable P, the logistic map and the tent map have P = 11, so they are repeated in a cycle of "(P-1) / 2 = 5". It turns out.
However, paying attention to the prime number {7,17,31,41,43} in FIG. 8, it can be confirmed that the maximum period when the prime number P is taken does not take "(P-1) / 2". For example, in "43", the maximum period of the tent map equation (8) is (43-1) / 2 = 21, and the numerical values of 1 to 21 are acquired once each to ensure uniformity within the period. However, since the cycle length is 7, it returns to 1 at {1,2,4,8,16,11,21} and is repeated, so it can be confirmed that all the values from 1 to 21 cannot be obtained uniformly.

Therefore, the following algorithm as in the second embodiment is adopted so as to guarantee that the value once taken at the maximum cycle length (P-1) / 2 is never taken again and a uniform value can be taken. ..

<Embodiment 2> Selection of prime numbers that satisfy uniformity in the maximum period In this embodiment, the tent mapping parameter selection means 20 shown in FIG. 1A maximizes that (P-1) / 2 is a prime number. It is characterized in that P is selected as a condition of the cycle length.
In FIG. 8, each prime number P of the equation (8) is applied, and “X _i ” or “PX _i ” for each iteration of the tent map starting from the initial value X ₀ = 1 is shown. As explained above, since the logistic map and the tent map are in a phase-conjugated relationship, the period length is "(P-1) / 2" and returns to "X _i = 1" in all prime numbers. You can confirm that it is there.

However, from the desire to acquire a uniform random number sequence, we want to obtain a sequence in which the value once taken is never taken again when "(P-1) / 2" is set as the maximum cycle. It is a thing. For example, the sequence of P = 43 has a cycle length of 7 in which the sequence becomes {1,2,4,8,16,11,21} within the cycle of (43-1) / 2 = 21, and the cycle length is 21. It is not a uniform sequence.

Here, when the initial value X ₀ = 3, which was not taken in the sequence as a trial, is set, the sequence becomes a sequence of period 7 of {3, 6, 12, 19, 5, 10, 20}. Examining the values not taken in the first and second sequences, it is found that "initial value X ₀ = 5" is applicable, and this sequence is {5,10,20,3,6,12,19. }, Which is a sequence with a period length of 7, and if these three are combined, the numerical values 1 to 21 can be taken one by one.

Further, when P = 43, “(P-1) / 2” becomes 21, but when 21 is factored into prime factors, it is decomposed into “3 × 7”, and when the initial value X ₀ = 1, the cycle length A case where 3 returns to "X ₃ = 1" or a cycle length of 7 returns to "X ₇ = 1" is conceivable.
Here, from the tent mapping equation (8), "X ₁ = 2" always follows "initial value X ₀ = 1", so if "(P-1) / 2>3", the cycle length is 1. Is impossible. Further, as described above, according to Fermat's little theorem (7), the value obtained by modulo with P at the period length "P-1" always takes "1".

Therefore, when "(P-1) / 2" is not a prime number but a composite number, the prime factorization is performed and "(P-1) / 2" is divided by the prime factor to shorten the divided cycle length. It is speculated that there may be a period. However, if "(P-1) / 2" is a prime number, the maximum period is "(P-1) / 2", and it can be guaranteed that a uniform sequence is obtained. Because, if "(P-1) / 2" is divisible by any prime factor, there is a possibility that it laps several times with the divided cycle length and returns to "1", but it is an indivisible numerical value (2 or more). If it is a prime number), the structure cannot be decomposed into a period smaller than "(P-1) / 2", and it is presumed that it returns to "1" at "(P-1) / 2". The relationship of phase conjugation is shown above, but from Fermat's little theorem, the period is "P-1" and the value obtained by using the P method (modulo) always returns to "1".

That is, if "(P-1) / 2" is a prime number, the initial value X ₀ = 1 is started, X _i = 1 is not taken within this cycle, and the original value X is circulated by X _p-1 . Return to _p-1 = 1. Here, as a prime factor, taking P = 37 as an example, it becomes “(P-1) / 2 = 18”, and when 18 is factored into prime factors, it becomes “2 × 3 × 3”. Therefore, a few prime factors can be mentioned, and when examining the cycle, dividing "(P-1) / 2 = 18" by the prime factor 2 gives 9 and may have a cycle length of 9, and other prime factors 3 may have a cycle length of 9. When "(P-1) / 2 = 18" is divided, it becomes 6, so it can be inferred that there is a possibility of having a cycle length of 6.

From the above, if "(P-1) / 2" is a prime number, it is possible to obtain a uniform sequence in which the number output once in the cycle is not output twice. As shown in the flowchart of FIG. 9, which corresponds to the process performed by the first embodiment of the tent mapping parameter selection means 20 shown in FIG. 1A, it is determined whether "(P-1) / 2" is a prime number and a prime number. If so, the value is adopted and applied to the equation (8). After executing equation (8) and outputting a sequence with a cycle length of "(P-1) / 2", another prime number P whose "P" and "(P-1) / 2" are prime numbers is selected. By repeatedly outputting a sequence with a cycle length of "(P-1) / 2" according to equation (8), the maximum cycle length is guaranteed and uniform values can be continuously output. It is understood.

In FIG. 10, the value of “(P-1) / 2” when the prime number P takes a value in the row direction, and the information on whether or not “(P-1) / 2” is a prime number (when it is a prime number). Shows a table listing the results of prime factorization of "(P-1) / 2") and "(P-1) / 2" in the "(P-1) / 2 is a prime number" column. The prime numbers shown in FIG. 10 are obtained by primality testing using an algorithm called "Elastotenness Sieve". This embodiment employs an algorithm that determines whether or not "(P-1) / 2" is a prime number, performs a mapping operation if it is a prime number, and outputs a random number sequence. In the mapping operation using one prime number, "(P-1) / 2" tent mapping operations are performed, and when this mapping operation is completed, "(P-1) / 2 is a prime number P" in FIG. In the column, the prime number candidate marked with "○" can be adopted next, and the mapping can be performed "(P-1) / 2" times to output a random number. As described above, the criterion of "(P-1) / 2 is a prime number" is a criterion of whether or not the prime number P can be adopted as a value P obtained by multiplying the maximum interval [0,1] of the tent map by P. It can be said that there is, and it is said to be "a criterion for adoption as a doubled value of the maximum section of the tent map".

○ Random number generation method Here, we will touch on the random number generation method. That is, a specific operation example of the random number generation means 30 in FIG. 1A is shown.
Consider using the value of the tent map X _i in equation (8) itself as a random number. In this case, the operation is started from the initial value X ₀ = 1 and a sequence of "X ₁ = 2, X ₂ = 4, ..." is obtained. That is, since it is limited to the result of multiplying by 2, the pattern of the sequence is limited to each cycle in order to use it as a random number, and the random number performance is poor. Therefore, the random number generation means 30 in FIG. 1A according to the present embodiment adopts a method of acquiring a random number from X _i of the tent map as follows.

Japanese Unexamined Patent Publication No. 2016-039418 introduces an A / D conversion circuit using a tent map. The method described in this publication acquires a bit "0" when "X _i " is less than 0.5 for each iteration of the mapping in which the interval [0, 1] of the initial value X ₀ is evenly divided, and "X". When _i "is 0.5 or more, bit" 1 "is acquired, and a bit string corresponding to the continuously obtained initial value X ₀ is obtained. Therefore, instead of using the output number sequence as a random number as it is, bit "0" is extracted when "X _i <0.5", and bit "1" when "0.5 ≤ X _i ". A method is conceivable in which a bit string is continuously obtained one bit at a time and used as a random number string according to the rule of extracting. That is, in the first random number generation method, the mapping results by the tent mapping are sequentially arranged to obtain a random number. According to this method, it can be estimated that the random number test will pass depending on how the initial value X ₀ is taken, so it is sufficient to set the initial value X ₀ that passes the random number test in advance. However, the random number generation speed becomes slow in the method of extracting random numbers of only one bit at a time of mapping. Therefore, in order to output a value that can be used as a random number sequence, a random number extraction method as shown in FIG. 11 can be considered.

In FIG. 11, assuming that P = 83 is given to the tent mapping formula (8), “(P-1) / 2 = 41” is obtained, and “a criterion for adoption as a doubling value of the maximum section of the tent mapping”. Is a prime number that satisfies. Therefore, it is possible to adopt a configuration in which the processing shown in the flowchart shown in FIG. 12 is performed so that the cycle length is determined to be 41. That is, as shown in step S21, the initial value AX ₀ and the initial value BX ₀ of the tent map are prepared. Further, as the tent map, two different tent maps A and tent map B shown in step S23 are prepared. In this way, "initial value AX ₀ = 1" is given as the tent map A prepared with two different initial values X ₀ , and "initial value BX ₀ = 23" is given to the other tent map B, and two at the same time. Execute (S21 to S23). Random number extraction is performed according to the reference shown in step S22. Here, as shown in the tent map of FIG. 13, when P in the equation (8) is “X _i <P / 4 (= 20.75)”, the bit “0” is output and “P / 4” is output. When (= 20.75) ≤ X _i <P / 2 (= 41.5) ", bit" 1 "is output, and" P / 2 (= 41.5) ≤ X _i <3P / 4 (= 62). When "0.25)", the bit "0" is output, and when "3P / 4 (= 62.25) ≤ X _i ", the bit "1" is output (S22). The example of FIG. 12 is an example in which two equations given two different initial values X ₀ = 1 and X ₀ = 23 are executed at the same time, and the output for two bits is obtained as a random number bit string. That is, this second random number generation method includes two series of tent maps, gives different initial values to each series, repeats the tent map, and repeats the tent map, one bit from one series for each tent map, for a total of 2 bits. Is repeated to obtain a random number sequence of a predetermined length by arranging them.

In addition, in FIG. 11, the result of "2 (P-AX _i )" and "2 (P-BX _i )" is "(P-AX)" so that it can be seen that the sequence takes 1 to 41 once each. The value of " _i )" or "(P-BX _i )" is displayed as a column of AX _i or BX _i .
In this way, instead of using X _i as a random number as it is, by extracting bits from X _i and executing calculations in parallel (simultaneously) with series with different initial values X ₀ , the period length corresponding to the prime number P can be increased. It is possible to generate more variations of random number patterns that can be confirmed and guarantee uniform values.

Other embodiments will be described. Three different initial values X ₀ are given, and three equations (8) are executed in parallel. The first initial value X ₀ extracts the lower bit digits 2nd and 6th to 2 bits, and the second initial value X ₀ extracts the lower bit digits 4th and 7th to 2 bits, and the third initial value. In X ₀ , 4 bits are extracted from the 1st, 3rd, 5th, and 8th low-order bit digits. By outputting these three outputs in total for 8 bits by the random number extraction unit of FIG. 1, it can be expected that a more distributed random number can be obtained. It is assumed that the number of digits can be included in the digit of the maximum value P of the closed interval [1, P-1] from the equation (8). Therefore, as the initial setting values, the number of operations of the equation (8) in parallel (simultaneously), each initial value X ₀ , and the bit digit for extracting a random number are given as the initial setting values. That is, the third random number generation method includes a series of three series of tent maps, gives different initial values to each series, repeats the tent map, and for each tent map, a predetermined bit from the first series, a second. A predetermined bit is obtained from a series and a predetermined bit is obtained from a third series, and the bits are arranged to form a random sequence having a predetermined length. That is, a random number sequence having a predetermined length is obtained by one mapping. Of course, the number of series may be 3 or more.

Further, a variation of random numbers may be added by changing these initial settings by switching the prime number P. As shown in FIG. 14, for example, when the prime number P = 83 is expanded in binary, it becomes “(1010011) ₂ ”. From this bit information, the lower 4 bits “(0011) ₂ ” are extracted, and P in the equation (8). = 83, and the initial value X ₀ = 1 is set, and the iterative operation of the equation (8) is performed. 4 bits are extracted from each "X _i " as in the column of "lower 4 bits" of "X _i " in FIG. 14, and exclusive-OR is performed with "(0011) ₂ " for the lower 4 bits taken from the prime number 83. Extract as a random number as shown in the rightmost column of FIG. 14. Every time the prime number is changed in this way, it is conceivable to increase the variation as a random number value by extracting the bit digit from the changed prime number. Since this fourth example has 4 bits, it is possible to obtain a predetermined length bit by mapping twice. Further, in the first to third random number generation methods, a predetermined bit is derived from the prime number P. Is taken out, an exclusive OR with the mapping value of each series is obtained, and these may be arranged by the first to third random number generation methods. Further, the present invention is not limited to a part of the value of P. An operation (not limited to exclusive-OR) may be performed on all the values of P and the mapping values of each series.

Further, in the above embodiment, in order to take a sequence that guarantees the cycle length "(P-1) / 2" in random number generation, a random number is obtained from the value in the case of P / 2≤X _i (P-X _i ). However, it is also possible to artificially double the cycle length of "X _i ". For example, taking the prime number 11 in FIG. 8 as an example, the sequence has a period length of {1, 2, 4, 3, 5}, but when P / 2 ≤ X _i (P-X _i ). If X _i is taken in the first order instead of taking, the sequence becomes {1, 2, 4, 8, 6}. Then, after the period is aligned in the tent map, the sequence {P-1, P-2, P-4, P-8, P-6} is obtained by subtracting all the sequences in the first order from P. .. Here, when P = 11 is applied, it becomes {10,9,7,3,5}, and together with the result of the first order above, {1,2,4,8,6,10,9,7, It is possible to take a cycle length of 10 called 3,5}. If it is applied to other P-values, it is conceivable that the cycle length "P-1" can be obtained, and a sequence of numbers whose cycle length is doubled can be obtained and a random number sequence can be obtained from them. In this fifth random number generation method, after obtaining a sequence of mapping values for the first cycle, a sequence of mapping values for one cycle is used as a "double value of the maximum interval of the tent mapping" and a prime number satisfying the criterion of adoption. The second sequence is obtained by subtracting from "P", the second sequence is connected to the sequence of mapping values for the first cycle, and the sequence is arranged, and a predetermined length is extracted from the sequence. The method of taking out is to take out a predetermined length from a position (position of a sequence) prepared in advance (if the number at the end does not reach the predetermined length, return to the beginning of the sequence and take out). This process is performed by changing the prime number P for each cycle.

In each embodiment, a method of selecting the prime number P in ascending order can be considered. For example, in order to increase the variation of random numbers, some ascending orders are skipped and selected, and the number of skipped numbers is changed every time by using the random numbers output so far, which can be considered as a method for improving the randomness. Further, the pseudo-random number generation device may include all of the above random number generation methods, and the operator or the user may select one method.

<Embodiment 3> Select a prime number that satisfies the uniformity in the maximum period by a mathematical formula. In the second embodiment, if "(P-1) / 2" is not a prime number, it is skipped and "(P-1) / 2" is It was an algorithm that searches for a prime number and obtains a sequence of numbers that are output once in one cycle and never output again from iterations of tent mapping. However, in the column of the tent map in FIG. 8, the cycle length is "(P-1) / 2" in the numerical values 13, the numerical value 19, the numerical value 29, and the numerical value 37 in which "(P-1) / 2" is not a prime number. It can be confirmed that it is. As described above, even if "(P-1) / 2" is not a prime number, it can be seen that there is a numerical value capable of obtaining a sequence in which the number output once in one cycle is not output twice. Therefore, in the following, an algorithm is adopted so that these can be used for random number generation.

In the tent map parameter selection means 20 according to the present embodiment, Km ≦ (log ₂ P-1) holds for the period length candidate Km calculated by the prime factor Cm obtained by factoring (P-1) / 2 into prime factors. It is characterized in that P is selected as a condition of the maximum cycle length. Further, the tent map parameter selection means 20 according to the present embodiment may determine the establishment of 2 ^Km ± 1≡0 (mod P), and if it is established, the P may not be used.

Focusing on the behavior of the tent mapping formula (8), the length of the cycle when starting from the initial value X ₀ = 1 of the tent mapping and returning to "PX _i = 1" is set to "prime number P = 7" in FIG. Consider a formula that traces and estimates in the example of. First, in the equation (8), the prime number P = 7 and the initial value X ₀ = 1 are set and started.

FIG. 15 shows the flow of calculation. Focusing on equation (8), when P / 2 = 3.5 or more, the power of 2 is executed until it becomes 3.5 or more in order to execute "7-X _i ". Since X ₀ = 1, "7-X ₂ " is executed at X ₁ = 2 (= 2 X ₀ ) and X ₂ = 4 (= 2 X ₁ ).

In FIG. 15, it is indicated by “X ₂ = 7-22”, and the index of ² here is indicated as “z ₁ = 2”. Since X ₂ = 3 and "7-X _i " is executed again, the power of 2 is executed until it becomes 3.5 or more, so "7" at X ₃ = 2 ¹ x 3 = 6 (= 2 X ₂ ). Since "-X ₂ " is executed, "7-6 = 1" and the initial value X ₀ = 1 are returned, it can be seen that the equation (8) is repeated with a cycle length of 3. Here, the index z ₂ = 1 of 2.

Applying the above flow to algebraic expressions,
7-2 ¹ (7-2 ² ) = 1
It turns out that holds true. Here, if the prime number 7 is the variable P and the exponent is z ₁ (= 2) and z ₂ (= 1) as shown in FIG. 15, then

When expanded next to

Is obtained, and when it is deformed by P, the following equation (9) is finally obtained.

Further, as another example, a prime number P = 43, an initial value X ₀ = 1 is set, and tracing is performed until X _i = 1 is returned. FIG. 16 shows the flow of calculation. Focusing on the equation (8), when P / 2 = 21.5 or more, the power of 2 is executed until it becomes 21.5 or more in order to execute "43-X _i ".

Since X ₀ = 1, "X ₁ = 2 (= 2 X ₀ ), X ₂ = 4 (= 2 X ₁ ), X ₃ = 8 (= 2 X ₂ ), X ₄ = 16 (= 2 X ₃ ), X ₅ "43-X ₅ " is executed at "= 32 (= 2X ₄ )". In FIG. 16, it is indicated by “X ₅ = 43-25”, and the index of 2 here is indicated as “z ₁ = ⁵ ”. Since X ₅ = 11, the power of 2 is executed until 21.5 (43/2) or more is executed in order to execute "43-X _i " again. Therefore, "43-X ₆ " is executed at X ₆ = 2 ¹ x 11 = 22 (= 2 X ₅ ), and "43-2 ¹ x 11 = 21" is obtained, and the index of 2 here is set to "". It is shown as "z ₂ = 1".

The next execution of "43-X _i " is X ₇ = 21 x 21 = 42 (= ² X ₆ ), and "43-X ₇ " is executed to be "43-42 = 1", where Let the exponent of 2 be "z ₃ = 1". Since the initial value X ₀ = 1 is returned, it is found that the equation (8) is repeated with a cycle length of 7, and it is expressed as a cycle length of 7 = z ₁ + z ₂ + z ₃ (= 5 + 1 + 1).

Applying the above flow to algebraic expressions,
43-2 ¹ (43-2 ¹ ( ^43-25 )) = 1
Is true. Similarly, if the prime number 43 is the variable P and the exponent is z ₁ , z ₂ , z ₃ as shown in FIG. 16, then

When deployed next to

Is obtained, and when it is deformed by P, the following equation (10) is finally obtained.

As described above, the initial value X ₀ = 1 was applied by the tent mapping formula (8), and the trace was made until the value returned to X _i = 1.
Next, assuming a sequence of "z ₁ , z ₂ , z ₃ , ..., z _r-1 , z _r ", equation (9) and equation (10) can be changed to equation (11) shown below. Can be predicted. here,

The one that expanded the formula as

For the last term "(-1) ^r ", the formula (9) is " ^{-2 Zm} " when the subscript "r" of z _r is odd, and "+2 ^Zm " when it is even. ) And equation (10). Finally, the following equation (11) is obtained.

As shown above, "Zm" is the value when "(P-1) / 2" is divided by the prime factor obtained by factoring "(P-1) / 2" into prime factors, with P as the prime number. There is a possibility that the period length "Zm" of the tent mapping formula (8) is set. For example, when P = 43, (P-1) / 2 = 21, and when this is factored into prime factors, two prime factors, “3” and “7”, are obtained. Divide "(43-1) / 2" by the obtained prime factors "3" and "7".

Then, since (43-1) / 2/3 = 7 and (43-1) / 2/7 = 3, the numerical values 7 and 3 are applied as “Zm”, and P = 43 is expressed in the equation (8). It can be seen that the period length may be 7 or 3 when the mapping is repeated by substituting into). In fact, looking at P = 43 in FIG. 8, it can be seen that the mapping is repeated with a period length of 7. Here, Zm is the sum of the exponents of the powers of 2 generated when the operation is performed by the equation (8), and is expressed by the equation (10A). Therefore, if the prime factor of "(P-1) / 2" is expressed by putting a numerical value in the subscript m of Zm, it will be confused with the exponent of the power of 2 in the equation (10A). When expressing the prime factor or cycle length of 1) / 2 ", it is expressed as" Km (m = 1, 2, 3, ...) ".

Here, considering the case where the cycle length is started from X ₀ = 1, 2 ^{s + 1} from Eq. (8) (“s + 1” and plus 1 are X when “PX _i = 1”. _{Since i} = 2 starts, 1 is added), and the minimum cycle length secured up to P is guaranteed to be "s" or more. This is because X _i is less than P until the operation of "P-X _i " is performed, so that the operation of "2 ^{s + 1} " can be performed until then. Therefore, the minimum cycle length can be taken up to "2 ^{s + 1} <P", and the following equation (12) can be obtained by taking the logarithm with the base of 2.

In the case of P = 43, since log ₂ P = 5.426265, “s = 4” (24 ^{+ 1} <43), so that the cycle length is at least 4 or more. Therefore, the candidates for the cycle length Km are K ₁ = 7, K ₂ = 3, but K ₂ = 3 is excluded (it is found that the cycle length 3 is impossible). When dividing "(P-1) / 2" by each prime factor in this way, Km of the cycle length candidate obtained by dividing from the smaller prime factors is calculated (in descending order), and " _Km " is calculated from the equation (12). It can be seen that if ≦ s ”, it may be excluded.

Since the calculation result of the formula on the left from the formula (11) always takes an integer, it can be seen that the value divided by the prime number P is a divisible number with no remainder in the calculation on the right. Therefore, when Km is a candidate for the period length obtained by dividing "(P-1) / 2" from the tent mapping equation (8) by the prime factor obtained by factoring it into prime factors, the equations (9) and (10) The following equation (13) is given from the form.

When the equation (13) holds, the theorem that the period length of the equation (8) is smaller than "(P-1) / 2" is derived.
In the tent mapping formula (8), when "Km ₌ (P-1) / 2", the mapping value Xi returns to the original value, so either "2 ^Km + 1" or "2 ^Km -1" must be used. It is divisible by the prime number P. By the way, at this time, it can be seen that it is equivalent to the Euler's criterion below.

<Euler's criterion>
When "P" is a prime number and "a" and "P" are relatively prime integers, the following holds.

It is called the Legendre symbol and is taken up as the basic theorem of elementary number theory. Although detailed explanation is omitted here, when "a" is "mod P" and quadratic residue, "+1" and "a" are "mod". It is known that "-1" is taken when P "is quadratic non-residue, and the following equation holds.

It was found that it was the same as the one in which "Km = (P-1) / 2" and "a = 2" were set in the equation (13), and Euler's criterion was set in the period of the tent mapping equation (8) which was converted into an integer calculation. It turns out that it applies.

In the random number generation by the tent mapping equation (8), it is desired to select a prime number P having a period length of "(P-1) / 2" and obtain a uniform "X _i ", so whether the equation (13) holds. After confirmation, when the remainder (modulo) becomes "0", it is found that a cycle length shorter than "(P-1) / 2" exists, so that the prime number P can be excluded. By acquiring the next prime number candidate and checking if there is a short cycle again, it is checked whether the cycle length is "(P-1) / 2", and the cycle is "(P-1) / 2". If it can be guaranteed that there is, the prime number P may be adopted and a random number may be generated.

FIG. 17 shows a flowchart of an embodiment of the P selection process including a determination for excluding an impossible short cycle according to the equation (12). In this P selection process, when the condition that "(P-1) / 2 is a prime number", which is the second embodiment, is satisfied, this P is selected. That is, it includes the processing of the flowchart shown in FIG.

Hereinafter, the process of selecting P will be described with reference to the flowchart of FIG. First, an arbitrary prime number is taken in as P (S31). Next, it is detected whether "(P-1) / 2" is a prime number (S32). If Yes, this value of P is adopted. If No in step S32, "(P-1) / 2" is factored into prime factors and the prime factor Cm is calculated (S33).

Next, "(P-1) / 2" is divided by each of the obtained prime factors Cm to obtain the cycle length candidate Km (S34). Here, the cycle length Km is
K ₁ = (P-1) / 2 / C1
K ₂ = (P-1) / 2 / C2
K ₃ = (P-1) / 2 / C3
・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・
K _m = (P-1) / 2 / Cm

Next, it is examined whether Km ≦ log ₂ P-1 holds (S35). When it becomes No,
It is determined that 2 ^Km ± 1≡0 (mod P) is established (S36). Here, when No, m of Km is incremented by 1 (S37), and the process returns to step S35 to continue the process.

If Yes in step S36, the process returns to step S31, the next prime number is set to "P", and the process is continued. Further, when Yes in step S35, P of this value is adopted.

When the number of Ps in the case of Yes in step S32 and the number of Ps in the case of Yes in step S35 are collected as many as necessary for random number generation, the process of FIG. 17 is stopped. The process proceeds to random number generation using the P selected here.

When the processing program based on the determination flowchart of FIG. 17 is implemented and "(P-1) / 2" is not a prime number, "(P-1) / 2" is factored into prime numbers to obtain Km, and the equation (13) is used. You can check if there is a short cycle. Here, the result of the prime number P determined to have a period shorter than "(P-1) / 2" is indicated by "x" in FIG. The column of 2 ^Km ± 1 in FIG. 10 shows the numerical value Km when the remainder becomes “0” as a result of modulo using the prime number P according to the equation (13).

As a result of the determination, using the prime number P whose mark shown in the above column in FIG. The generated "X _i " can be output to generate a random number. Compared with the second embodiment, it is necessary to calculate Km and to confirm whether the prime number P can guarantee the period length and uniformity according to the equation (13). However, since the number of prime numbers that can be used can be increased, there is an advantage that the range of selection of prime numbers can be expanded and many prime numbers can be used.

In equation (13), since it is a power of 2, the number of digits becomes enormous as the exponent becomes larger, but in binary numbers, the exponent should be shifted to the left, and the prime number P is expanded into binary numbers to divide the bit values. You just have to check if it is divisible (whether there is a remainder). Therefore, when considering the calculation in the computer implementation, it is presumed that the amount of calculation does not matter so much because the division can be implemented by the subtraction operation.

In addition, the larger the exponent, the larger the digit becomes. It has the property of being equal to the remainder of the original multiplication (Q ³ ), and can be calculated with a reduced number of digits. This method is shown in FIG.

As an example, the result of ²⁷ + 1 from the column of 2 ^Km ± 1 in the row of "prime number P = 41" in FIG. 10 shows that it is divisible by the prime number P = 43.
In other words, since it is ²⁷ + 1≡0 (mod43), when the index is honestly multiplied, the remainder of 129≡0 (mod43) and 129 divided by 43 is obtained. Here, when "+1" is transferred to the right side in the above equation, it becomes ²⁷ ≡ -1 (mod 43). In this equation, "-1" means "43-1", so in order to confirm whether it is ²⁷ ≡ 42 (mod 43), 1 is multiplied by 2 with the remainder as shown in FIG. By taking (modulo) and calculating one by one, it is possible to perform calculations without increasing the number of digits.

However, it is said that the larger the digit, the longer it takes to calculate the prime factorization of the composite number by multiplying the prime numbers of large digits. Therefore, in reality, it is recommended to select a prime number with a number of digits that does not take much calculation time on the current computer.

The conventional pseudo-random number sequence using chaos has not been theoretically constructed in terms of uniformity, and there is a concern that the numerical values may be biased. On the other hand, in the embodiment shown above, since the period can be estimated, the uniformity can be guaranteed and a good quality pseudo-random number sequence can be generated.

Further, in the above embodiment, it is confirmed whether the prime number can guarantee the uniformity within one cycle, the random number sequence is output using the prime number, and another prime number that guarantees the uniformity is adopted after the lap, and the random number sequence is adopted. Is to generate. For this reason, it is known that prime numbers exist infinitely, and it can be expected to have the effect of being able to generate a pseudo-random number sequence having a substantially unlimited period length.

All of the pseudo-random number generators of the above embodiments can be realized by a computer performing processing by a program corresponding to the flowchart shown in FIG. 1B. Hereinafter, the operation will be described with reference to FIG. 1B. First, the capacity (random number sequence length) "M" of the random numbers to be acquired (generated) is set (S11).

Next, a prime number is acquired as "P", and a numerical value that realizes the maximum period "(P-1) / 2" when tent mapping is performed is set (S12). Next, the maximum calculation accuracy P and the initial value X ₀ are set in the tent map, and the register i is set to 0 (S13). Next, if the result of the tent mapping calculation is not obtained, step S14 is skipped and the tent mapping calculation in step S15 is executed (S15). In this step S15, the value of the register i is incremented by 1, and the value of the register j is incremented by L. Here, L indicates the length of a random number created by using the result of one random number operation.

Next, it is detected whether the value of the register i is "(P-1) / 2" (S16), and if it becomes No, the random number generation process described above is performed by the random number generation means 30 (S14). When the tent mapping for one cycle is completed by repeating the processes from step S14 to step S6, the tent map is branched to Yes in step S16, and it is detected whether the value of the register j exceeds the random number capacity (random number sequence length) "M". (S17), if No, the process returns to step S12 and the process is continued, and if Yes in step S17, the process ends.

10 Tent mapping calculation means 20 Tent mapping parameter selection means 30 Random number generation means 40 Parameter transition control means

Claims (18)

A tent mapping calculation means that performs mapping calculation by tent mapping,
In order to enable the tent mapping by an integer operation, it is detected whether the equation (P-1) / 2 using the prime number P when multiplying the maximum interval of the tent mapping by an integer is a prime number, and (P- 1) When / 2 is not a prime number, (P-1) / 2 is factored into prime factors to obtain a prime factor, and based on whether (P-1) / 2 is a prime number or the prime factor satisfies a predetermined conditional expression. , The tent mapping parameter selection means that selects the prime number P and gives it to the tent mapping calculation means, and
A pseudo-random number generation device including a random number generation means for generating a sequence of predetermined lengths based on a calculation result by the tent mapping calculation means. The pseudo-random number generator according to claim 1, wherein the tent mapping parameter selection means selects P when (P-1) / 2 is a prime number. The tent map parameter selection means sets P when the conditional expression Km ≦ (log ₂ P-1) created by Km calculated by the prime factor Cm obtained by factoring (P-1) / 2 into prime factors holds . The pseudo-random number generator according to claim 1, wherein the pseudo-random number generator is selected. The pseudo-random number generation device according to claim 3, wherein the tent mapping parameter selection means determines the establishment of 2 ^Km ± 1≡0 (mod P), and if it is established, the P is not used. The first aspect of the present invention is characterized in that the tent map calculation means includes a parameter transition control means for controlling so as to give a new P from the tent map parameter selection means each time the tent map calculation means performs a calculation for one cycle of the tent map. 4. The pseudo-random number generator according to any one of 4. The pseudo-random number generation device according to any one of claims 1 to 5, wherein the random number generation means sequentially arranges mapping results by tent mapping to obtain random numbers. The tent map calculation means includes two series of tent maps, gives different initial values to each series, and repeats the tent map.
One of claims 1 to 6, wherein the random number generation means repeats a process of obtaining a total of 2 bits, one bit from one series for each tent map, and arranges them to form a random number sequence having a predetermined length. Pseudo-random number generator described in. The tent map calculation means includes three series of tent maps, gives different initial values to each series, and repeats the tent map.
The random number generation means obtains a predetermined bit from a first series, a predetermined bit from a second series, and a predetermined bit from a third series in each tent map, and arranges them to form a random number sequence having a predetermined length. The pseudo-random number generator according to any one of claims 1 to 6 . Any of claims 1 to 8, wherein the random number generation means performs an operation on the result of the tent mapping operation and all or a part of the value of P, and obtains a random number sequence having a predetermined length from the operation result. The pseudo-random number generator according to item 1. Computer,
Tent mapping calculation means that performs mapping calculation by tent mapping,
In order to enable the tent mapping by an integer operation, it is detected whether the equation (P-1) / 2 using the prime number P when multiplying the maximum interval of the tent mapping by an integer is a prime number, and (P- 1) When / 2 is not a prime number, (P-1) / 2 is factored into prime factors to obtain a prime factor, and based on whether (P-1) / 2 is a prime number or the prime factor satisfies a predetermined conditional expression. , The tent mapping parameter selection means, which selects the prime number P and gives it to the tent mapping calculation means,
A pseudo-random number generation program characterized in that it functions as a random number generation means for generating a sequence of predetermined lengths based on the calculation result by the tent mapping calculation means. The pseudo-random number generation program according to claim 10, wherein the computer functions as the tent mapping parameter selection means to select P when (P-1) / 2 is a prime number. .. When the conditional expression Km ≦ (log ₂ P-1) created by Km calculated by the prime factor Cm obtained by factoring (P-1) / 2 into the prime factor Cm is satisfied by using the computer as the tent mapping parameter selection means. The pseudo-random number generation program according to claim 10 , further comprising functioning to select P. 12. The computer is characterized in that, as the tent mapping parameter selection means, it is made to determine the establishment of 2 ^Km ± 1≡0 (mod P), and if it is established, the computer is made to function so as not to use the P. Pseudo-random number generator described in. The computer is characterized by functioning as a parameter transition control means for controlling the tent map parameter selection means to give a new P each time the tent map calculation means performs a calculation for one cycle of the tent map. The pseudo-random number generation program according to any one of claims 10 to 13. The pseudo-random number generation program according to any one of claims 10 to 14, wherein the computer, as the random number generation means, sequentially arranges mapping results by tent mapping and functions to generate random numbers. The computer is provided with two series of tent maps as the tent map calculation means, and each series is given a different initial value to function so as to repeat the tent map.
The present invention is characterized in that, as the random number generation means, the computer is repeatedly processed to obtain a total of 2 bits, one bit from one series for each tent map, and arranged so as to form a random number sequence having a predetermined length. The pseudo-random number generation program according to any one of 10 to 15. The computer is provided with three series of tent maps as the tent map calculation means, and each series is given a different initial value to function so as to repeat the tent map.
Using the computer as the random number generating means, each tent map obtains predetermined bits from the first series, predetermined bits from the second series, and predetermined bits from the third series, and arranges them to form a random number sequence having a predetermined length. The pseudo-random number generation program according to any one of claims 10 to 15 , characterized in that it functions. As the random number generation means, the computer is characterized in that it performs an operation on the result of a tent mapping operation and all or a part of the value of P, and causes the computer to function as a random number sequence having a predetermined length from the operation result. The pseudo-random number generation program according to any one of claims 10 to 17.

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