KR101918741B1 - Method for distinction of safe prime number - Google Patents

Abstract

The present invention discloses a safe prime number determining method which combines the greatest common divisor (GCD) test using a property that a random number r is not a prime number and a Miller-Rabin (MR) test determining the prime number according to a result value performing a modular exponentiation computation for the random number r, if the greatest common denominator of the random number r and the prime numbers is not equal to 1. A prime number determining device comprises a GCD computation module and a modular exponentiation computation module.

Description

{METHOD FOR DISTINCTION OF SAFE PRIME NUMBER}

The present invention relates to a method for determining a safety factor, which is not only a prime factor but also a number obtained by dividing 2 by 1, which is a prime number.

Large primes are widely used in standard public key cryptosystems such as RSA (Rivest Shamir Adleman), ElGamal, Digital Signature Standard (DSS), and ECC (Elliptic Curve Cryptosystems) as the basis of current cryptography.

Today, there is a greater need to keep these systems safe from more advanced malicious attacks. By way of illustration, the initial RSA used a 512-bit key, but now requires a key of 1024 bits to 2048 bits.

Generally, when generating or discriminating a prime number, first generate an arbitrary odd random number and check whether it is a prime number. If the prime number is a prime number, the value is returned. If the prime number is not a prime number, . It is very time consuming to check whether the number is a prime number.

On the other hand, computing environments using information protection algorithms such as RSA encryption algorithms are rapidly changing in various forms, and it is necessary to study a method for determining safety fractions more quickly.

Korean Registered Patent No. 10-1731921, published on April 25, 2017.

The problem to be solved is that the GCD test (Greatest Common Divisor Test), which uses a property that the random number r is not a prime number when the greatest common divisor of r is 1 and the random number r is not a prime number, (Miller-Rabin Test) that discriminates the prime numbers according to the number of the users.

The problems to be solved are not limited to those mentioned above, and another problem to be solved which is not mentioned can be clearly understood by those skilled in the art from the following description.

)로 선정하는 단계(단, i는 1에서 s까지의 자연수)와, n-bit 크기의 홀수 난수(r)에 대해, r₁(단, r₁=r)과 r₂(단, r₂=

) 및 상기 s개의 테스트용 소수(

)를 이용하여, 두 수의 최대공약수가 1인지를 판별하는 GCD 테스트를 수행하는 단계와, 상기 GCD 테스트를 통과하는 경우에, 상기 r₁ 및 상기 r₂에 대해 m회의 모듈러 지수 연산을 수행하여 산출된 결과값이 '1' 또는 'r-1'인지를 판별하는 MR 테스트를 수행하는 단계와, 상기 MR 테스트를 통과하는 경우에, 상기 난수(r)를 안전소수로 판별하는 단계를 포함한다.According to the first embodiment, a safety factor determining method performed by a safety factor determining apparatus includes: grouping prime numbers that are multiplied by k prime numbers in sequence and multiplied by a value smaller than m-bit, Divide the number of prime numbers belonging to each prime group into prime numbers

(Where i is a natural number from 1 to s) and an odd random number r of n-bit size are r ₁ (where r ₁ = r) and r ₂ (where r ₂ =

) And the s test small numbers (

), Performing a GCD test to determine if the greatest common divisor of the two numbers is 1; and, if the GCD test passes, performing a modular exponentiation operation on the r ₁ and r ₂ times, Performing an MR test to determine whether the calculated result value is '1' or 'r-1'; and if the MR test is passed, determining the random number (r) as a safety decimal number .

)로 선정(단, i는 1에서 s까지의 자연수)하며, n-bit 크기의 홀수 난수(r)에 대해, r₁(단, r₁=r)과 r₂(단, r₂=

) 및 상기 s개의 테스트용 소수(

)를 이용하여, 두 수의 최대공약수가 1인지를 판별하는 GCD 테스트를 수행하는 GCD 연산 모듈과, 상기 GCD 테스트를 통과한 상기 난수(r)에 의한 상기 r₁ 및 상기 r₂에 대해 m회의 모듈러 지수 연산을 수행하여 산출된 결과값이 '1' 또는 'r-1'인지를 판별하는 MR 테스트를 수행하고, 상기 MR 테스트를 통과하는 경우에 상기 난수(r)를 안전소수로 판별하는 모듈러 지수 연산 모듈을 포함한다.According to the second embodiment, the prime number discriminating apparatus divides prime numbers that are multiplied by k prime numbers in order and multiplied by a value smaller than m-bit but satisfy a condition as large as possible, into groups of s number of prime numbers, The product of the prime numbers in the group is the test fraction (

(Where r is a natural number from 1 to s) and r ₁ (where r ₁ = r) and r ₂ (where r ₂ =

) And the s test small numbers (

), A GCD operation module for performing a GCD test to determine whether the greatest common divisor of the two numbers is 1, and a GCD operation module for performing a GCD test on the r ₁ and r ₂ by the random number (r) Modulo exponentiation to perform a MR test to determine whether the resultant value is '1' or 'r-1', and when the MR test is passed, a modulo (r) And an exponentiation module.

According to the embodiment of the present invention, the combination of the GCD test and the MR test can discriminate whether the random number r is a safety decimal number or not, thereby determining the safety decimal number faster than before.

1 is a functional block diagram of a safety decimation apparatus according to an embodiment of the present invention.
FIG. 2 is a flowchart illustrating a method of determining a safe decimal number according to an embodiment of the present invention.
FIG. 3 is a flowchart for explaining an example of a GCD test used for discriminating a safety factor.
FIG. 4 is a flowchart for explaining another example of the GCD test used for determining the safety factor.
5 is a graph comparing the execution times of the GCD-GCD-MR-MR and GCD-MR-MR tests and the conventional TD-MR-MR test according to an embodiment of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS The advantages and features of the present invention and the manner of achieving them will become apparent with reference to the embodiments described in detail below with reference to the accompanying drawings. It should be understood, however, that the invention is not limited to the disclosed embodiments, but may be embodied in various forms and should not be construed as limited to the embodiments set forth herein, To fully disclose the scope of the invention to a person skilled in the art, and the scope of the invention is only defined by the claims.

In describing embodiments of the present invention, a detailed description of well-known functions or constructions will be omitted unless otherwise described in order to describe embodiments of the present invention. The following terms are defined in consideration of the functions in the embodiments of the present invention, which may vary depending on the intention of the user, the intention or the custom of the operator. Therefore, the definition should be based on the contents throughout this specification.

1 is a functional block diagram of a safety decimation apparatus according to an embodiment of the present invention.

The safety factor determining apparatus 100 according to the embodiment of FIG. 1 includes a GCD computing module 110 and a modular exponentiation computing module 120.

The GCD operation module 110 divides k prime numbers into s prime groups and multiplies prime numbers in each group to generate s test prime numbers. In this case, the s number of fractional groups satisfy the condition as large as possible but the product of the prime numbers belonging to each group is smaller than m-bit.

, i=1,2,3,…,s)로 생성한다. 예를 들어, k=7개의 소수 P₁, P₂, P₃, P₄, P₅, P₆, P₇이 있을 때 P₁부터 차례대로 소수를 곱하여 그 곱의 크기가 3-bit(m=3) 미만의 최대값이 되는 소수 P₁, P₂, P₃을 소수 그룹1로, 이와 유사하게 소수 P₄, P₅를 소수 그룹2로, 소수 P₆, P₇을 소수 그룹3으로 하여, 3개(s=3)의 소수 그룹으로 나누고, 테스트용 소수 =P₁×P₂×P₃, =P₄×P₅, =P₆×P₇를 생성한다.At this time, the GCD operation module 110 performs the process of grouping the k number of prime numbers into a small number of groups until k number of prime numbers are exhausted. Then, the GCD operation module 110 multiplies the product of the prime numbers within the group by the prime number group

, i = 1, 2, 3, ... , s). For example, k = 7 minority _{P 1, P 2, P 3} , P 4, P 5, P 6, P 7 is multiplied by a prime number in order from P ₁ when the product is 3-bit (m size = 3), the prime numbers P ₁ , P ₂ , and P _{3 are set} to the prime group 1, and similarly, the prime numbers P ₄ and P _{5 are set} to the prime group 2, the prime numbers P ₆ and P _{7 are set} to the prime group 3 , And divides the result into three groups of three (s = 3) prime numbers to generate a prime number for test = P ₁ P ₂ P ₃ , = P ₄ P ₅ , = P ₆ P ₇ .

Meanwhile, the GCD operation module 110 generates or receives an odd random number r of n-bit size. FIG. 1 shows an example of receiving a random number r generated in a separate random number generator (not shown).

) 및 s개의 테스트용 소수(

)를 이용하여, 두 수의 최대공약수가 1인지를 판별하는 GCD 테스트(Greatest Common Divisor Test)를 수행한다. 여기서, GCD 연산 모듈(110)은 r₁과 테스트용 소수(

)의 최대공약수가 1일 때에, r₂와 테스트용 소수(

)의 최대공약수가 1이면 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다. 또는, GCD 연산 모듈(110)은 r₁과 r₂를 곱한 값과 테스트용 소수(

)의 최대공약수가 1일 경우에 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다.The GCD calculation module 110 calculates r ₁ (r ₁ = r) and r ₂ (where r ₂ =

) And a small number of test samples (

), And performs a GCD test (Greatest Common Divisor Test) to determine whether the greatest common divisor of the two numbers is one. Here, the GCD calculation module 110 calculates r ₁ and a test fraction

) Is 1, r ₂ and the test decimal (

) Is 1, it can be judged that the random number (r) has passed the GCD test. Alternatively, the GCD operation module 110 may calculate a value obtained by multiplying r ₁ by r ₂ ,

) Is 1, it can be determined that the random number (r) has passed the GCD test.

The modular exponentiation computation module 120 performs m modular exponentiation operations on r ₁ and r ₂ by the random number (r) passed the GCD test in the GCD operation module 110. At this time, the modulus exponentiation module 120 performs an MR test (Miller-Rabin Test) to determine whether the result value calculated in the modular exponentiation operation is not '1' or 'r-1'. Here, the modular exponentiation computing module 120 determines the random number r as a safety decimal number when passing the MR test. At this time, the modular exponentiation module 120 can sequentially perform the MR test for r ₁ and r ₂ . For example, a second-order MR test for r ₂ may be performed if r ₁ has passed the first MR test.

FIG. 2 is a flowchart illustrating a method of determining a safe decimal number according to an embodiment of the present invention.

)를 선정하는 단계(S210)를 수행할 수 있다. GCD 연산 모듈(110)이 테스트용 소수(

)를 선정하는 구체적인 과정은 도 1에서 설명한 바와 같다.Referring to FIG. 2, the GCD operation module 110 included in the prime number discrimination apparatus 100 shown in FIG. 1 includes a test small number

(Step S210). When the GCD operation module 110 receives a test small number (

) Is as described in Fig.

Thereafter, the GCD operation module 110 may perform step S220 of generating or receiving an odd random number r of n-bit size. FIG. 2 shows an example in which the GCD operation module 110 generates an odd random number r.

) 및 s개의 테스트용 소수(

)를 이용하여, 두 수의 최대공약수가 1인지를 판별하는 GCD 테스트를 수행하는 단계(S230)를 수행할 수 있다. 여기서, 단계 S230에 의한 GCD 테스트는 m-bit보다 작지만 최대한 큰 테스트용 소수(

)를 이용하므로 이하에서 'm-bit GCD 테스트'라고 한다.The GCD calculation module 110 calculates r ₁ (r ₁ = r) and r ₂ (where r ₂ =

) And s test prime numbers (

) To perform a GCD test to determine whether the greatest common divisor of the two numbers is 1 (S230). Here, the GCD test by the step S230 is smaller than m-bit, but the largest test small number (

), It is called 'm-bit GCD test' below.

)의 최대공약수가 1일 때에, r₂와 테스트용 소수(

)의 최대공약수가 1이면 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다. 이에 대한 보다 구체적인 설명은 도 3을 참조하여 아래에서 다루기로 한다.In step S230, GCD calculation module 110 as an example, r ₁ and a small number of test (

) Is 1, r ₂ and the test decimal (

) Is 1, it can be judged that the random number (r) has passed the GCD test. A more detailed description thereof will be described below with reference to Fig.

)의 최대공약수가 1일 경우에 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다. 이에 대한 보다 구체적인 설명은 도 4를 참조하여 아래에서 다루기로 한다.Alternatively, in step S230, the GCD operation module 110 may calculate, as another example, a value obtained by multiplying r ₁ by r ₂ and a value obtained by multiplying the test decimal

) Is 1, it can be determined that the random number (r) has passed the GCD test. A more detailed description thereof will be described below with reference to FIG.

If the GCD test is not passed in step S240, the GCD operation module 110 returns to step S220 to re-generate or re-input the odd random number r to sequentially perform the following processes again.

However, if the GCD test passes in step S240, the flow proceeds to step S250 performed by the modular exponentiation module 120. [

The modulus exponentiation computing module 120 included in the prime number discriminating apparatus 100 of FIG. 1 may perform the MR test on the random number r passed the GCD test (S250). More specifically, modular exponentiation module 120 performs m modular exponentiation operations on r ₁ and r ₂ . The modulus exponentiation module 120 performs an MR test to determine whether the resultant value calculated in the modular exponentiation operation is not '1' or 'r-1'. At this time, the modular exponentiation module 120 can sequentially perform the MR test for r ₁ and r ₂ . For example, a second-order MR test for r ₂ may be performed if r ₁ has passed the first MR test.

If it is determined in step S260 that the MR test has not been passed, the process returns to step S220. In step S260, the GCD operation module 110 generates the odd random number r again or receives the odd random number r again.

The algorithm for identifying the prime numbers using the MR test can be represented as steps 1 through 5 as follows.

Step 1: Create or receive a random number (r)

W = 2 x ^h , where w is an odd number, h > 0,

Step 3: Determine the number of repetitions (t), the number of executions (i) = 1

Step 4: while i ≤ t do

   Step 4-1: arbitrary number (a) selection, where 1 < a < r

Step 4-2: modular value (b) = a ^w mod r

Step 4-3: If the modular value (b) = 1 mod r, the process proceeds to Step 4-6

Step 4-4: for j = 0 to h-1 do

    Step 4-4-1: If b = (r-1) mod r, proceed to Step 4-6

Step 4-4-2: b = b ² mod r

Step 4-5: r is determined to be a synthetic number, and the process returns to the first step

Steps 4-6: i = i + 1

Step 5: Determine r as a prime number

On the other hand, the MR test is a test method for compensating for the fact that the Fermat test always passes a small number of determinations for the number of chambers, and is a kind of known minor test method, so a detailed description will be omitted.

In step S260, the modular exponentiation calculating module 120 determines that the random number r is a safety decimal number if it is determined that the MR test is passed (S270).

FIG. 3 is a flowchart for explaining an example of a GCD test used for discriminating a safety factor.

)의 최대공약수를 산출하는 단계(S310)를 수행할 수 있다. 그리고, GCD 연산 모듈(110)은 단계 S310에서 산출된 최대공약수가 1인가를 판정(S320)하고, 단계 S320에서 최대공약수가 1로 판정되면, r₂와 테스트용 소수(

)의 최대공약수를 산출(S330)할 수 있다. 그리고, GCD 연산 모듈(110)은 단계 S330에서 산출된 최대공약수가 1인가를 판정(S340)하고, 단계 S340에서 최대공약수가 1로 판정되면, 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다(S350). 반면에, 단계 S320 및 S340에서 최대공약수가 1이 아닌 것으로 판정되면 GCD 테스트를 불통과한 것으로 판단할 수 있다(S360).Referring to FIG. 3, the GCD operation module 110 included in the decimation apparatus 100 shown in FIG. 1 includes r ₁ and test decimals

(S310) of calculating the greatest common divisor of the number Then, when the GCD calculation module 110 is the greatest common factor of 1 is applied to the determination (S320), and the greatest common factor is determined by 1 in step S320 calculates in step S310, r ₂ and a few test (

(S330). &Lt; / RTI &gt; Then, the GCD operation module 110 determines whether the greatest common divisor is one (S340). If it is determined in step S340 that the greatest common divisor is 1, the GCD operation module 110 determines that the random number r has passed the GCD test (S350). On the other hand, if it is determined in steps S320 and S340 that the greatest common divisor is not 1 (S360), it can be determined that the GCD test is disrupted.

As shown in FIG. 3, when the second GCD test is performed in steps S330 and S340 after performing the first GCD test in steps S310 and S320, the execution time of the entire m-bit GCD test will be described.

First, GCD should be performed on r ₁ and, if the result is 1, GCD for r ₂ and GCD should be performed. Here, let X be the probability that the first GCD result is 1. Let R be the probability that the result of GCD is 1 when GCD is performed on r ₁ and r ₁ , and r ₁ and r ₂ are 1. At this time, the probabilities X and Y are as shown in the following equation (1).

Where P _i is a prime number and P ₁ <P ₂ <... <P _k follows the rules.

Therefore, the probability that the result of GCD is 1 when GCD is performed on r ₂ and is given by Equation 2 below.

와 r₁, r₂의 GCD가 1일 경우

에 대해서 GCD를 수행하여야 한다. 이때, r₁과

에 대해서 GCD를 수행했을 때의 1일 확률은 다음의 수학식 3과 같다.next,

And the GCD of r ₁ and r ₂ is 1

The GCD should be performed. At this time, r ₁ and

The probability of one day when the GCD is performed is expressed by Equation (3).

에 대해 GCD를 수행하였을 때에 GCD의 결과가 1일 확률은 다음의 수학식 4와 같다.Then, r ₂ and

The probability that the result of the GCD is 1 at the time when the GCD is performed is expressed by Equation (4).

와 r₁에 대해 GCD를 수행하였을 때와

와 r₂에 대해 GCD를 수행하였을 때에 GCD의 결과가 1일 확률은 각각 다음의 수학식 5와 같다.The last one

And r ₁ were performed for GCD

And r ₂ , the probability of the GCD result being 1 is given by Equation (5) below.

Based on Equations (1) to (5), the execution time of the m-bit GCD test according to the embodiment of FIG. 3 can be expressed as Equation (6) below.

Here, T _{GCD (m)} is the one time GCD test.

In addition, when the execution time of the safety factor determination method applied in the embodiment of FIG. 3 is summed with the execution time of the known MR test, it can be expressed as Equation (7) below.

Here, T (n, k) denotes the time taken to generate one n-bit safe decimal number through an n-bit random number r and k decimals. N _trial is the number of times t must be tried until r is a safety _factor , T _rnd is the random number generation time, and T _MR is the time of the MR test.

FIG. 4 is a flowchart for explaining another example of the GCD test used for determining the safety factor.

)의 최대공약수를 산출(S420)하고, 단계 S430에서 최대공약수가 1로 판정되면, 난수(r)가 GCD 테스트를 통과한 것으로 판단할 수 있다(S440). 반면에, 단계 S430에서 최대공약수가 1이 아닌 것으로 판정되면 GCD 테스트를 불통과한 것으로 판단할 수 있다(S450).Referring to FIG. 4, the GCD operation module 110 included in the prime number discriminating apparatus 100 illustrated in FIG. 1 may calculate a value obtained by multiplying r ₁ by r ₂ (S410). Thereafter, the GCD operation module 110 multiplies the value obtained by multiplying r ₁ and r ₂ calculated at step S410 and the test decimal number (

(S420). If it is determined in step S430 that the greatest common divisor is 1, it can be determined that the random number r has passed the GCD test (S440). On the other hand, if it is determined in step S430 that the greatest common divisor is not 1 (S450), it is determined that the GCD test is not performed.

As shown in FIG. 4, when the GCD test is performed only once in steps S410 and S420, the execution time of the entire m-bit GCD test will be described.

the probability that the result of GCD is 1 when the GCD is performed with respect to the value obtained by multiplying r ₁ by r ₂ and the value obtained by multiplying r ₁ by r ₂ is as shown in the following equation (8).

Based on Equation (8), the execution time of the m-bit GCD test according to the embodiment of FIG. 4 can be expressed as Equation (9) below.

In addition, when the execution time of the safety factor determination method applied in the embodiment of FIG. 4 is summed with the execution time of the known MR test, it can be expressed as Equation (10) below.

5 is a graph comparing the execution times of the GCD-GCD-MR-MR and GCD-MR-MR tests and the conventional TD-MR-MR test according to an embodiment of the present invention. 601 is a TD-MR-MR test, 602 is a GCD-GCD-MR-MR test, and 603 is a GCD-MR-MR test.

There are deterministic and probabilistic methods for determining the prime numbers. The deterministic test method ensures that the random number that passes the test is a prime number with a probability of 1, for example, there is a TD test (Trial Division Test). Furthermore, the probabilistic test method guarantees a probability of 1- (0.5) s for a very large s, and a random number passing through the test is a prime number, for example, an MR test or the like exists.

When determining the safety factor, a combination of minor test methods is used for a quicker minority test. The most common combination is the combination of the TD test and the MR test.

)에 대해서 TD 테스트를 한번에 수행한 뒤, r₁과 r₂ 각각에 대해 MR 테스트를 수행하는 조합이다.According to the prior art, there is a TD-MR-MR test method as a method for discriminating the safety factor. In this TD-MR-MR test, r ₁ (r ₁ = r) and r ₂ (where r ₂ =

) Is a combination of performing a TD test at a time and then performing an MR test for each of r ₁ and r ₂ .

In FIG. 5, the experimental environment of the PC was the Eclipse IDE for Java Developers Mars.1 Release (4.5.1) in Microsoft Windows8.1, 3.20GHz CPU, and 8.00GB RAM. Experimental method is to generate the number of safety factors 10,000 times when m is 32bit, 128bit, 512bit, 1024bit and 2048bit, and the average execution time is measured.

The fastest execution time of the TD-MR-MR method was 6.68 seconds at k = 984, and the fastest execution time of the GCD-MR-MR method was 6.33 seconds at m = 32 bits and k = 1102. The fastest execution time of the GCD-GCD-MR-MR method was m = 128 bits and 178 seconds when k = 114. Therefore, it can be seen that the GCD-MR-MR test can be used to generate the safety factor at the earliest in the PC environment.

Combinations of each step of the flowchart and each block of the block diagrams appended to the present invention may be performed by computer program instructions. These computer program instructions may be loaded into a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus so that the instructions, which may be executed by a processor of a computer or other programmable data processing apparatus, And means for performing the functions described in each step are created. These computer program instructions may also be stored in a computer usable or computer readable memory capable of directing a computer or other programmable data processing apparatus to implement the functionality in a particular manner so that the computer usable or computer readable memory It is also possible for the instructions stored in the block diagram to produce a manufacturing item containing instruction means for performing the functions described in each block or flowchart of the block diagram. Computer program instructions may also be stored on a computer or other programmable data processing equipment so that a series of operating steps may be performed on a computer or other programmable data processing equipment to create a computer- It is also possible that the instructions that perform the processing equipment provide the steps for executing the functions described in each block of the block diagram and at each step of the flowchart.

Also, each block or each step may represent a module, segment, or portion of code that includes one or more executable instructions for executing the specified logical function (s). It should also be noted that in some alternative embodiments, the functions mentioned in the blocks or steps may occur out of order. For example, two blocks or steps shown in succession may in fact be performed substantially concurrently, or the blocks or steps may sometimes be performed in reverse order according to the corresponding function.

The foregoing description is merely illustrative of the technical idea of the present invention, and various changes and modifications may be made by those skilled in the art without departing from the essential characteristics of the present invention. Therefore, the embodiments disclosed in the present invention are intended to illustrate rather than limit the scope of the present invention, and the scope of the technical idea of the present invention is not limited by these embodiments. The scope of protection of the present invention should be construed according to the following claims, and all technical ideas within the scope of equivalents should be construed as falling within the scope of the present invention.

100: fractional discrimination device
110: GCD operation module
120: modular exponentiation module

Claims (6)

A method for distinguishing safety numbers performed by a safety factor discriminator,
The prime numbers that are smaller than m-bit but satisfy the largest condition are grouped and divided into s prime groups, and the product of the prime numbers belonging to each prime group is divided into test prime numbers

), Where i is a natural number from 1 to s,
r ₁ (where r ₁ = r) and r ₂ (where r ₂ = r) are set for an odd random number r of n-bit size,

) And the s test small numbers (

), Performing a GCD test (Greatest Common Divisor Test) to determine whether the greatest common divisor of the two numbers is 1,
When passing the GCD test, an MR test (Miller-Rabin (R)) for determining whether the calculated result value is '1' or 'r-1' by performing m modular exponent operations on the r ₁ and r ₂ Test)
Determining the random number (r) as a safety decimal number when passing the MR test
How to identify safety factor.
The method according to claim 1,
Passing the GCD test is to pass the r ₁ and the test prime number (

) Is 1, the r ₂ and the test fractional number (

) &Lt; / RTI &gt; is 1, the GCD test is passed.
How to identify safety factor.
The method according to claim 1,
Passing the GCD test is a value obtained by multiplying r ₁ by r ₂ ,

) &Lt; / RTI &gt; passes the GCD test when the greatest common divisor of &lt; RTI ID = 0.0 &gt;
How to identify safety factor.
The method according to claim 1,
The step of performing the MR test may include sequentially performing an MR test on the r ₁ and the r ₂
How to identify safety factor
21. A computer-readable medium having stored thereon instructions for performing the steps of determining a safety factor when executed by one or more processors,
The above-
The prime numbers that are smaller than m-bit but satisfy the largest condition are grouped and divided into s prime groups, and the product of the prime numbers belonging to each prime group is divided into test prime numbers

), Where i is a natural number from 1 to s,
r ₁ (where r ₁ = r) and r ₂ (where r ₂ = r) are set for an odd random number r of n-bit size,

) And the s test small numbers (

), Performing a GCD test (Greatest Common Divisor Test) to determine whether the greatest common divisor of the two numbers is 1,
When passing the GCD test, an MR test (Miller-Rabin (R)) for determining whether the calculated result value is '1' or 'r-1' by performing m modular exponent operations on the r ₁ and r ₂ Test)
And determining the random number (r) as a safety decimal number when passing the MR test
A computer readable recording medium.
The prime numbers that are smaller than m-bit but satisfy the largest condition are grouped and divided into s prime groups, and the product of the prime numbers belonging to each prime group is divided into test prime numbers

(Where r is a natural number from 1 to s) and r ₁ (where r ₁ = r) and r ₂ (where r ₂ =

) And the s test small numbers (

), A GCD operation module for performing a GCD test (Greatest Common Divisor Test) for determining whether the greatest common divisor of two numbers is 1,
An MR test for determining whether the calculated result value is '1' or 'r-1' by performing m modular exponent operations on the r ₁ and r ₂ by the random number r passed the GCD test A modulo exponentiation module for performing the Miller-Rabin Test and determining the random number (r) as a safety decimal number when passing the MR test
Fractional discrimination device.

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