KR101300915B1 - Method of generating pseduo-random sequences based on digit derivatives - Google Patents

Abstract

PURPOSE: A number derivative-based pseudo random number producing method effectively performs a calculating process by using a polynomial differentiation, a polynomial sum, and a multiplication modulus. CONSTITUTION: After a number derivative is defined, an ergodic function is calculated using the number derivative (120). A pseudo random number sequence is generated based on the ergodic function (130). The bit values of a promising pseudo random number are set up and a polynomial expression for calculating a function value is selected by substituting the bit values in the ergodic function. The function value is obtained by applying the polynomial expression to the ergodic function continuously and the random number sequence is obtained based on the function value. [Reference numerals] (110) Define a number derivative {D_n(x)}_N>=0; (120) Calculate an ergodic function F(x) using the number derivative D_n(x); (130) Generate a pseudo random number sequence using the ergodic function F(x); (AA) Start; (BB) End

Description

Pseudo random number generation method based on numeric derivative {METHOD OF GENERATING PSEDUO-RANDOM SEQUENCES BASED ON DIGIT DERIVATIVES}

The present invention relates to a method for generating pseudo random numbers based on the characterization of an ergodic function represented by a numeric derivative.

A random number is an arbitrary sequence that doesn't depend on a particular sequence or each term. When you generate this random number on a computer, it has a long period because it is generated by some algorithm. In this sense, a random number is not a random number, but a random number that does not interfere with use as a random number is called a pseudo random number.

One way to generate this pseudorandom number is to apply classical p-base ergodic theory based on Van der Put and Mahler basis. Recently, the researchers proposed the characterization conditions of the Ergodic function represented by the Van der Put basis and Carlitz polynomial on the functional body instead of the p-base number, and proposed a pseudo random number generation method and program based on this. This program may have difficulty in choosing an extension factor or implementing this basis when using the van der Put basis. On the other hand, when the Carlitz polynomial is used, there is a difficulty in executing a program such as storing an ignition expression when obtaining a function value of an Ergodic function.

As a new alternative to secure the shortcomings of the existing pseudorandom methods and programs based on the characterization of ergodic functions on functional bodies, that is, a method of generating pseudorandom numbers based on numeric derivatives, and I would like to propose a program. The method proposed in the present invention has a computational efficiency unlike the conventional method because it is related to the derivative of the polynomial and the addition and the modulus operation of the polynomial.

CLAIMS 1. A method for generating pseudo random numbers, _comprising : defining a numeric derivative {D _n (x)} _n≥0 ; Obtaining an ergodic function f (x) using the numeric derivative D _n (x); And generating a pseudo random number sequence using the ergodic function f (x).

에 대해 숫자도함수 D_n(x)는

으로 정의되며, 여기서 숫자도함수 D_n(x)에서 분리된 도함수 D _i 는

으로 주어지는 고계 미분연산자이다.On one side, the numeric derivative D _n (x) is a binary extended function sequence of discrete high-order derivatives defined above on the power series F ₂ [[T]] with the finite field F ₂ = {0,1}. Nonnegative integer expressed

For the numeric derivative D _n (x) is

Where the derivative D _i separated from the numeric derivative D _n (x) is

It is a high-order differential operator given by.

을 갖는다. In another aspect, any continuous function above power series F ₂ [[T]] is a series expression given by a numeric derivative.

Respectively.

가 에르고딕이기 위한 특성화 조건으로, 확장계수 c_n을 결정하는 상기 특성화 조건은, 첫 번째로,

,

이고, 두 번째로, 1보다 큰 모든 자연수 n에 대해

(여기서 [x]는 x보다 크지 않은 최대정수)이며, 마지막 세 번째로, 2보다 큰 모든 자연수 n에 대해

인 것이다.In another aspect, the continuous function represented by the number derivative above the power series F ₂ [[T]].

As the characterization condition for is ergodic, the characterization condition for determining the expansion coefficient c _n is, firstly,

,

And secondly, for all natural numbers n greater than 1

(Where [x] is the largest integer not greater than x), and lastly, for all natural numbers n greater than 2

It is

In another aspect, in the step of generating a pseudo random number sequence by using the Ergodic function, the number of bits n (n ≧ 80) of the pseudo random number to be generated is determined, and one polynomial is selected as an input value. The values of the functions are computed by successive assignments to the Gothic function, and the resulting function values are n-bit polynomial sequences that yield pseudorandom sequences with a uniformly distributed period of 2 ⁿ .

를 임의로 선택하며, 다항식 v₀에 상응하는 벡터

로 나타낸다. In another aspect, the polynomial is an initial input value, wherein the polynomial is less than the number n of bits of the pseudorandom number

Selects randomly, the vector corresponding to polynomial v ₀

Respectively.

In another aspect, a function value obtained by applying the polynomial input value v ₀ to the ergodic function f (x) consecutively k times is

이며,

Is,

Finally, the pseudo random number sequence

이다.

to be.

A computer readable medium containing instructions for controlling a computer system to generate a pseudo random number, the instructions comprising: defining a numeric derivative {D _n (x)} _{n ≧ 0} ; Obtaining an ergodic function f (x) using the numeric derivative D _n (x); And generating a pseudo random number sequence using an ergodic function f (x), and controlling the computer system by a pseudo random number generation method.

Pseudorandom numbers play an important role in cryptography as plaintext to ciphertext and as a key used to decrypt ciphertext. In public key cryptography, digital signatures, authentication, and personal identification are used. In stream ciphers, random sequences are used as keys. Since pseudorandom numbers are widely used not only in cryptography but also in the fields of number theory, numerical analysis, and computer algorithm, it is very important to develop a program for generating pseudo random numbers.

Accordingly, the present specification proposes a new pseudo random number generation method based on a numeric derivative, and can be implemented and executed in computer simulation based on this, and can be compared and analyzed as an efficient alternative to the existing pseudo random number generator.

1 illustrates a flowchart of a method for generating pseudo random numbers according to an embodiment of the present invention.
2 is a flowchart illustrating a step of obtaining a pseudo random number sequence using an ergodic function in an embodiment of the present invention.
3 illustrates pseudo random number generation program code according to an embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings.

Pseudorandom number generator is an indispensable component of cryptography, and in most cases, pseudorandom numbers are generated through practical calculations associated with mathematical functions. Pseudorandom numbers are selected in some way and can be managed and controlled, but they are not predictable, so they appear random to third parties.

The present invention proposes a method for generating a pseudorandom number consisting of a polynomial sequence uniformly distributed from the characterization conditions of an ergodic function represented by a numeric derivative on a function body. Numerical derivatives deal with multiplication of polynomial derivatives, so it can be judged that the implementation of programs based on them has the advantages and efficiency of computation compared to the existing programs. Specific methods and contents proposed by the present invention are as follows.

1 illustrates a flowchart of a method for generating pseudo random numbers according to an embodiment of the present invention. CLAIMS What is claimed is: 1. A method of generating pseudo random numbers using characterization of an Ergodic function on a function body, the method _comprising : defining a number derivative {D _n (x)} _n≥0 (110); Obtaining 120 an ergodic function f (x) using the numeric derivative D _n (x); And generating a pseudo random number sequence 130 using the ergodic function f (x).

에 대해 숫자도함수 D_n(x)는

로 정의된다.In step 110, we define a numerical derivative that is computationally efficient, on which the invention is based, where the numeric derivative D _n (x) is a power series F ₂ [[T] with finite field F ₂ = {0,1}. ]] A non-negative integer expressed as binary, with the separated higher derivative binary extended function string defined above.

For the numeric derivative D _n (x) is

.

으로 주어지는 고계 미분연산자이다. Where the derivative D _i separated from the numeric derivative D _n (x)

It is a high-order differential operator given by.

을 갖는데, 멱급수환 F₂[[T]]위의 숫자도함수로 표현된 연속함수

가 에르고딕이기 위한 특성화 조건으로, 확장계수 c_n을 결정하는 조건은 다음과 같이 세 가지이다.In step 120 the continuous function f (x) is a series

A continuous function represented by the number derivative above the power series F ₂ [[T]].

As the characterization condition for is ergodic, there are three conditions for determining the expansion coefficient c _n .

이고,One.

ego,

(여기서 [log₂n]는 log₂n보다 크지 않은 최대정수)이며,2. for all natural numbers n greater than 1

(Where [log ₂ n] is the largest integer not greater than log ₂ n),

인 것이다.3. For all natural numbers n greater than 2

It is

It is also easy to choose the expansion coefficient c _n that satisfies the given sufficient condition that the function f (x) becomes an Ergodic function, so there is little difficulty in the early stages of program implementation. Finding the value of a function for any polynomial input value depends on the cost of calculating the value of the separated high derivative. Since the separated higher derivatives are simply related to the higher derivatives of the polynomials and the modulus calculations for addition and multiplication of the polynomials, the calculation of the function values of the ergodic functions can be seen as a no burden. Therefore, unlike the conventional basis, the function value calculation expressed in terms of the number derivative can be executed at no cost.

It is also easy to choose an expansion factor c _n that satisfies the given sufficient condition that the function f (x) becomes an ergodic function, so there is little difficulty in the early stages of program implementation. Finding the value of a function for any polynomial input value depends on the cost of calculating the value of the separated high derivative. Since the separated higher derivatives are simply related to the higher derivatives of the polynomials and the modulus calculations for addition and multiplication of the polynomials, the calculation of the function values of the ergodic functions can be seen as a no burden. Therefore, unlike the conventional basis, the function value calculation expressed in terms of the number derivative can be executed at no cost.

Step 130 of generating a pseudo random number sequence using an ergodic function may be performed according to the step of FIG. 2, in step 210 of determining a bit number n of a pseudo random number to be generated, and substituting the ergodic function. Selecting the polynomial to calculate the function value (220), applying the polynomial to the Ergodic function continuously to calculate the function value (230), and obtain a pseudo random number sequence of the calculated function value n bits (240) Can be.

를 임의로 선택하며, 다항식 v₀에 상응하는 벡터

로 나타내고 에르고딕 함수에 대입할 수 있다.The number of bits n of the pseudorandom number to be generated is determined to be at least 80, and the polynomial to calculate the function value in step 220 is an initial input value of the polynomial, and the polynomial whose degree is smaller than the number n of bits of the pseudorandom number.

Selects randomly, the vector corresponding to polynomial v ₀

Can be assigned to an Ergodic function.

의 함수 값을 얻을 수 있으며, 이렇게 해서 얻게 되는 n비트의 벡터 값의 의사 난수열, 혹은 다항식 수열은

이며, 균일하게 분포된 벡터 값 수열로, 그 주기가 2ⁿ이다. Furthermore, if we apply the polynomial input value, i.e., the vector value v _0, to the ergodic function f (x) in succession k times,

You can get the function value of, and the pseudo random or polynomial sequence of the n-bit vector value

Is a uniformly distributed vector value sequence whose period is 2 ⁿ .

In an embodiment, Figure 3 illustrates a programming code using Pari / GP. The code shown in FIG. 3 can be implemented as a program for performing each step of the pseudo random number generation method, and the result value is an n-bit pseudo random number sequence as a sequence of vector values.

According to one embodiment of the invention, a computer readable medium is provided that includes instructions for controlling a computer system to generate a pseudo random number. The command may further comprise defining the numeric derivative {D _n (x)} _{n ≧ 0} described above; Obtaining an ergodic function f (x) using the numeric derivative D _n (x); And instructions for controlling the computer system by a pseudo random number generation method including generating a pseudo random number sequence by using the ergodic function f (x), and may be recorded on a computer readable medium.

In addition, in the step of generating a pseudo random number sequence by using the ergodic function of the computer-readable medium, the number of bits n (n≥80) of the pseudo random number to be generated is determined, and substituted into the ergodic function to assign a function value. Select a polynomial to be calculated, apply the polynomial to the ergodic function continuously to calculate the function value, and obtain a pseudo-random sequence of n bits of the calculated function value.

Computer-readable media may include, alone or in combination with the program instructions, data files, data structures, and the like. The program instructions recorded on the media may be those specially designed and constructed for the purposes of the present invention, or they may be of the kind well-known and available to those having skill in the computer software arts. Meanwhile, the program according to the present embodiment may also be configured as a computer-based program or a mobile terminal-only application (eg, a smart device terminal-only application or a feature phone VM).

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed embodiments, but, on the contrary, This is possible.

Therefore, the scope of the present invention should not be limited to the described embodiments, but should be determined by the equivalents of the claims, as well as the claims.

Claims (9)

In the method for generating a pseudo random number,
Defining a numeric derivative {D _n (x)} _{n ≧ 0} ;
Obtaining an ergodic function f (x) using the number derivative D _n (x); And
Generating a pseudo random sequence using the ergodic function f (x)
Pseudo random number generation method comprising a. The method of claim 1,
The numeric derivative D _n (x) is a binary extended function sequence of the separated higher derivatives defined on the power series F ₂ [[T]] with a finite field F ₂ = {0,1},
Nonnegative integer represented in binary

For the numeric derivative D _n (x) is

Is defined as
Where the derivative D _i separated from the number derivative D _n (x) is

Being high differential operator given by
Pseudo random number generation method characterized in that. The method of claim 2,
The Ergodic function f (x) obtained using the number derivative D _n (x) is
Series of arbitrary powers above the power series F ₂ [[T]], given by a numeric derivative

Having
Pseudo random number generation method characterized in that. The method of claim 3,
Continuous function represented by the number derivative of the power series F ₂ [[T]]

As a characterization condition for which is an ergodic, the condition for determining the expansion coefficient c _n is

,

ego,
For all natural numbers n greater than 1

(Where [x] is the largest integer not greater than x),
For all natural numbers n greater than 2

Thing
Pseudo random number generation method characterized in that. 5. The method of claim 4,
In the step of generating a pseudo random number sequence using the ergodic function,
Determine the number of bits n (n≥80) of pseudorandom numbers to generate,
Select a polynomial to calculate a function value by substituting the ergodic function,
Applying the polynomial to the ergodic function successively to calculate the function value,
Obtaining a pseudo-random number sequence of the calculated function value n bits
Pseudo random number generation method characterized in that. The method of claim 5,
The polynomial is an initial input value and is a polynomial whose order is smaller than the number n of bits of the pseudorandom number.

Selects randomly,
Vector corresponding to polynomial v ₀

Represented by
Pseudo random number generation method characterized in that. The method according to claim 6,
The function value obtained by applying the polynomial input value v ₀ to the ergodic function f (x) consecutively k times is

Is,
The pseudo random sequence

2 ⁿ periods of uniform distribution of n-bit polynomial sequences
Pseudo random number generation method characterized in that. A computer readable medium containing instructions for controlling a computer system to generate a pseudo random number, the computer readable medium comprising:
The command
Defining a numeric derivative {D _n (x)} _{n ≧ 0} ;
Obtaining an ergodic function f (x) using the number derivative D _n (x); And
Generating a pseudo random sequence using the ergodic function f (x)
And control the computer system by a pseudo random number generation method. 9. The method of claim 8,
In the step of generating a pseudo random number sequence using the ergodic function,
Determine the number of bits n (n≥80) of pseudorandom numbers to generate,
Select a polynomial to calculate a function value by substituting the ergodic function,
Applying the polynomial to the ergodic function successively to calculate the function value,
Obtaining a pseudo-random number sequence of the calculated function value n bits
And a computer readable medium.

Images