CN102769528A - Quick large number decomposition method based on cryptographic technology application - Google Patents

Abstract

The invention discloses a quick large number decomposition method based on cryptographic technology application. The creative quick large number decomposition method organically combining the congruence theory and the composite number distribution rule includes the steps of compressing positive integers, setting up an almost prime number group, selecting a maximum dereferencing range, generating the almost prime number group, decomposing a large number, setting digits, deleting composite numbers and the like. The method different from traditional large number decomposition methods can be used for quick factorization of any large number selected from n without depending on computing speed, and quick decomposition can be realized by using computer software for simple identification processing instead of a great amount of time-consuming complex operations. Therefore, evaluation and improvement of relevant public key system safety can be greatly promoted, and the quick large number decomposition method is widely applicable to the field of information safety and cryptographic technology application.

Description

Big several quick decomposition methods based on the cryptological technique application

Affiliated technical field

The present invention relates to information security and cryptological technique application, particularly relate to a kind of big several quick decomposition methods of using based on cryptological technique.

Background technology

In the Modern Cryptology, the fail safe of password is primary.Because the complexity that the algorithm of mathematical problem is found the solution can be described through computational complexity theory, and can be and break a code and computation complexity provides actual measure.Therefore, current most of cryptographic system usually can be summed up as and find the solution certain mathematical problem.And some the typical mathematical problems in the computational complexity theory provide the basis of the high strength cryptographic system of the practical safety of design to people.The public key cryptosystem that for example designs based on the knapsack problem in the NPC class is based on RSA public key cryptosystem of big several factorization problems of np problem etc.It is a great problem of art of mathematics that big number decomposes; In cryptography, 1977, Rivest; Samir and Adleman unite a kind of common key cryptosystem based on Euler's theorem in the number theory of proposition; Be called for short the RSA public key cryptosyst, the RSA public-key cryptosystem is exactly on the basis of this assumption: finding two big prime numbers is not very difficult relatively, but a big but difficulty very of prime factor form that number resolves into it of closing.This system is moved as follows:

C＝M ^e(mod?N) M＝C ^d(mod?N)

Wherein, M is expressly; C is a ciphertext; N=pq is a modulus, and p is different big prime numbers with q; E is disclosed encryption exponent (key), and d is privately owned decryption exponent (key) and satisfies ed ≡ 1 (mod φ (N)), (N, e) necessary disclosing, but d (also having φ (N)) need to be keep secret.Because function f: M → C is a trap-door one-way function, because it is an easy for calculation by quick exponentiation algorithm, and its contrary f ^-1: C → M is difficult to calculate, and for the people who does not know decruption key (trap door information) d, they will have to n is carried out factorization in order to find d, and calculates φ (n), yet for those people that know d, f ^-1Calculating as the same simple of the calculating of f, rsa cryptosystem thought that Here it is.Because the fail safe of RSA is higher relatively, therefore in current public key cryptosyst, be used widely.The fail safe of RSA is based on and is difficult to big number is carried out factor decomposition, and the latter is a mathematical famous difficult problem.Because big number decomposes the practical uses in the RSA public key cryptosyst, caused the dual current demand that it is decoded and its fail safe is assessed.At present; The most effective factoring algorithm; With tagsort running time; Mainly can be classified as following two big types: 1. mainly depend on the size of Integer N to be decomposed running time, the size that does not extremely depend on the factor p that finds mainly depends on the size of p among the N (factor of the N that finds) 2. running times.At present, adopted factorization method has a lot, and the simplest factorization method is a trying division method, and it is the complete prime factor breakdown that obtains n through all possible factor of attempting: n=p ₁P ₂P _t, p ₁≤p ₂≤≤p _tIt is effectively that trying division method detects getting rid of the little factor, but the method can not be used for decomposing fully, only if n is a very little number, such as is n＜10 ⁸Also have Fermat factoring method, this method be for one given greater than 1 odd number n, available this algorithm is confirmed being no more than of n

The maximum factor.And the general purpose integer factorization method that is widely used at present has following three kinds, that is: continued fraction method (being called for short CFRAC), secondary sieve method (being called for short QS) and number field sieve method (being called for short NFS).Continued fraction method (being called for short CFRAC) is first modern general factorization method.Secondary sieve method (QS) is at first proposed in nineteen eighty-two by Carl POmerance, because the complexity of computing, the time of computing is the matter of utmost importance that all kinds of algorithms are paid close attention to always.Nineteen eighty-three, Davis, Holdredge and Simmons successfully select for use the postsearch screening method to decompose 69 decimal numbers; Lerntra in 1989 and Manasse utilize this method to have decomposed 106 decimal number, in April, 1994, Atkins for dispensed hundreds of a good distance away work stations again; Graff; Lenstra and Leyland utilize the postsearch screening method to decompose to be called 129 metric numbers of RSA-129 once more, have organized 600 experts, 1600 computer networkings to calculate 9 months, succeed; The fail safe that RSA is described thus is as cryptographic system, and n should be greater than 200.The last century end and the beginning of this century; Big number decomposes has obtained new progress again, and 1999, RSA-155 (512bits) was successfully decomposed; Spent five months (about 8000MIPS) and 224CPU hours is on a Cray C916 computer that 3.2G central authorities internal memory arranged, to accomplish; 2002, RSA-158 was also decomposed by successful factor.Although the trial work of above-mentioned factoring has all obtained success, complexity that it cracks and difficult degree are also seen some from this.Yet, also have no a kind of algorithm can solve big several resolution problem fully at present, above-mentioned several kinds of algorithms mostly are the specific formations of utilizing big number; To the trial property decomposition that big number carries out, all can not be successful under a lot of situation, therefore; As far as encryption design person; As long as can avoid the assumed conditions of these several kinds of algorithms, theoretically, his password of design is exactly comparatively safe.To this; The applicant has proposed patent name in 2011: " a kind of prime number family that is applicable to that information encryption is used generates method fast "; Number of patent application is: 201110253413.7 patent of invention, purpose are exactly to avoid the assumed conditions of several kinds of algorithms of prior art, utilize prime number family to generate method fast; Design comparatively safe password, the present invention is exactly the improvement patent of on this patented technology, carrying out.

Summary of the invention:

The object of the invention is exactly deficiency and the defective to prior art, provides a kind of RSA of being applicable to public key cryptography to decode and fail safe is reappraised to RSA PKI system, big several quick decomposition methods of using based on cryptological technique.

In order to achieve the above object, the present invention adopts following technical scheme; Adopt congruence theory and the combination of closing several regularities of distribution, generate a kind of new, creationary big several quick decomposition methods, its concrete steps are following, a kind of big several quick decomposition methods of using based on cryptological technique:

Step 1 is compressed positive integer, sets up the residue system that contracts of mould M=30.Choosing M=30 is mould, asks its residue class to positive integer, and makes its residue system that contracts, by the Euler function

Get

Thereby can form eight arithmetic progression,

Step 2 is set up almost prime family.With coprime eight types of mould M=30 in respectively take out one and represent number a ₁, ..., a ₈, they are followed successively by

1、7、11、13、17、19、23、29

So the residue system table that contracts of all available mould M=30 of the prime number p more than 7 goes out, promptly

$P=\left\{\begin{array}{c}{a}_1\\ \·\\ \·\\ \·\\ a_8\end{array}\right.\left(\mathrm{mod}30\right)---\left(1.2\right)$

Whole numerical value that the present invention representes following formula are defined as almost prime family, and remember and make Kp, so have

Kp＝a+30(n-1)

Wherein n >=1, a＜30 and (a, 30)=1

Step 3, according to the permission and the actual needs in Computer Storage space, can select 30n is the maximum occurrences scope;

Step 4 generates the almost prime Kp of family ₁, Kp ₂..., Kp _s, wherein, Kp _s≤30n-1;

Step 5 is with intending the big several M substitution formula that decompose: Kp=a+30 (n-1)

That is: M=a+30 (n-1)

Thereby can find out this numerical value M allocation really in the almost prime table, i.e. the position of place row (n) and capable (a);

Step 6 is set this numerical digit, and promptly when computer was carried out delete program and deleted this numerical digit, program can stop and sending prompting automatically;

Step 7 adopts elimination method, deletes the number that closes in the almost prime family, according to the Kp of almost prime family ₁, Kp ₂..., Kp _mNumerical values recited (wherein

), operate as follows successively: according to containing Kp _iThe factor close several characteristic distributions, will contain Kp _iThe number that closes of the factor is all deleted, and until the scope 30n that will select, and needs any complex calculation;

Step 8, when deletion action ran to deletion numerical value M position, Automatic Program was out of service and send prompting;

Step 9, what just carrying out before program stops is that certain prime factor of deletion closes several operations, this prime factor is the factor of numerical value M;

Step 10 is also handled its another factor as stated above, till it also is prime factor until affirmation.

The quick decomposition method of big number of the present invention; Be to adopt congruence theory and understanding in depth based on involutory several regularity of distribution property; A kind of new big several quick decomposition methods of creating, maximum characteristics of this method are, in n, generate virtual dangerous plain table family table in advance, through to closing the grasp of several regularities of distribution in the virtual dangerous plain table family table; The reverse generation factor of finding out big number; Must not rely on the leak that computational speed and given big number itself have fully, just can carry out the Turbo Factor decomposition by big number, thereby the Turbo Factor of having realized no complex calculation on computers decompose of choosing arbitrarily in the n.Be applicable in big several quick decomposition method described; Eight arithmetic progression that the residue system that contracts of mould M=30 is generated are arranged by following mode and are generated almost prime family table (in the positive integer scope, screen out 2,3,5 and their multiple, but the prime number more than 7 being all in it) as follows:

Almost prime family table

According to containing Kp _iThe factor (except 1) close the characteristic distributions of number in almost prime family, can demonstrate,prove below the generality conclusion:

1. with certain K _pIn numerical table, repeat to occur lucky and this K of its Cycle Length (occupying columns) by the cycle variation for closing of factor is several _pNumerical value equate;

2. in each cycle, must there be eight and only have eight with this K _pFor closing of factor counted (at cycle beginning, K _pSelf having occupied one closes numerical digit and puts);

3. in each cycle, these eight are closed several each row that are evenly distributed in, and promptly in the one-period, each row can only occur one with this K _pThe number that closes for factor;

4. in delegation, with this K _pDeciding, to take advantage of the number that closes of factor, the numerical value of its another factor then be to increase and increase progressively 30 along with the cycle.

To intend decomposing big number M arbitrarily, just can in virtual almost prime family table, find out its pairing row and row immediately, use above-mentioned rule, can reversely find out the prime factor that forms this big number, and need any complex calculation.

Adopt the big several quick decomposition methods of the present invention; Be different from the big number of tradition decomposition method; Promptly must rely on computational speed and utilize the specific formation of big number just can decompose a certain big number; But must not rely on the leak that computational speed and given big number itself have fully, just can be to of choosing arbitrarily in the n big number carry out Turbo Factor and decompose; Be different from the big decomposition method of counting of tradition and must carry out a large amount of consuming time huge many computings, but appliance computer software differentiates that through simple processing can realize, needs any complex calculation, therefore can realize quick decomposition; Thereby can greatly promote the assessment and the improvement of the fail safe of associated public key system, be widely used in information security and cryptological technique application.

Description of drawings

Fig. 1 decomposes overview flow chart fast for big number of the present invention;

Fig. 2 is an operation principle block diagram of the present invention

Embodiment

Below in conjunction with accompanying drawing 1 and Fig. 2 and specific embodiment the present invention is done further detailed explanation; According to the flow process of accompanying drawing 1 and the fundamental diagram of accompanying drawing 2 a certain big number is decomposed applying examples fast:

(1) confirms that intending the big number that decomposes is M=10 ¹⁰+ 1;

(2) substitution formula: 10 ¹⁰+ 1=30 (n-1)+a, wherein, what n represented is the columns of these big several M in almost prime family table, what a represented is the first number that these big several M are expert in almost prime family table;

(3) calculate (10 ¹⁰+ 1)/and 30=333333333+11/30, thus confirm that the position of this numerical value M in the possibility table of primes is: n=333333334 row, a=a ₃=11, promptly this numerical value is on the row of " 11 " at the first number;

(4) start to calculate delete program, successively deletion contain 7,11,13 ... Prime factor close number;

(5) when containing in deletion in the several process of closing of 101 factors, the computer delete program stops, and has sent prompting, thereby confirms that 101 is 10 ¹⁰A prime factor of+1, and M is arranged ₁=(10 ¹⁰+ 1)/101=99009901;

(6) calculate 9900990 ¹/ ₃₀=330033+1/30, thus confirm this numerical value M ₁Position in almost prime family table is: n=330034 row, a=a ₁=1,, promptly this numerical value is on the row of " 1 " at the first number;

(7) continue delete program, restart the calculating delete program to 99009901, successively deletion contain 7,11,13 ... Prime factor close number;

(8) close in several processes when contain 3541 prime factors in deletion, calculate delete program and stop, and send prompting, thereby confirm that 3541 also is 10 ¹⁰A prime factor of+1, and M is arranged ₂=9900990 ¹/ ₃₅₄₁=27961;

(9) continue delete program; To 27961; Successively deletion contain 7,11,13 ..., until the number that closes of

prime factor; Computational process stops, thereby confirms that 27961 is prime numbers;

(10) calculate output final result, that is: M=10 ¹⁰+ 1=101 * 3541 * 27961;

At present, the basic principle of the rsa cryptosystem system that widely uses needs to accomplish the following step:

(1) selects or selects two big prime number p and q inequality;

(2) calculate N=pq;

(3), be not more than N and the integer number relatively prime with N according to Euler's function

(4) select an integer e and (p-1) (q-1) relatively prime, and e less than (p-1) (q-1);

(5) calculate d:d * e ≡ 1 (mod (p-1) (q-1)) with following formula;

(6) record of p and q is destroyed;

(N e) is PKI, and (N d) is private key, and (N is secret d), and (N e) passes to user B to user A, and (N d) treasures with his private key with his PKI.

Encrypting messages: suppose that user B wants to send a message m to user A, he knows N and e that user A produces.He converts m into an integer M less than N with the form that user A appoints in advance at use, can each word be converted into the Unicode sign indicating number of this word such as him, these numerals is connected together form a numeral then.If his information is very long, he can be divided into several sections with this information, converts each section into M then.He can be encrypted as C with M with following this formula:

M ^e≡C(mod?N)

Calculate c and uncomplicated.User B just can pass to user A with it after calculating C.

Decrypt: user A obtains just can utilizing his key d to decode behind the message C of user B.He can convert C into M with following this formula:

C ^d≡M(mod?N)

After obtaining M, he can restore original information m again.

Utilize the inventive method can be fast big several N of rsa cryptosystem system to be decomposed, thereby find out key d, and its ciphertext is decoded.

Be the simple application example that the rsa cryptosystem system is decoded below:

(1) because (N e) is PKI, and PKI is disclosed, if be (143,7);

(2) N is decomposed, thereby can obtain two prime number: p=11, q=13;

(3) calculate secret Euler's function: φ (n)=(p-1) * (q-1)=120;

(4), thereby can confirm d owing to e=7.Make (d * e) mod 120=1, and d＜120, right value is d=103;

Because 103 * 7=721=6 * 120+1=1 (mod120)

Thereby the private key of confirming this group number is (143,103);

If that utilize that the rsa cryptosystem system sends expressly is x=85, through PKI (n e)=(143,7) calculates secret value:

y＝x ^e(mod?N)＝85 ⁷mod143＝123；

After receiving ciphertext y=123, utilize (n, d)=(143,103) calculate expressly:

x＝y ^d(mod?N)＝123 ¹⁰³mod143＝85。

Claims (2)

1. big several quick decomposition methods of using based on cryptological technique: its decomposition step is following:

Step 1 is compressed positive integer, sets up the residue system that contracts of mould M=30.Choosing M=30 is mould, asks its residue class to positive integer, and makes its residue system that contracts, by the Euler function:

Obtain:

thus eight arithmetic progression can be formed;

Step 2 is set up prime number family, with coprime eight types of mould M=30 in respectively take out one and represent number a ₁, ..., a ₈, they are followed successively by:

1、7、11、13、17、19、23、29；

So the residue system table that contracts of all available mould M=30 of the prime number p more than 7 goes out, that is:

$P=\left\{\begin{array}{c}{a}_1\\ \·\\ \·\\ \·\\ a_8\end{array}\right.\left(\mathrm{mod}30\right);$

Whole numerical value that the present invention representes following formula are defined as almost prime family, and remember and make Kp, so have:

Kp＝a+30(n-1)；

Wherein: n >=1, a＜30 and (a, 30)=1;

Step 3, according to the permission and the actual needs in Computer Storage space, can select 30n is the maximum occurrences scope;

Step 4 generates the almost prime Kp of family ₁, Kp ₂..., Kp _s, wherein, Kp _s≤30n-1;

Step 5, with big several M substitution formula of intending decomposition: Kp=a+30 (n-1),

That is: M=a+30 (n-1);

Thereby can find out this numerical value M allocation really in the almost prime table, i.e. the position of place row (n) and capable (a);

Step 6 is set this numerical digit, and promptly when computer was carried out delete program and deleted this numerical digit, program can stop and sending prompting automatically;

Step 7 adopts elimination method, deletes the number that closes in the almost prime family; According to the Kp of almost prime family ₁, Kp ₂..., Kp _mNumerical values recited (wherein

), operate as follows successively: according to containing Kp _iThe factor close several characteristic distributions, will contain Kp _iThe number that closes of the factor is all deleted, until the scope 30n that will select;

Step 8, when deletion action ran to deletion numerical value M position, Automatic Program was out of service and send prompting;

Step 9, what just carrying out before program stops is that certain prime factor of deletion closes several operations, this prime factor is the factor of numerical value M;

Step 10 is also handled its another factor as stated above, till it also is prime factor until affirmation.

2. big several quick decomposition methods of using based on cryptological technique according to claim 1 are applied to information security and cryptological technique application.

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