JP2011123693A - Method, apparatus and program for determining period of integer sequence - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To determine a period of an integer sequence prepared by using a bijective function based on the set of natural numbers having an arbitrary size. <P>SOLUTION: A permutation preparation part 4 prepares a permutation A(i) which is the array of output data (images corresponding to an original image) with respect to the input of a bijective function (mapping). A permutation storage part 5 stores a permutation S(i, j) indicating an output state with respect to the input of all bijective functions of an arbitrary size m (m is a natural number of ≥2). A period calculation part 6 repeats a procedure for further inputting an output data sequence with respect to the input of an arbitrary bijective function to calculate a period of an integer sequence to be prepared. A period storage part 10 stores the period calculation result of the period calculation part 6 which is specified by a permutation number setting part 11. The permutation number setting part 11 stores the existence number of all bijective functions of the arbitrary number m of data. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to an integer sequence period determination method, a period determination apparatus, and a period determination program, and more particularly to an integer sequence period determination method, a period determination apparatus, and a period determination program used for pseudorandom numbers and the like.

  In recent years, pseudo-random number sequences used in cryptography, the Monte Carlo method, and the like are required to have a long cycle in order to increase the difficulty of cryptographic analysis. In addition, an M-sequence or the like has often been used as a pseudo-random sequence with a guaranteed sequence length so far, but it is considered to use a generation method using a chaos, a feedback function, or the like as other means. (See, for example, Non-Patent Document 1 and Non-Patent Document 2). Non-Patent Document 1 discloses that a pseudo-random number sequence having a long period is generated using a shift register. Non-Patent Document 2 discloses general cryptography that generates a pseudo-random number sequence.

  Further, in order to increase the difficulty of cryptographic analysis and generate a highly secure pseudo-random number sequence, a cryptographic processing device configured to perform processing using a plurality of different S boxes (see, for example, Patent Document 1), linear There is known a pseudo-random number generation device configured to make it difficult to estimate an output pseudo-random number sequence by using a nonlinear function as a filter for M-sequence output using a feedback shift register (see, for example, Patent Document 2).

  Further, in order to extend the sequence repetition period of the random number generator in the system based on the availability of the pseudo-random number sequence, an RNS (Residue Number System) residue value is obtained using a plurality of relatively prime numbers as a modulus, A method of extending the sequence length by combining them into an output sequence is also known (see, for example, Patent Document 3).

Solomon Wolf Golomb, “Shift Register Sequences”, San Francisco, Holden-Day, 1967. Bruce Schneier, “Applied Cryptography”, Second Edition, John Wiley & Sons, Inc. 1996.

JP 2008-58828 A JP-A-8-202535 JP 2009-3925 A

  However, when generating a pseudo-random number sequence using a generation method using chaos, a feedback function, etc., the generated pseudo-random number sequence length may not be clear, and the generated pseudo-random number sequence may be There is a case where a pseudo-random number sequence that is not suitable for the purpose of use must be used for some reason.

  In addition, the devices described in Patent Documents 1 and 2 do not have a function of determining the period of the generated pseudorandom number sequence and obtaining the length of the period or the number of periods, and use the values used for the period. It does not have a function to distinguish as finished. Furthermore, the method described in Patent Document 3 also does not disclose a function for determining the period and number of the pseudo-random number sequence and obtaining a complete pseudo-random number without using a remainder. .

  Thus, conventionally, it is not known to determine the period of a pseudo-random number sequence. From this, it is not easy to obtain a complete pseudo-random number with a guaranteed sequence length.

  The present invention has been made in view of the above points, and in order to generate a pseudo-random number sequence, an integer that can determine the period of an integer sequence generated using a bijection function based on a set of natural numbers of an arbitrary size. It is an object of the present invention to provide a sequence period determination method, a period determination apparatus, and a period determination program.

  Another object of the present invention is to perform a calculation to obtain the sum of the cycle lengths in a bijection function using a natural number as input / output and an output feedback such as a conversion table or S box, and how many each cycle length exists. It is another object of the present invention to provide an integer sequence period determination method, a period determination apparatus, and a period determination program that can determine whether or not the result has a maximum length period.

  In order to achieve the above object, the integer sequence period determining method according to the first aspect of the present invention is such that the control unit has m integers whose values from 0 to m-1 (where m is an integer of 2 or more) do not overlap. M with an integer sequence input (original image) and output (image)! (M! Is the factorial of m. That is, 1 × 2 ×... X (m−1) × m.) The state of the output for the m types of inputs for the bijection function (mapping). The first step of generating all as permutations and storing them in the first storage unit, and the control unit stored in the first storage unit m! For each permutation of each bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. The number of conversions is calculated as a cycle, and when the cycle is smaller than m, it is determined that there is a non-appearing input value, and the output value when the non-appearing input value is given to the permutation is In addition, converting the output value as an input value is repeated until the same output value as the input value that has not yet appeared is obtained, and obtaining the number of conversions as a cycle means that all the input values and all the unconverted values are obtained. The second step is sequentially performed on the input values that appear, and the control unit obtains m! And a third step of storing the period from the minimum period to the maximum period of each bijective function in the second storage unit.

  In order to achieve the above object, the integer sequence period determining method according to the second aspect of the invention is such that the control unit has m number of values from 0 to m−1 (where m is an integer equal to or greater than 2). M with an input (original image) and an output (image) of an integer sequence of integers! Among the plurality of bijection functions (mappings), the first storage unit stores in the first storage unit the output state of the desired one bijection function with respect to the input of the integer series consisting of the m integers as a permutation. 1 and the control unit obtains an output value when the first input is given among the m inputs, and further sets the output value as an input value. The second step is performed for all of the m inputs, and the control unit stores in the second storage unit that the number of repetitions is repeated as a cycle and stored in the second storage unit. A third step of determining whether or not the generated period is the m, and determining that the period is the maximum period of the desired one bijective function when the period is the m. It is characterized by including.

  In order to achieve the above object, an integer sequence period determining apparatus according to a third aspect of the present invention includes a first and second storage unit, and 0 to m−1 (where m is an integer of 2 or more). M having an input (original image) and an output (image) of an integer sequence consisting of m integers whose values do not overlap. Permutation generation and storage means for generating all the states of outputs for the m kinds of inputs for the bijection functions (mappings) as permutations and storing them in the first storage unit, and storing them in the first storage unit M done! For each permutation of each bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. The number of conversions is calculated as a cycle, and when the cycle is smaller than m, it is determined that there is a non-appearing input value, and the output value when the non-appearing input value is given to the permutation is In addition, converting the output value as an input value is repeated until the same output value as the input value that has not yet appeared is obtained, and obtaining the number of conversions as a cycle means that all the input values and all the unconverted values are obtained. Period calculation and accumulation means sequentially performed for the input value of appearance, and the m! Obtained by the period calculation and accumulation means. Storage means for storing a period from the minimum period to the maximum period of each bijective function in the second storage unit.

  In order to achieve the above object, an integer sequence period determining apparatus according to a fourth aspect of the present invention includes first and second storage units, and 0 to m−1 (where m is an integer of 2 or more). M having an input (original image) and an output (image) of an integer sequence consisting of m integers whose values do not overlap. A permutation that stores, in the first storage unit, a state of an output with respect to an input of an integer sequence composed of the m integers of a desired one of the bijection functions (mapping) among a plurality of bijection functions (mappings). The storage means and the output value when the first input is given among the m inputs are obtained, and the output value is further set as the input value until the same output value as the first input value is obtained, and Storing the repetition count as a cycle in the second storage unit, the cycle calculation and accumulation means for all the m inputs, and whether the cycle stored in the second storage unit is m Maximum period determining means for determining whether or not the period is the maximum period of the desired one-shot bijection function when the period is m.

  In order to achieve the above object, an integer sequence period determination program according to a fifth invention causes a computer to execute the first to third steps of the integer sequence period determination method according to the first invention. Features.

  Furthermore, in order to achieve the above object, an integer sequence period determination program according to a sixth aspect causes a computer to execute the first to third steps of the integer sequence period determination method according to the second aspect. Features.

  According to the present invention, an integer sequence generated by feeding back an output to an input using a bijection function (mapping) having a set of natural numbers of an arbitrary size as an input (original image) and an output (image). And the maximum period of the integer series can be determined.

1 is a block diagram of a first embodiment of an integer sequence period determination device of the present invention. FIG. It is explanatory drawing of an example of a bijection function. It is explanatory drawing of the outline | summary of pseudorandom number preparation by this invention. It is a flowchart for outline | summary description of the 1st Embodiment of this invention. It is a flowchart for outline | summary description of permutation creation of the input-output data of the bijection function of the 1st Embodiment of this invention. It is a flowchart for outline | summary description which determines each period from each permutation data which the 1st Embodiment of this invention produced. It is a flowchart (the 1) explaining operation | movement of the 1st Embodiment of this invention. It is a flowchart (the 2) explaining operation | movement of the 1st Embodiment of this invention. It is a flowchart explaining the detailed operation | movement of step S1 in FIG. It is a figure explaining the function feedback for the determination operation | movement of the period by this invention. It is a block diagram of 2nd, 3rd embodiment of the period determination apparatus of the integer series of this invention. It is a flowchart (the 1) explaining operation | movement of the 2nd Embodiment of this invention. It is a flowchart (the 2) explaining operation | movement of the 2nd Embodiment of this invention. It is a flowchart explaining the operation | movement of the 3rd Embodiment of this invention.

  Next, each embodiment of the present invention will be described in detail with reference to the drawings.

(First embodiment)
FIG. 1 shows a block diagram of a first embodiment of an integer sequence period determining apparatus according to the present invention. As shown in the figure, the integer sequence period determining apparatus 1A of the present embodiment includes a control unit 2A that centrally controls the entire apparatus 1A, a data number storage unit 3, a permutation creation unit 4, a permutation storage unit 5, a period The calculation unit 6, the input designation unit 7, the output storage unit 8, the cycle accumulation unit 9, the cycle storage unit 10, the permutation number setting unit 11, and the search position designation unit 12 are configured.

  The control unit 2A, the permutation creation unit 4, the period calculation unit 6, the permutation number setting unit 11, and the search position designation unit 12 are configured by, for example, a central processing unit (CPU). Further, the data number storage unit 3, the output storage unit 8, the cycle accumulation unit 9, and the cycle storage unit 10 are configured by a storage device, and the input designation unit 7 is configured by a predetermined input device.

  The data number storage unit 3 stores the number M of continuous natural number input / output data (original image and image) possessed by a bijection function that is a bijection and a bijection function. The permutation creating unit 4 creates a permutation A (i) that is a sequence of output data (images with respect to the original image) with respect to the input of the bijection function. In this case, the permutation A (i) is created so that the values of the output data do not overlap and all values appear.

  The permutation storage unit 5 outputs all bijective functions (mappings) of arbitrary size (number of data M) m (m is a natural number of 2 or more) created as described later (for the original image). The first storage unit stores a permutation S (i, j) indicating the state of the image. As will be described later, the period calculation unit 6 repeatedly performs a procedure in which output data (an image for the original image) for an arbitrary bijective function input is further input, and calculates a period P of the integer sequence to be created.

  The input designation unit 7 designates an input position (original image position) INP. The output storage unit 8 stores the value OUTP of the output data (image) of the cycle calculation unit 6 at the input position specified by the input specification unit 7. The period accumulating unit 9 stores the accumulated result PS of the period P calculated by the period calculating unit 6.

  The cycle storage unit 10 is a second storage unit that stores the cycle calculation result PR (PM, M) of the cycle calculation unit 6 specified by the permutation number setting unit 11. In the present embodiment, the cycle storage unit 10 stores the number of existence for each cycle of all bijective functions having an arbitrary size (number of data M) m. The permutation number setting unit 11 stores the existence number PM of all bijection functions of an arbitrary number of data m. The search position designating unit 12 designates the search position POS of the permutation so far created for the value A (i) of any permutation created by the permutation creating unit 4.

  Among the above configurations, the control unit 2A, the data number storage unit 3, the permutation creation unit 4, the input designation unit 7, the permutation number setting unit 11, and the search position designation unit 12 are m! A permutation generation and storage unit is configured to generate all the states of outputs for m types of input / output for each bijection function as permutations and store them in the permutation storage unit 5 as the first storage unit. In addition, the control unit 2A, the data number storage unit 3, the cycle calculation unit 6, the input designation unit 7, the output storage unit 8, the cycle accumulation unit 9, and the permutation number setting unit 11 include a cycle calculation and accumulation unit, The memory | storage means which memorize | stores the determined period in the period memory | storage part 10 which is a memory | storage part is comprised.

  By the way, as shown in FIG. 2, m natural numbers x (for example, data set X = {x | x = natural number, 0 ≦ x ≦ (m−1)}) are input (original image) and output (original image). The number of all bijection functions (mappings) f, which is an image), is m! It is known to exist. In the generation of a pseudo-random number sequence by conversion using a bijection function in which input data and output data correspond one-to-one, the present invention maps m natural numbers to conversion by the function as shown in FIG. The input data (0-m-1) that is the original image and the output data (0-m-1) that becomes the image, and the feedback sequence (pseudorandom number sequence) that feeds back the output data as input data, All cycles are determined.

  In the present invention, m! The output states for the m types of inputs for each of the bijection functions are all obtained by listing them as permutations. This means that 0 to (m−1) is the permutation of the output of the first bijection function, and the output sequence of the bijection function after that is the position of the last output in the previous permutation. By using the backtracking and incremental search techniques, the next permutation can be obtained.

  Next, in the present invention, the permutation (function) obtained by the above method is converted from the first input value (for example, 0) to the same output value (for example, 0) as the input, and the number of conversions is calculated. Calculate as a period. Further, similar conversions are sequentially performed for input values that do not appear as output values, and the number of conversions until the first input value is returned is obtained as a cycle. When all the input values appear, all the cycles and the number of appearances of the cycle are obtained. Further, it is possible to obtain all the cycles and the number of appearances of any one bijective function by the same procedure as described above.

  The bijection function in the present invention corresponds to a simple permutation table of arbitrary arrangements of m natural numbers, a conversion table (substitution table) such as a Latin square, an S box, a block cipher, and a feedback stream cipher. The integer sequence for which the period is determined in the present invention corresponds to an OFB (Output Feed Back) mode of block cipher or an output sequence of feedback stream cipher.

  Here, the relationship between the input and output data of all bijective functions of size 2 (m = 2) is shown below.

f1 = [0 1]: 0 → 0, 1 → 1
f2 = [1 0]: 0 → 1 → 0
The value on the left side of the above → indicates an input value, and the value on the right side indicates an output value (the same applies to other examples described later). Therefore, the relationship f1 of the first input / output data shows a case where 0 is output when 0 is input, and a case where 1 is output when 1 is input. is there. The second input / output data relationship f2 indicates that 1 is output when 0 is input, and 0 is output when 1 is input, and the period 2 is one. Show. Therefore, it can be concluded that all bijective functions of an integer sequence of m = 2 have two periods 1 and one period 2 (maximum period). This example can also be represented by the following table.

同様に、大きさ３（ｍ＝３）の全ての全単射関数の入出力データの関係は、下記に示される。

Similarly, the relationship between input and output data of all bijective functions of size 3 (m = 3) is shown below.

f1 = [0 1 2]: 0 → 0,1 → 1,2 → 2 (hence three periods 1)
f2 = [0 2 1]: 0 → 0, 1 → 2 → 1 (hence, 1 period 1 and 1 period 2)
f3 = [1 0 2]: 0 → 1 → 0,2 → 2 (hence, 1 period 1 and 1 period 2)
f4 = [1 2 0]: 0 → 1 → 2 → 0 (hence one cycle 3)
f5 = [2 0 1]: 0 → 2 → 1 → 0 (hence one cycle 3)
f6 = [2 1 0]: 0 → 2 → 0,1 → 1 (hence, 1 period 1 and 1 period 2)
Therefore, in the case of m = 3, it can be concluded that all bijective functions have 6 periods 1, 3 periods 2, and 2 periods 3 (maximum period). This example can also be represented by the following table.

同様に、ｍ個の自然数（整数）からなる整数系列の全ての全単射関数の入出力データの関係は、下記の表３で表すことができる。

Similarly, the relationship of input / output data of all bijection functions of an integer series consisting of m natural numbers (integers) can be expressed in Table 3 below.

図１に示す第１の実施形態の整数系列の周期判定装置１Ａは、入出力データの大きさが任意のｍの場合の全ての全単射関数の全ての整数系列の作成とその系列の全周期を判定する装置の例である。まず、本実施形態の動作の概要について説明する。

The integer sequence period determining apparatus 1A of the first embodiment shown in FIG. 1 creates all integer sequences of all bijection functions when the size of input / output data is an arbitrary m, and all the sequences of the sequences. It is an example of the apparatus which determines a period. First, an outline of the operation of the present embodiment will be described.

  FIG. 4 shows a flowchart of the outline of the present embodiment. As shown in the figure, the integer sequence period determining apparatus 1A first creates permutation data of the output of one bijection function (step S151). Subsequently, the integer sequence period determination apparatus 1A determines all the periods of the maximum partial period for the period of the feedback sequence by repeating the feedback from the output of the bijection function to the input from the permutation data created in step S151. Long) is determined (step S152).

  FIG. 5 shows details of the permutation data creation operation in step S151 of FIG. As shown in FIG. 5, in the permutation data creation process, permutation data of the first function is created (step S201), and then permutation data of the second function is created from the first data (step S202). In the same manner, m! The permutation data of the second function is m! -Created from the first permutation data (step S203).

  FIG. 6 shows details of the operation for determining the period from the permutation data created in step S152 of FIG. As shown in FIG. 6, in the period determination process, the periods of the permutation data of the first function are accumulated and added to the period storage unit 10 (step S301). Subsequently, each period of the permutation data of the second function is accumulated and added to the period storage unit 10 (step S302). In the same manner, m! The respective periods of the permutation data of the th function are accumulated and added to the period storage unit 10 (step S303).

  Next, the operation of the integer series period determination device 1A of the present embodiment will be described in more detail. 7 and 8 are flowcharts for explaining the outline operation of the present embodiment described in FIG. 4 in more detail. FIG. 9 is a flowchart for explaining the detailed operation of the permutation (integer series data) creation method created in step S1 in FIG. Step S1 in FIG. 7 is a processing step whose outline operation has been described with reference to the flowchart of FIG. 5. Step S2 and subsequent steps in FIG. 7 and each step in FIG. 8 are detailed operations of the process whose outline operation has been described with reference to the flowchart of FIG. Is described.

  First, the control unit 2A in FIG. (Here, m = 2) permutations A (0) to A (m-1) are set in the permutation storage unit 5 (step S1 in FIG. 7). The creation of the permutation in step S1 is performed by setting an ascending natural number (0 to m-1) not set from A (0) to the last position A (m-1) of the permutation creation unit 4 as the first position of the output data. It can be created by the backtrack technique and the incremental search technique of FIG. 9, which are set in order so that no overlap occurs.

  When the permutation of the output data of the function whose period is to be determined is completed, the data of permutations A (0) to A (m-1) of the permutation creation unit 4 is m! Set to permutations S (m!, 0) to S (m!, M-1) in the first permutation storage unit 5. Furthermore, m! The remaining m! -1 of all the sequences S (m! -1,0) to S (m! -1, m-1), ..., S (1,0) to S ( 1, m-1) also uses the set of the previous series data S (m!, 0) to S (m!, M-1) and backtracks from the data at the last position m-1. By doing so, it is possible to obtain a new next m! -1st sequence S (m! -1,0) to S (m! -1, m-1). All m with each series of data! It is possible to create series data S (m!, 0) to S (m!, M-1), ..., S (1,0) to S (1, m-1) for a set of functions is there.

  Note that all the permutations for determination set in the permutation storage unit 5 are prepared and prepared in advance. However, as another method, the same permutation may be performed in order before one cycle determination. By using only one permutation storage unit as a storage location to be used by setting in advance one permutation storage unit 5 that is a storage location of the series data by creating the method, the amount of storage unit used M! It is also possible to reduce by -1.

  Here, an example in which the input / output data size (size) m of the bijection function is 2 will be described.

  A method for creating a permutation (integer series data) in step S1 will be described with reference to the flowchart of FIG. First, the control unit 2A causes the permutation number setting unit 11 to include the number of sequences of the bijection function (number of permutations) m! (Here, m! = 2) is set (step S101). Subsequently, the control unit 2A sets the data number m (here, m = 2) in the data number storage unit 3 (step S102), and further sets the value INP of the input designation unit 7 to indicate the head position of the data. After initialization to 0 (step S103), the permutation data value A (INP) of the permutation creation unit 4 at the position indicated by the input designation unit 7 is set to 0 (step S104). As a result, the permutation data at the first set position is A (INP) = A (0) = 0.

  Next, the control unit 2A determines whether or not the value INP of the input specifying unit 7 is 0, which is the first setting position of the data (step S105). If the first setting position is 0, the control unit 2A proceeds to step S106. If it is not the position 0, the process proceeds to step S108 to check for data duplication. Here, since the value INP of the input designation unit 7 is set to the first setting position 0 in step S103, the process proceeds to step S106, and whether the value INP of the input designation unit 7 is the last setting position m-1 (= 1). Judge whether. Here, since the value INP of the input designation unit 7 is the first setting position 0 and not the last setting position m−1 (= 1), the value INP of the input designation unit 7 is set to 1 as the next data setting position. After the addition (step S107), 0 is set as setting data to the value A (INP) (here, A (1)) of the permutation creation unit 4 (step S104). As a result, A (INP) = A (1) = 0.

  Next, the control unit 2A determines whether or not the value INP of the input specifying unit 7 is 0 as the position of the top data (step S105). Here, INP is set to 1 in step S107 and is not the leading position 0, so the process proceeds to step S108, and the search position POS indicated by the search position specifying unit 12 is initialized to 0. Subsequently, the control unit 2A determines whether the value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is equal to the value A (POS) of the permutation creation unit 4 indicated by the search position designation unit 12. (Step S109). Here, the value A (INP) of the set data is A (1) = 0, the data A (POS) at the search position is A (0), and A (0) is the first step S104. Since it is set to 0 in the process, it is determined that both are equal and overlap (Yes in step S109).

  Thereby, the control unit 2A determines whether or not the value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is equal to the maximum set value m−1 of the data (step S115). At this time point, A (INP) = A (1) = 0 and is not equal to 1 (= m−1), so the value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is incremented by 1. Then (step S119), the process returns to step S105. As a result, A (1) = 1.

  Subsequently, the control unit 2A determines whether or not the value INP of the input designating unit 7 is the head setting position 0 (step S105). Here, since the head setting position 0 is not 0, the process proceeds to step S108. After the search position POS indicated by the position specifying unit 12 is initialized to the first set position 0, the value A (INP) of the permutation generating unit 4 indicated by the input specifying unit 7 is again the permutation generating unit indicated by the search position specifying unit 12. It is determined whether or not it is equal to the value A (POS) of 4 (step S109). At this time, since A (INP) is A (1) = 1 and A (POS) is A (0) = 0, it is determined that they are not equal (No in step S109).

  Thereby, the control unit 2A adds 1 to the search position value POS of the search position specifying unit 12 (step S110), and then whether the value POS of the search position specifying unit 12 and the value INP of the input specifying unit 7 are equal to each other. It is determined whether or not (step S111). If the result of determination in step S111 is that POS and INP are equal, it means that there is no duplication in all data, and the process proceeds to step S106, and if not, the process proceeds to step S109. At this time, since both POS and INP are 1, it is determined that both are equal. Thereby, the control unit 2A proceeds to step S106, and determines whether the value INP of the input designating unit 7 is the last setting position 1 (= m−1) of the data.

  In step S106, since it is determined that they are equal, creation of data of one function is completed, and then the control unit 2A performs PM of the permutation number setting unit 11 (this is m! = Set in step S101). 2), the function data already created in the permutations S (PM, 0) to S (PM, m-1), that is, S (2,0) and S (2,1). Is set as values A (0) to A (m−1) of the permutation creation unit 4 in which A is set, that is, A (0) = 0 and A (1) = 1 as the permutation values of the first function data (Step S112).

  Subsequently, the control unit 2A determines whether or not the value PM indicated by the permutation number setting unit 11 is 1, which is the set number of the last function sequence. -1 has been created. If it is not 1, data for the remaining functions is created (step S113). At this time point, the value PM indicated by the permutation number setting unit 11 is 2, not the value 1 of the last function data, so the value PM indicated by the permutation number setting unit 11 is subtracted by 1, and the setting position of the next function data (Step S114), it is determined whether or not the value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is the maximum set value m-1 of the data (step S115).

  At this time, the value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is A (1) = 1 and is equal to the maximum value m−1. After subtracting 1 and returning the setting position of the permutation creating unit (step S116), it is determined whether or not the value INP of the input designating unit 7 is the first setting position 0 of the data (step S117). At this time, since the value INP of the input designating unit 7 is 0 and leading, the control unit 2A continues to the value A (INP) of the permutation creating unit 4 indicated by the input designating unit 7, that is, A (0) is the maximum value. It is determined whether or not m-1 (step S118). At this time point, A (0) is 0, which is not equal to the maximum value m−1 (= 1), so the control unit 2A sets the function setting data that is the value of the permutation creation unit 4 indicated by the input designation unit 7 A (INP) is incremented by 1 as the next order value (step S119), and the process returns to step S105. Thereby, the set value A (0) of the function data is set to 1.

  In step S105, the control unit 2A determines that the value INP of the input designation unit 7 is the setting position 0 of the first data, and then in step S106, the value INP of the input designation unit 7 is the maximum set value 1 of the data. In step S107, the data setting position value INP of the input designation unit 7 is incremented by 1 to set it to 1 as the next setting data position value, and then in step S104. Returning to FIG. 4, the function data setting value A (INP) of the permutation creation unit 4 indicated by the input designation unit 7 is set to zero. As a result, the second setting data A (INT) = A (1) = 0.

  Subsequently, the control unit 2A determines whether or not the data setting position value INP of the input specifying unit 7 is the leading position 0 (step S105). At this time, INP is 1, and the leading position value is 0. Therefore, the search position value POS of the search position specifying unit 12 is set to 0 (step S108), and the value A (INP) of the permutation creating unit 4 indicated by the input specifying unit 7 is indicated by the search position specifying unit 12. It is determined whether it is equal to the value A (POS) of the permutation creation unit 4 (step S109). Here, since the set data A (INP) is A (1) = 0 and the data at the search position is A (POS) = A (0) = 1, and they are not equal, the process proceeds to step S110. After the search position value POS of the search position specifying unit 12 is set to the next value 1, it is determined whether or not the search position value POS is equal to the value INP of the input specifying unit 7 in step S111. At this time, since POS = INP = 1, the process proceeds to step S106 to determine whether or not the value INP of the data setting position of the input designating unit 7 is equal to 1 (= m−1). At this time, since the data setting position INP = 1 is the last position, the process proceeds to step S112, and the permutations S (PM, 0) to S (PM, m-1) of the permutation storage unit 5, that is, the second position. The values A (0) = 1 and A (1) = 0 of the setting data of the permutation creation unit 4 are set in S (1,0) and S (1,1) as permutation values of the function data.

  Then, the control unit 2A determines whether or not the value PM of the permutation number setting unit 11 is 1 of the set number of the last function sequence (step S113). At this time, since the value PM of the permutation number setting unit 11 is the setting number 1 of the last function, the control unit 2A has created all the number sequences, and thus ends the processing.

  In this way, the permutations S (PM, 0) to S (PM, m-1) of the permutation storage unit 5, that is, S (2,0), S (2,1), S (1,0), S The value (permutation) of the permutation creation unit 4 is stored in (1,1).

  Returning to FIG. 7 and FIG. After the processing in step S1, the control unit 2A sets the permutation number setting unit 11 to m! As the existence number (number of output series) PM of all bijection functions having an arbitrary number of data m. After setting (= 2) (step S2 in FIG. 7), the number M (= m = 2) of input / output data (original image and image) is set in the data number storage unit 3 (step S3 in FIG. 7). .

  Subsequently, the control unit 2A sets each cycle (1 to m!) Of each sequence (1 to m!) To PR (1,1) to PR (m!, M) for setting the number of each cycle in the cycle storage unit 10. ) Is set as 0 as the initial value (step S4 in FIG. 7), and is the first (PM = 2) series data of the bijection function stored in the permutation storage unit 5 and indicated by the permutation number setting unit 11. The permutation S (2,0) to S (2, m-1) values (S (2,0) = 0, S (2,1) = 1) in the period calculation unit 6 m (m = 2) Are respectively set as calculation data P (0) to P (m-1) (P (0) = 0, P (1) = 1) (step S5 in FIG. 7).

  Subsequently, the control unit 2A sets the value PS of the period accumulating unit 9 to 0 (step S6 in FIG. 7), and then sets the head position corresponding to the first input data (0 as the head data) as the input designation unit. 7 is set to the value INP (INP = 0) (step S7 in FIG. 7). Next, the control unit 2A outputs the output data corresponding to the input data (data is 0) at the position of the period array P (1) to P (m) of the period calculation unit 6 indicated by the value INP of the input designation unit 7. P (INP) (data is 0) is set as the value OUTP of the output storage unit 8 (step S8 in FIG. 7).

  Subsequently, the control unit 2A determines whether or not the value OUTP of the output data in the output storage unit 8 has already been used for the series (step S9 in FIG. 7). When used, the value is -1 as a used mark. If it has been used, the process proceeds to step S10 in FIG. 7, and if it has not been used, the process proceeds to step S15 in FIG. At this time, since OUTP is 0 and not −1 (used), the control unit 2A proceeds to step S15, and the period calculation unit 6 at the position of the value INP (here, 0) indicated by the input designation unit 7 The value P (INP) is updated to -1 as a used mark to indicate that it has been used for the series.

  Subsequently, the control unit 2A sets the value OUTP (here, 0) of the output data in the output storage unit 8 to the value INP of the input designating unit 7 (step S16 in FIG. 8). Subsequently, the control unit 2A adds 1 to the value PS of the cycle accumulating unit 9 to update the number of cycles to 1 (step S17 in FIG. 8), and then the value INP (0 in this case) of the input designating unit 7 is changed. The value P (INP) (here, −1) of the output data corresponding to the input data at the position of the cycle calculation unit 6 shown is set to the value OUTP of the output storage unit 8 (step S18 in FIG. 8).

  Next, the control unit 2A determines whether or not the value OUTP of the output storage unit 8 is −1 indicating that the value has already been used for the series (step S19 in FIG. 8), and has been used (−1). If so, the process proceeds to step S23, and if not used, the process proceeds to step S20. Here, since it has been used, the process proceeds to step S23, and among the m values PR (PM, 1) to PR (PM, m) in the period storage unit 10 indicated by the value PM of the permutation number setting unit 11, the period 1 is added to the period value of the position PR (PM, PS) corresponding to the value PS of the accumulating unit 9 (here, PR (2,1) = PR (2,1 +1) After updating the number of periods, the process proceeds to step S6 in FIG. 7 for calculation of the next period.

  In step S6, the control unit 2A clears the value PS of the period accumulating unit 9 to zero. Thereafter, the control unit 2A sets the head position corresponding to the first input data (0 as the head data) as the value INP of the input specifying unit 7 (step S5 in FIG. 7). Subsequently, the control unit 2A outputs the output data value P (INP) (here, -1) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP (here, 0) of the input designation unit 7. The value OUTP of the storage unit 8 is set (step S8 in FIG. 7).

  Next, the control unit 2A determines whether or not the value OUTP in the output storage unit 8 is −1 indicating that the value has already been used for the series (step S9 in FIG. 7), and has been used (−1). If so, the process proceeds to step S10 in FIG. 7, and if not used, the process proceeds to step S15 in FIG. Since it is already used at this point, the control unit 2A proceeds to step S10 and determines whether or not the value INP of the input designating unit 7 is a value (m-1) indicating the last position of the sequence. At this time, the value INP is 0 and not the last position 1 (= m−1). Therefore, after updating the value INP to 1 (step S11 in FIG. 7), the process returns to step S8 and is input again. The value P (INP) (here 1) of the output data corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP (here 1) of the designation unit 7 is set as the value OUTP of the output storage unit 8.

  Subsequently, in step S9, the control unit 2A determines whether or not the value OUTP in the output storage unit 8 has already been used for the series. Here, since the value OUTP of the output storage unit 8 is 1 and not −1, the control unit 2A proceeds to step S15 in FIG. 8 and the cycle of the position (here, 1) indicated by the value OUTP of the output storage unit 8. The data value P (INP) of the calculation unit 6 is updated to -1 as a used mark to indicate that it has been used in the series. Subsequently, the control unit 2A sets the value OUTP (here, 1) in the output storage unit 8 to the value INP in the input designating unit 7 (step S16), and then adds 1 to the value PS in the period accumulating unit 9. The number of cycles is updated (step S17). As a result, the number of periods becomes 1. Subsequently, the control unit 2A outputs the output data value P (INP) (here, -1) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP (here, 1) of the input designation unit 7. The value OUTP of the storage unit 8 is set (step S18).

  Subsequently, the control unit 2A determines whether or not the value OUTP of the output storage unit 8 is −1 (step S19). At this time, since the value OUTP is set to −1 in step S18, the control unit 2A determines that it is −1 (used) in step S19, and then in step S23, the number of permutations. Of the m values PR (PM, 1) to PR (PM, m) of the period storage unit 10 indicated by the value PM of the setting unit 11, a position PR (PM, PS) corresponding to the value PS of the period accumulating unit 9 1 is added to the value of the period (here, PR (2,1) = PR (2,1) +1) in which 1 is added to the position of period 1 and the period 1 becomes two, and the number of periods is updated. Then, the process proceeds to step S6 in FIG.

  Subsequently, the control unit 2A clears the value PS of the period accumulating unit 9 to 0 in step S6, sets the leading position (0 as the leading data) to the value INP of the input designation unit 7 in step S7, In S8, the output data P (INP) (here, −1) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP of the input designating unit 7 is set as the value OUTP of the output storage unit 8.

  Subsequently, in step S9, the control unit 2A determines whether or not the value OUTP of the output storage unit 8 is -1. At this time, since the value OUTP is −1, the control unit 2A determines that it is −1 (used) in step S9, and then proceeds to step S10 where the value INP of the input designating unit 7 is a series. It is determined whether or not the value is 1 (= m−1) indicating the final position. At this time, since the value INP of the input designation unit 7 is the first position 0 and not the last position 1, the control unit 2A proceeds to step S11 and adds 1 to the value INP of the input designation unit 7. Then, the process proceeds to step S8. As a result, the value INP becomes 1.

  Subsequently, the control unit 2A outputs output data P (INP) (here, -1) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP of the input designation unit 7 in step S8. In step S9, it is determined whether the value OUTP of the output storage unit 8 is -1. At this time, since the value OUTP is −1, the control unit 2A determines that it is −1 (used) in step S9, and then proceeds to step S10 where the value INP of the input designating unit 7 is a series. It is determined whether or not the value is 1 (= m−1) indicating the last position.

  At this time, the value INP of the input designating unit 7 is 1, which is equal to the value 1 indicating the last position of the sequence, so all the periods existing in one function have been obtained. It progresses to step S12 and it is determined whether the value PM of the permutation number setting part 11 is the value 1 corresponding to the last function. At this time, the value PM of the permutation number setting unit 11 is 2 (= m!) Set in step S2 and not 1. Therefore, the control unit 2A subtracts 1 from the value PM of the permutation number setting unit 11. After setting to 1 (step S13 in FIG. 7), the process returns to step S5, and the permutation S which is all series data of the bijection function stored in the permutation storage unit 5 indicated by the value PM of the permutation number setting unit 11 again. M data set in (PM, 0) to S (PM, m-1) (here, PM = 1, so as described above, the two data S (1, 0) = 1, S (1,1) = 0), the calculation data P (0) to P (m-1) for the period of the period calculation unit 6, that is, P (0) = 1, P (1) Set each as = 0.

  Subsequently, the control unit 2A clears the value PS of the period accumulating unit 9 to 0 in step S6, sets the head position (0 as the head data) to the value INP of the input designation unit 7 in step S7, and then performs step S8. Here, the output data P (INP) (here 1) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP of the input designation unit 7 is set to the value OUTP of the output storage unit 8. Subsequently, the control unit 2A proceeds to step S9 to determine that the value OUTP is not −1 (used), and then proceeds to step S15 in FIG. 8 to calculate the cycle of the position of the value INP indicated by the input designation unit 7 The value P (INP) of 6 is updated to -1 as a used mark to indicate that it has been used for the series.

  Subsequently, the control unit 2A sets the value OUTP (here, 1) of the output data in the output storage unit 8 to the value INP of the input designating unit 7 in step S16, and the value PS of the period accumulating unit 9 in the next step S17. 1 is added to update the number of periods to 1 (the number of periods becomes 1), and the position input of the period calculation unit 6 indicated by the value INP (here, 1) of the input designation unit 7 in the next step S18. The output data P (INP) (here, 0) corresponding to the data is set to the value OUTP of the output storage unit 8.

  Next, the control unit 2A determines whether or not the value OUTP of the output storage unit 8 is −1 indicating that the value has already been used for the series, and if it is used (−1), the process proceeds to step S23. If not used, the process proceeds to step S20. Here, since OUTP is 0 and not −1 (not used), the process proceeds to step S20, and the value P of the output data of the period calculation unit 6 at the position indicated by the value INP (here, 1) of the input designation unit 7 is reached. Set 0 which is (INP) to the value INP of the input designation unit 7.

  Subsequently, the control unit 2A updates the output data value P (0) of the cycle calculation unit 6 at the position indicated by the value INP (here, 0) of the input designating unit 7 to −1 and has already been used for the series. After a certain fact (step S21 in FIG. 8), the value PS of the cycle accumulating unit 9 is incremented by 1 to update the cycle number, and the process proceeds to step S18. As a result, the period value PS becomes 2.

  In step S18, the value of the output data (INP = 0 and P (INP) is -1) corresponding to the input data at the position P (INP) of the period calculation unit 6 indicated by the value INP of the input designating unit 7 is an output storage unit. Set the value 8 to OUTP. Subsequently, the control unit 2A determines whether or not the value OUTP of the output storage unit 8 is −1 in step S19. Here, since the value OUTP of the output storage unit 8 is −1, the process proceeds to step S23, and the m period values PR (PM, 1) ˜ of the period storage unit 10 indicated by the value PM of the permutation number setting unit 11. Of PR (PM, m), 1 is added to the value PR (PM, PS) corresponding to the value PS of the period accumulating unit 9 (here, 1 is added to the position of period 2 and PR (1,2) After the number of periods is updated, the process proceeds to step S5 in FIG.

  Subsequently, the control unit 2A proceeds again to the step S8 through the above-described steps S6 and S7, where the input data of the position of the value P (INP) of the period calculation unit 6 indicated by the value INP of the input specifying unit 7 (here In this case, the output data corresponding to 0) (here, -1) is set to the value OUTP of the output storage unit 8. Subsequently, the control unit 2A proceeds to step S9, determines that the value OUTP of the output storage unit 8 is −1 and has been used, then proceeds to step S10, and the value INP of the input designating unit 7 is 0. Therefore, since it is determined that the position is not the final position 1 (= m−1) in the series, the process proceeds to step S11, and the value INP of the input designating unit 7 is incremented by 1 to set INP to 1.

  Subsequently, the control unit 2A outputs the output data corresponding to the input data (here 1) of the position of the value P (INP) of the period calculation unit 6 indicated by the value INP (here 1) of the input designation unit 7 in step S8. After setting (here, -1) to the value OUTP of the output storage unit 8, the process proceeds to step S9, where the value OUTP of the output storage unit 8 is -1 and determined to be used, and then the process proceeds to step S10. Thus, it is determined that the value INP of the input designating unit 7 is 1, and that it is the last position 1 (= m−1) in the series.

  Thus, the control unit 2A determines whether the value PM indicated by the permutation number setting unit 11 is 1 at the last position of the data (step S12 in FIG. 8). At this point, since PM is 1 by the process in step S13 described above, it is determined in step S12 that PM is 1, and it is determined that all series of processes have been performed, and the process ends (FIG. 8 step S14). When all the processes are completed, the number of all the existences of each period length of each series of the bijection function is obtained in the period storage unit 10. Here, m = 2, and the cycle storage unit 10 stores a value indicating that two cycles 1 and one cycle 2 are stored.

  Thus, according to the present embodiment, the period can be obtained by the feedback of the function as shown in FIG. In FIG. 10, m! For each permutation data created by each of the bijection functions, the first input data is a value of 0 which is the smallest of m input data from 0 to m−1 of each bijection function. The output data is further fed back as input data, and the number of feedbacks until the input data returns to 0, which is the first input data, is obtained as a sequence period. A value that does not exist in the output data, such as −1, is set in the period calculation unit 6 so that it can be determined that the output data that has appeared at this time has appeared.

  If all m data appear in the sequence, the maximum period is reached, and a complete pseudo-random sequence is created. However, when output data that does not appear remains, the period is of a partial series, and another partial period exists in the remaining data that does not appear. If unappeared output data remains, the smallest input data among the non-appearing output data is set as the first input data of a new period, and a new sequence period until the input data is returned to This procedure is repeated until all output data appears in the series. And m! With respect to all permutation data (functions), the number of periods and the number of existence of each period are obtained by this procedure.

  In this way, according to the present embodiment, it is possible to determine the entire period of all integer series of all bijective functions. Then, it is possible to create a pseudo-random number sequence or conversion table using a bijection function with an arbitrary period.

(Second Embodiment)
FIG. 11 shows a block diagram of a second embodiment of the integer sequence period determining apparatus according to the present invention. As shown in the figure, the integer sequence period determination device 1B of the present embodiment includes a control unit 2B that centrally controls the entire apparatus 1B, a data number storage unit 3, a permutation creation unit 4, a permutation storage unit 5, and a cycle. The calculation unit 6, the input designation unit 7, the output storage unit 8, the cycle accumulation unit 9, and the cycle storage unit 10, and when compared with the integer series cycle determination device 1 A of the first embodiment, a permutation number setting unit 11 and the search position designation unit 12 are not provided.

  The control unit 2B, the permutation creation unit 4, and the cycle calculation unit 6 are configured by, for example, a central processing unit (CPU). Further, the data number storage unit 3, the output storage unit 8, the cycle accumulation unit 9, and the cycle storage unit 10 are configured by a storage device, and the input designation unit 7 is configured by a predetermined input device.

  The data number storage unit 3 stores the number M of continuous natural number input / output data (original image and image) possessed by a bijection function that is a bijection and a bijection function. The permutation creating unit 4 sets a permutation A (i), which is data prepared in advance for determining the period, and is an array of output data (images relative to the original image) with respect to the input of the bijective function. In this case, it is assumed that the permutation A (i) is created so that all values appear without overlapping the values of the output data. The permutation storage unit 5 is a first storage unit that stores in advance a permutation S (j) that is an array of output data (images with respect to the original image) with respect to the input of a bijection function having an arbitrary number of data m. As will be described later, the period calculation unit 6 repeatedly performs a procedure in which output data (an image with respect to the original image) with respect to an input of an arbitrary bijection function is further input, and calculates the period P of the integer series to be determined.

  The input designation unit 7 designates an input position (original image position) INP. The output storage unit 8 stores the value OUTP of the output data (image) of the cycle calculation unit 6 at the input position specified by the input specification unit 7. The period accumulating unit 9 stores the accumulated result PS of the period P calculated by the period calculating unit 6. The cycle storage unit 10 is a second storage unit that stores the cycle calculation result PR (PM, M) of the cycle calculation unit 6. In the present embodiment, the cycle storage unit 10 stores the existence numbers for all cycles including a partial cycle of a bijection function having an arbitrary number of data m (here, m = 3).

  Among the above configurations, the control unit 2B, the data number storage unit 3, the permutation creation unit 4, the cycle calculation unit 6, the input designation unit 7, and the cycle accumulation unit 9 are m! Among the bijective functions, a permutation storage unit 5 which is a first storage unit by generating an output state for an input of an integer sequence consisting of m integers of a desired bijective function as a permutation. The permutation storage means for storing the information is configured. In addition, the control unit 2B, the cycle calculation unit 6, the input designation unit 7, the output storage unit 8, and the cycle accumulation unit 9 calculate the cycle of the permutation stored in the permutation storage unit 5 and serve as the second storage unit. The period calculation and accumulation means stored in the storage unit 10 is configured.

  Next, the operation of this embodiment will be described with reference to the flowcharts of FIGS.

  In FIG. 11, the control unit 2B first sets the data number m (here, m = 3) in the data number storage unit 3 (step S31 in FIG. 12). Subsequently, the control unit 2B sets each value of the periods PR (0) to PR (m-1) stored in the period storage unit 10 to 0 (step S32 in FIG. 12), and then the permutation creation unit 4 Are set as output series data A (0) to A (m-1) corresponding to input data (natural numbers from 0 to m-1) as input / output data of the bijection function (step S33 in FIG. 12). . Specifically, the output series data A (0) to A (m-1) is data of arbitrary combinations of non-overlapping natural numbers from 0 to m-1, and here m = 3. , A (0) = 2, A (1) = 0, A (2) = 1.

  Subsequently, the control unit 2B converts the three output series data A (0) to A (2) set in the permutation creation unit 4 into the permutations S (0) to S (m-1) in the permutation storage unit 5. (Step S34 in FIG. 12). Here, the above-described permutation storage unit 5 stores three permutations of S (0) = 2, S (1) = 0, and S (2) = 1.

  Subsequently, the control unit 2B sends the permutations S (0) to S (m-1) stored in the permutation storage unit 5 to the cycle calculation unit 6 for calculation data P (0) to P (m-1). As shown in FIG. 12, the function output data is set (step S35 in FIG. 12). Here, three values of P (0) = 2, P (1) = 0, and P (2) = 1 are set in the period calculation unit 6 described above.

  Subsequently, the control unit 2B sets (clears) the cycle value PS of the cycle accumulating unit 6 to 0 (step S36 in FIG. 12), and then the position of the leading data corresponding to the first input data (for example, 0). Is set as the value INP of the input designation unit 7 (step S37 in FIG. 12). Next, the control unit 2B sets the value P (INP) of the output data corresponding to the input data of the cycle calculation unit 6 indicated by the value INP of the input designating unit 7 as the value OUTP of the output storage unit 8 (FIG. 12). Step S38). Here, since INP is 0, the input data is P (0), and the output data corresponding to it is 2 as described above, so OUTP is set to 2.

  Subsequently, the control unit 2B compares whether or not the value OUTP in the output storage unit 8 has already been used for the series (step S39 in FIG. 12). If it is used, it is -1 as a used mark. Here, since OUTP is 2 and not −1, the control unit 2B determines that it has not been used, and proceeds to step S43 of FIG. 13 to input the period calculation unit 6 indicated by the value INP of the input designation unit 7 The value P (INP) of the output data corresponding to the data is set to -1, indicating that it has been used for the series. As a result, P (0) = − 1.

  Next, the control unit 2B sets the value OUTP (here, 2) in the output storage unit 8 to the value INP in the input designating unit 7 (step S44 in FIG. 13). Subsequently, the control unit 2B adds 1 to the value PS of the cycle accumulating unit 9 to update the number of cycles (step S45 in FIG. 13). As a result, the cycle number PS becomes 1. Subsequently, the control unit 2B sets the value P (INP) of the output data corresponding to the input data of the period calculation unit 6 indicated by the value INP (here, 2) of the input designating unit 7 as the value OUTP of the output storage unit 8. Setting is made (step S46 in FIG. 13). Here, since INP is 2, the output data corresponding to P (2) is 1, as described above, so OUTP is set to 1.

  Next, the control unit 2B determines whether or not the value OUTP in the output storage unit 8 is −1 indicating that it has been used (step S47 in FIG. 13). Here, since OUTP is 1 and not −1 as described above, the control unit 2B determines the value of the input data at the position of the period calculation unit 6 indicated by the value INP (here, 2) of the input designation unit 7. Output data (here, 1) corresponding to P (INP) is set to the value INP of the input designating unit 7 (step S48 in FIG. 13). Subsequently, the control unit 2B sets the value P (INP) of the output data corresponding to the input data of the period calculation unit 6 indicated by the value INP (here, 1) of the input designating unit 7 to -1, and It shows that it has been used (step S49 in FIG. 13). As a result, P (1) = − 1.

  Next, the control unit 2B adds 1 to the cycle value PS of the cycle accumulating unit 9 to 2 and updates the number of cycles (step S50 in FIG. 13), and then returns to step S46. The value P (INP) of the output data corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP (here 1) is set as the value OUTP of the output storage unit 8. Here, since INP is 1, the output data corresponding to P (1) is -1, so OUTP is set to -1.

  Next, the control unit 2B determines whether or not the value OUTP in the output storage unit 8 is −1 indicating that it has been used (step S47 in FIG. 13). Here, since OUTP is −1 as described above, the control unit 2B determines that it has been used, and the cycle of the position of the cycle storage unit 10 corresponding to the value PS (2 here) indicated by the cycle accumulation unit 9 After adding 1 to the value PR (PS) and updating the number of periods (step S51 in FIG. 13), the process proceeds to step S36 in FIG. As a result, PR (2) becomes 1. That is, the number of periods of period 2 is determined to be one.

  Subsequently, in step S36, the control unit 2B sets (clears) the value PS of the period accumulating unit 6 to 0, and then sets the leading position (for example, 0) corresponding to the first input data to the value INP of the input designating unit 7. (Step S37), and further, the output data P (INP) corresponding to the input data at the position of the cycle calculation unit 6 indicated by the value INP of the input designation unit 7 is set as the value OUTP of the output storage unit 8 (step S37). S38). Here, since INP is 0, the input data is P (0), and the output data corresponding to it is −1 as described above, so OUTP is set to −1.

  Subsequently, in step S39, the control unit 2B determines that the value OUTP of the output storage unit 8 is −1 and has already been used for the series, and proceeds to step S40. In step S40, the control unit 2B determines whether or not the value INP of the input specifying unit 7 is 2 indicating the last position (m-1) in the sequence. Here, since INP is 0 and not 2, the control unit 2B adds 1 to the value INP of the input designating unit 7 to 1 (step S42 in FIG. 12), and then proceeds to step S38 and proceeds to step S38. The output data P (INP) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP is set as the value OUTP of the output storage unit 8. Here, since INP is 1, the input data is P (1), and the output data corresponding to it is 1 as described above, so OUTP is set to 1.

  Subsequently, the control unit 2B determines whether or not the value OUTP in the output storage unit 8 is −1 indicating that it has been used (step S39). Here, since OUTP is 1 and not −1, the control unit 2B determines that it has not been used, and proceeds to step S43 in FIG. 13 to indicate the position of the period calculation unit 6 indicated by the value INP of the input designation unit 7 The output data P (INP) corresponding to the input data value is set to -1 to indicate that it has been used for the series. As a result, P (1) = − 1.

  Thereafter, the control unit 2B determines that the value OUTP of the output storage unit 8 is −1 indicating that it has been used in step S47 after performing the above-described processing of steps S44, S45, and S46. Proceeding to S51, 1 is added to the period value PR (PS) of the position of the period storage unit 10 corresponding to the value PS (1 in this case) indicated by the period accumulating unit 9 to update the number of periods. As a result, PR (1) becomes 1. That is, it is determined that the number of periods 1 is 1.

  Thereafter, the control unit 2B determines that the value OUTP of the output storage unit 8 is −1 indicating that the value has been used in step S39 after performing the above-described processes of steps S36, S37, and S38. Proceeding to S40, it is determined whether or not the value INP of the input specifying unit 7 is 2 indicating the last position (m-1) of the sequence. Here, since INP is 0 and not 2, the control unit 2B adds 1 to the value INP of the input specifying unit 7 to 1 (step S42), and then proceeds to step S38, where the value INP of the input specifying unit 7 is The output data corresponding to the input data P (INP) at the position of the cycle calculation unit 6 shown is set to the value OUTP of the output storage unit 8. Here, since INP is 1, the input data is P (1), and the output data corresponding to it is −1 as described above, so OUTP is set to −1.

  Subsequently, in step S39, the control unit 2B determines that the value OUTP of the output storage unit 8 is −1 and has already been used for the series, and proceeds to step S40. In step S40, the control unit 2B determines whether or not the value INP of the input specifying unit 7 is 2 indicating the last position (m-1) in the sequence. Here, since INP is 1 and not 2, the control unit 2B adds 1 to the value INP of the input specifying unit 7 to 2 (step S42), and then proceeds to step S38, where the value INP of the input specifying unit 7 is The output data P (INP) corresponding to the input data at the position of the cycle calculation unit 6 shown is set to the value OUTP of the output storage unit 8. Here, since INP is 2, the input data is P (2), and the corresponding output data is −1 as described above, so OUTP is set to −1.

  Subsequently, in step S39, the control unit 2B determines that the value OUTP of the output storage unit 8 is −1 and has already been used for the series, and proceeds to step S40. In step S40, the control unit 2B determines whether or not the value INP of the input specifying unit 7 is 2 indicating the last position (m-1) in the sequence. Here, since INP is 2, the control unit 2B determines that the value INP of the input designation unit 7 has reached the last position in the sequence and has processed all input / output data, and ends all processing. (Step S41 in FIG. 12). In this way, in the periods PR (1) to PR (3) stored in the period storage unit 10, all existing numbers of each period length of this function are set. In this embodiment, m = 3. As described above, one PR (1) indicating the period number 1 as each period number of the bijective function and PR (2) indicating the period number 2 are provided. ) Is one.

  In this way, according to the present embodiment, it is possible to obtain all the periods including a partial period for an arbitrary data series of a bijective function having an arbitrary number of input / output data.

(Third embodiment)
Next, a third embodiment of the integer sequence period determining apparatus according to the present invention will be described. The hardware configuration of the cycle determination device of this embodiment is the same as that of the integer sequence cycle determination device 1B shown in FIG. 11, but the software operation by the control unit 2B is the same as that of the second embodiment as described below. Different.

  In the present embodiment, when a bijection function including a natural number of size 3 (m = 3) is given, a method for confirming whether the bijection function has a maximum period 3 (= m) is a certain method. The initial value (for example, 0) is given as an input value to the bijection function, and the conversion of the bijection function is repeated until the output (for example, 0) returns to the initial value, and the cycle that is the number of repetitions is 3 (= If m), all input / output values are covered and determined to be the maximum cycle, and if the number of conversion iterations is less than 3 (= m), it is determined not to be the maximum cycle.

  Next, a software operation using the hardware resources of this embodiment will be described with reference to the flowchart of FIG. As described above, the maximum period when m = 3 is 3. In the present embodiment, for example, it is determined whether or not the data of the output 2 when the input is 0, the output 0 when the input is 1, and the output 1 when the input is 2 is the maximum period 3 as in the case of f5.

  In FIG. 11, the control unit 2B first sets the number of data m (here, m = 3) in the data number storage unit 3 (step S61 in FIG. 14). Subsequently, the control unit 2B sets (clears) the value PS of the period accumulating unit 9 to 0 (step S62), and then inputs the input data (0 to m as input / output data of the bijection function to the permutation generation unit 4). Output series data A (0) to A (m-1) corresponding to natural numbers up to -1 is set (step S63). Specifically, the output series data A (0) to A (m-1) is data of arbitrary combinations of non-overlapping natural numbers from 0 to m-1, and here m = 3. , A (0) = 2, A (1) = 0, A (2) = 1. Here, A (0) = 2 indicates output 2 when input 0, A (1) = 0 indicates output 0 when input 1, and A (2) = 1 indicates output 1 when input 2.

  Subsequently, the control unit 2B converts the three output series data A (0) to A (2) set in the permutation creation unit 4 into the permutations S (0) to S (m-1) in the permutation storage unit 5. (Step S64). Here, the above-described permutation storage unit 5 stores three permutations of S (0) = 2, S (1) = 0, and S (2) = 1. These three permutations describe the output states for the three input values of a desired bijective function.

  Subsequently, the control unit 2B sets the permutations S (0) to S (m-1) stored in the permutation storage unit 5 as the periods P (0) to P (m-1) in the period calculation unit 6, respectively. (Step S65). Here, three cycles of P (0) = 2, P (1) = 0, and P (2) = 1 are set in the cycle calculation unit 6 described above. Here, P (0) = 2 is the period of output 2 when input 0, P (1) = 0 is the period of output 0 when input 1 and P (2) = 1 is the period of output 1 when input 2 Indicates the period.

  Subsequently, the control unit 2B sets a value indicating the head position (in this case, the input position 0) corresponding to the first input data to the value INP of the input specifying unit 7 (step S66). As a result, INP becomes zero. Subsequently, the control unit 2B outputs the output data P (INP) corresponding to the input data at the position of the period calculation unit 6 indicated by the value INP (here, 0) of the input designation unit 7 to the value OUTP of the output storage unit 8. (Step S67). Here, since INP is 0, the input data is 0, and the output data P (0) corresponding to it is 2 as described above, so OUTP is set to 2.

  Subsequently, the control unit 2B sets the value OUTP (here, 2) in the output storage unit 8 to the value INP in the input designation unit 7 (step S68), and then adds 1 to the cycle value PS of the cycle accumulating unit 9. Then, the number of cycles is updated (step S69). As a result, INP becomes 2 and PS becomes 1.

  Next, the control unit 2B determines whether or not the value OUTP (here, 2) in the output storage unit 8 is the value 0 of the first input data (step S70). If the value of OUTP is equal to the value of the first input data, the process proceeds to step S71, and if not equal, the process returns to step S67. At this time, since OUTP is 2 and not 0, the control unit 2B returns to step S67 and corresponds to the input data at the position of the period calculation unit 6 indicated by the value INP (here, 2) of the input designating unit 7. The output data P (INP) is set to the value OUTP of the output storage unit 8. Here, since the input data INP is 2, the output data corresponding to it is 1 as described above, so OUTP is set to 1.

  Subsequently, the control unit 2B sets the value OUTP (here, 1) in the output storage unit 8 to the value INP in the input designating unit 7 (step S68), and then adds 1 to the value PS in the cycle accumulating unit 9. The number of cycles is updated (step S69). As a result, INP becomes 1 and PS becomes 2.

  Next, the control unit 2B determines again whether or not the value OUTP (here, 1) in the output storage unit 8 is the value 0 of the first input data (step S70). At this time, since OUTP is 1 and not 0, the control unit 2B returns to step S67 and again enters the input data INP at the position of the period calculation unit 6 indicated by the value INP (here, 1) of the input designation unit 7. Corresponding output data P (INP) is set to the value OUTP of the output storage unit 8. Here, since INP is 1, the output data P (1) corresponding to the input data is 0 as described above, so OUTP is set to 0.

  Subsequently, the control unit 2B sets the value OUTP (here, 0) in the output storage unit 8 to the value INP in the input designating unit 7 (step S68), and then adds 1 to the value PS in the period accumulating unit 9. The number of cycles is updated (step S69). As a result, INP becomes 0 and PS becomes 3.

  Next, the control unit 2B determines again whether or not the value OUTP (here, 0) in the output storage unit 8 is the value 0 of the first input data (step S70). At this time, OUTP is 0, and it is determined that it is the first input value. Subsequently, the control unit 2B determines whether or not the cycle value PS of the cycle accumulating unit 9 is the sequence length m (step S71). At this time, since the value PS of the cumulative calculation unit 9 is equal to the sequence length m (= 3), the control unit 2B determines that it is the maximum length (maximum cycle) (step S72). That is, in the case of m = 3, it is determined that the data of the output 2 when the input is 0, the output 0 when the input is 1, and the output 1 when the input is 2 is the maximum cycle, and the cycle length PS is obtained as 3.

  When it is determined in step S71 that the value PS of the period accumulating unit 9 is not the sequence length m, it is determined that the input data at that time is not the maximum length (step S73).

  In this way, according to the present embodiment, it is possible to determine the maximum period length of the bijective function having an arbitrary input / output data size. Then, a pseudo random number sequence, a conversion table, or the like can be created using a bijective function having a maximum period.

  By the way, in the above embodiment, the case where the number of data (sequence length) m is 2 or 3 has been described.

この表４から分るように、入出力データ数（系列長）ｍを有する全単射関数の平均周期長はｍ／２となる。これにより、ｍが２¹²⁸の場合では、全単射関数の数は２¹²⁸！個となり、周期長は１が２¹²⁸！個で、２が２¹²⁸！／２個、３が２¹²⁸！／３個で、最大周期となる周期長２¹²⁸は２¹²⁸！／２¹²⁸個存在する。ここで、平均周期長は２¹²⁸／２となる。

As can be seen from Table 4, the average period length of the bijection function having the number of input / output data (sequence length) m is m / 2. Thus, in the case m is 2 ^128, the number of bijective functions 2 ^128! The cycle length is 1 ^{128 128} ! 2 with 2 ¹²⁸ ! / 2 pieces, 3 are 2 ¹²⁸ ! / 3, the maximum cycle length 2 ¹²⁸ is 2 ¹²⁸ ! / 2 There are ¹²⁸ pieces. Here, the average period length is 2 ^128/2 .

Further, 128-bit Considering the case of a block cipher, since the number of 128-bit key is 2 ^128, 128-bit block cipher m = 2 ¹²⁸ bijective functions all functions 2 ^128! 2 ¹²⁸ of the number said to be used to select the key. The maximum period ²¹²⁸ bijective function of m = 2 ¹²⁸ 2 ^128! / 2 Since there are ^128, considering the total number of keys for 128-bit block ciphers, all keys have the maximum period.
2 ¹²⁸ ! / 2 ¹²⁸ × 2 ^{128/2 128} ! = 1
The probability of the maximum period among all keys is at most one.

  However, the maximum cycle described here means that each combination of 128-bit output data appears once, and when one arbitrary bit of the output data is extracted, the sequence is M-sequence. Therefore, it is not always possible to obtain a bit sequence having a maximum length. For this purpose, further selection from a sequence having the maximum period is necessary.

  Note that a pseudo-random number sequence having a longer period can be generated by using a combination of two or more integer sequences of bijection functions having the maximum period determined in the above embodiment. For example, if the maximum length sequences having different periods of m1 and m2 are combined with a combination of disjoint periods having no divisor other than 1, the maximum period is m1 × m2.

  The present invention is not limited to the above embodiment. For example, the processing of each step of the first embodiment described with reference to FIGS. 7, 8, and 9 and the description with reference to FIGS. The present invention also includes an integer series period determination program that executes the processing of each step of the second embodiment and the processing of each step of the third embodiment described with reference to FIG. 14 by software processing by a computer. . The integer series period determination program is executed by the control unit 2A in FIG. 1 or the control unit 2B in FIG.

1A, 1B Integer period determination device 2A, 2B Control unit 3 Data number storage unit 4 Permutation creation unit 5 Permutation storage unit 6 Period calculation unit 7 Input designation unit 8 Output storage unit 9 Period accumulation unit 10 Period storage unit 11 Number of permutations Setting section
12 Search position specification part

Claims (9)

The control unit m has an input (original image) and an output (image) of an integer sequence composed of m integers with non-overlapping values from 0 to m−1 (where m is an integer of 2 or more). A first step of creating all permutations of output state data for the m types of inputs for a plurality of bijection functions (mappings) and storing them in a first storage unit;
The control unit stores the m! Stored in the first storage unit. For each of the permutations of the bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. Until the number of conversions is calculated as a period, and when the period is smaller than the m, it is determined that there is an unappeared input value, and an unappeared input value is given to the permutation All the above inputs are obtained by repeatedly converting until the same output value as the input value that does not appear is obtained, and obtaining the number of conversions as a cycle. A second step for sequentially performing the value and all said unoccurring input values;
The control unit determines the m! Determined in the second step. And a third step of storing, in a second storage unit, the period from the minimum period to the maximum period of each bijective function. The first step includes
Said m! After creating the permutation data of the first bijection function among the bijection functions (mappings), the second to m! 2. The integer sequence period determining method according to claim 1, wherein the permutation data of the bijection function up to the th is created from the permutation data of the bijection function of the immediately preceding bijection function. The control unit m has an input (original image) and an output (image) of an integer sequence composed of m integers with non-overlapping values from 0 to m−1 (where m is an integer of 2 or more). Among the plurality of bijection functions (mappings), the first storage unit stores in the first storage unit the output state of the desired one bijection function with respect to the input of the integer series consisting of the m integers as a permutation. 1 step,
The control unit obtains an output value when the first input is given out of the m inputs, and further sets the output value as an input value until the same output value as the first input value is obtained. And storing the number of repetitions as a cycle in the second storage unit for all input values not appearing in the cycle;
The control unit determines whether the cycle stored in the second storage unit is the m, and when the cycle is the m, the cycle is set to the desired one bijection. And a third step of determining that the cycle is the maximum cycle of the function. First and second storage units;
M having an input (original image) and an output (image) of an integer sequence consisting of m integers with non-overlapping values from 0 to m-1 (where m is an integer of 2 or more). Permutation generation and storage means for generating all permutations of the output states for the m types of input / output for a single bijection function and storing them in the first storage unit;
The m! Stored in the first storage unit. For each of the permutations of the bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. Until the number of conversions is calculated as a period, and when the period is smaller than the m, it is determined that there is an unappeared input value, and an unappeared input value is given to the permutation All the above inputs are obtained by repeatedly converting until the same output value as the input value that does not appear is obtained, and obtaining the number of conversions as a cycle. Periodic calculation and accumulation means for sequentially performing values and all the non-appearing input values;
The m determined in the second step! And a storage unit that stores the period from the minimum period to the maximum period of each bijective function in a second storage unit. The permutation generation and storage means includes
Said m! After creating the permutation data of the first bijection function among the bijection functions (mappings), the second to m! The permutation data of the first bijection function up to the second is created from the permutation data of the bijection function of the immediately preceding one, and the created permutation data is stored in the first storage unit. The integer sequence period determining apparatus according to claim 4. First and second storage units;
M having an input (original image) and an output (image) of an integer sequence consisting of m integers with non-overlapping values from 0 to m-1 (where m is an integer of 2 or more). Permutation storage means for storing, in the first storage unit, a state of an output with respect to an input of an integer sequence composed of the m integers of one desired bijection function among the plurality of bijection functions as a permutation; ,
Obtaining an output value when the first input is given among the m inputs, and further setting the output value as an input value is repeated until the same output value as the first input value is obtained, and the number of repetitions. A period calculation and accumulating unit for storing all of the input values that have not appeared in the period as a period in the second storage unit;
It is determined whether or not the period stored in the second storage unit is the m, and when the period is the m, the period is the maximum period of the desired single bijection function. An integer sequence period determining device comprising: a maximum period determining unit that determines that there is an integer sequence. On the computer,
M having an input (original image) and an output (image) of an integer sequence consisting of m integers with non-overlapping values from 0 to m-1 (where m is an integer of 2 or more). A first step of generating all permutations of output states for the m types of input / output for the bijective functions and storing them in a first storage unit;
The m! Stored in the first storage unit. For each of the permutations of the bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. A second step of repeatedly performing conversion until it is obtained, performing the conversion as a cycle for all input values, and storing the obtained cycle in the second storage unit;
When it is determined that the period obtained in the second step is smaller than the m, it is determined that there is an input value that does not appear, and an output value when an input value that does not appear is given to the permutation is obtained. Obtaining the output value as an input value, iteratively transforming until the same output value as the non-appearing input value is obtained, and obtaining the number of conversions as a period for all the non-appearing input values And a third step of sequentially storing the obtained period in the second storage unit;
Based on the period obtained in the second and third steps stored in the second storage unit, the m! And a fourth step of determining a period from the minimum period to the maximum period of each bijection function. The first step includes
Said m! After creating the permutation data of the first bijection function among the bijection functions (mappings), the second to m! 2. The program for determining a period of an integer sequence according to claim 1, wherein the permutation data of the bijection function up to the th is created from the permutation data of the bijection function in the immediately preceding order. On the computer,
M having an input (original image) and an output (image) of an integer sequence consisting of m integers with non-overlapping values from 0 to m-1 (where m is an integer of 2 or more). A first step of storing, in the first storage unit, a state of an output for an input of an integer sequence composed of the m integers of a desired one of the bijective functions among a plurality of bijective functions as a permutation; When,
The m! Stored in the first storage unit. For each of the permutations of the bijection function, an output value when a certain input value is given is obtained, and the output value is set as the input value, and the same output value as the first input value is obtained. Until the number of conversions is calculated as a period, and when the period is smaller than the m, it is determined that there is an unappeared input value, and an unappeared input value is given to the permutation To obtain the output value of the output, and further to convert the output value as the input value until the same output value as the input value that does not appear is obtained, and to obtain the number of conversions as the cycle A second step for all unoccurring input values;
The m determined in the second step! And a third step of storing the period from the minimum period to the maximum period of each bijective function in a second storage unit.

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