CN100512104C - A public key encrypting method for information encryption - Google Patents

Abstract

The invention discloses a public key encrypting method, using Chebyshev polynomial as a basis, redefining the Chebyshev polynomial in the finite domain, and thus forming a Chebyshev polynomial in the finite domain. It uses the semigroup property of the Chebyshev polynomial in the finite domain to construct a public key encrypting method for message encryption, avoiding the defect of the original Chebyshev polynomial at the interval [-1, 1]. Its remarkable effects: the generation of secret key pair is easier and only need select large integer; the operation has the compactness similar with that of encrypting the RSA and ElGamal public keys; it can adopt the fast calculating method used by the encryption of the RSA and ElGamal public keys; and it has the more complexity in attacking than the encryption of the RSA and ElGamal public keys.

Description

A kind of key encrypt method that is used for information encryption

Technical field

The present invention relates to information encryption in network and the information security field, particularly a kind of key encrypt method that is used for information encryption.

Background technology

Universalness along with network application, current society has stepped into informationalized epoch, the processing of increasing routine work and affairs need depend on the transmission of the network information, such as ecommerce, E-Government, Web bank, online working etc., be very easy to people's work and life, but also brought serious network security and information security issue simultaneously.At this problem, general solution is to adopt PKIX PKI technology to come the fail safe of protecting network information at present.

In the PKI technology, a core technology the most basic is exactly a key encrypt method.After Diffie in 1976 and Hellman proposed to have the Diffie-Hellman algorithm of intention, the various key encrypt methods that up to the present people constructed still were confined to the difficult problem that big number decomposes a difficult problem and asks discrete logarithm.Typical key encrypt method has: based on the RSA key encrypt method of a big number decomposition difficult problem with based on the ElGamal key encrypt method of asking a discrete logarithm difficult problem.Raising along with the computer hardware processing speed, the development of network technology and parallel computing, and various decomposition factors and the appearance of asking the new method of discrete logarithm, as the number field sieve method, make and be faced with very big threat based on the big various key encrypt methods of counting decomposition and discrete logarithm, always in the length that constantly increases its key, to increase the difficulty that key is decomposed.Like this, can cause the heavy load of encryption and decryption calculating on the one hand, strengthen the memory space of key on the other hand.The latest development of announcing from present RSA website has realized the decomposition to 576 big numbers as can be known, and therefore 1024 and 2048 s' RSA key is only safe in following a period of time.For new key encrypt method,, therefore can utilize short key to reach the degree of safety of 1024 and 2048 s' RSA key, as the ECC key encrypt method owing to people lack effective attack means to the deficiency of its research and understanding.

2003, the paper " Public-Key Encryption Based on Chebyshev Maps " that L.Kocarev and Z.Tasev deliver in international conference " The 2003 IEEE International Symposiumon Circuits and Systems ", introduced and utilized the Chebyshev multinomial interval [1,1] chaotic characteristic on and semigroup characteristic have been constructed a kind of key encrypt method.But because the Chebyshev multinomial on interval [1,1] can be represented as the form of trigonometric function, therefore utilize periodicity and its invertibity on interval [1,1] of trigonometric function, this key encrypt method is cracked very soon.

Summary of the invention

The present invention expands the key encrypt method of L.Kocarev and Z.Tasev and improves, and has avoided the mode that cracks of trigonometric function, makes it have more fail safe.

In the present invention, key encrypt method has still utilized the polynomial semigroup characteristic of Chebyshev, but to expanding between its map section, make its operation interval by original [1,1] become finite field, be formed with the Chebyshev multinomial in the confinement, all values are integer.The Chebyshev multinomial just can not be represented with trigonometric function more like this, has avoided the weakness of former Chebyshev multinomial on interval [1,1], has guaranteed the fail safe of key encrypt method.

Technical scheme of the present invention is as described below:

Chebyshev is polynomial on the finite field is defined as follows:

If F _PBe finite field, P is a prime number, Z _nBe integer item, n is an integer.Then at finite field F _POn the Chebyshev multinomial just like giving a definition:

Make n ∈ Z _n, variable x ∈ F _P, multinomial T then _n(x): F _P→ F _PRecurrence relation be defined as:

T _n(x)≡(2xT _n-1(x)-T _n-2(x))(modP) n≥2

And initial value T is arranged ₀(x) ≡ 1 (mod P), T ₁(x) ≡ x (mod P).

Can get finite field F by top definition _POn the Chebyshev multinomial as follows:

T ₀(x)≡1(modP) T ₁(x)≡x(modP)

T ₂(x)≡(2x ²-1)(modP) T ₃(x)≡(4x ³-3x)(modP)

T ₄(x)≡(8x ⁴-8x ²+1)(modP) …

Here claim these multinomials to be successively: the 0th Chebyshev multinomial, the 1st Chebyshev multinomial, the 2nd Chebyshev multinomial ...

The Chebyshev multinomial has following semigroup characteristic on the top finite field:

$T_r\left(T_s\left(x\right)\left(\mathrm{mod}P\right)\right)\left(\mathrm{mod}P\right)=T_{\mathrm{rs}}\left(x\right)\left(\mathrm{mod}P\right)=T_s\left(T_r\left(x\right)\left(\mathrm{mod}P\right)\right)\left(\mathrm{mod}P\right)$

$\⇒T_r\left(T_s\left(x\right)\left(\mathrm{mod}P\right)\right)\left(\mathrm{mod}P\right)=T_{\mathrm{rs}}\left(x\right)\left(\mathrm{mod}P\right)=T_s(T_r\left(x\right))\left(\mathrm{mod}P\right)r,s\∈Z$

The available following exponential manner of the polynomial value of Chebyshev advances calculation on the top finite field:

$\left[\begin{array}{c}{T}_{n-1}\left(x\right)\\ T_n\left(x\right)\end{array}\right]\≡{\left[\begin{array}{cc}0& 1\\ -1& 2x\end{array}\right]}^{n-1}\left[\begin{array}{c}1\\ x\end{array}\right]\left(\mathrm{mod}P\right)$

Utilize above-mentioned technology, the information encryption and decryption process of key encrypt method is as described below among the present invention:

The both sides that suppose communication are A and B, and B will be information

Send A in the mode of encrypting, then utilize the process of key encrypt method encryption and decryption information of the present invention as follows:

1) A picked at random integer $\mathrm{SK}\∈Z_n^{*}$ With $x\∈F_P^{*};$

2) A calculating K=T _SK(x) (modP);

3) A SK as private key, PK={x, K} is as PKI;

4) B is by the public key management center or directly obtained the PKI PK={x of A by A, K};

5) B obtains the PKI PK={x of A, behind the K}, and the picked at random integer $R\∈Z_n^{*};$

6) B utilizes the PKI element x calculating K 1=T of A _R(x) (mod P);

7) B utilizes the PKI element K calculating K 2=T of A _R(K) (mod P);

8) B calculates C1=MK2 (modP);

9) B allows information encrypted ciphertext C={C1, K1}, and transmit ciphertext C and give A;

10) A receives information encrypted C={C1, behind the K1}, with the private key SK calculating K 2=T of oneself _SK(KI) (mod P);

11) A calculates M=C1 (K2) ^-1(modP), reduction information encrypted M;

Above-mentioned steps is the encryption and decryption overall process that protected information utilizes key encrypt method of the present invention, wherein the 1st) go on foot the 3rd) step for key to production process; The 4th) going on foot the 9th) step is for the information encryption process; The 10th) to the 11st) step is for the decrypts information process.

The polynomial semigroup characteristic of Chebyshev has guaranteed among the present invention information encrypted by correct reduction and recovery on the finite field, promptly guarantees in the said process the 7th) K2 and the 10th that calculates in the step) K2 that calculates in going on foot is identical.

Implement technique scheme, can realize that beneficial effect of the present invention is:

The defective of having avoided L.Kocarev and Z.Tasev key encrypt method to be attacked has kept the good characteristic of Chebyshev multinomial in public key encryption simultaneously again.Compare with the ElGamal public key encryption with the RSA public key encryption, key encrypt method of the present invention has following characteristics:

1) the right generation of key is easier, need not to seek big prime number and primitive element, only needs an ordinary integer to get final product.

2) key encrypt method of the present invention has the terseness that is similar to RSA and ElGamal public key encryption.

3) key encrypt method of the present invention can employing and RSA and the similar quick calculation method of ElGamal public key encryption.

4) cracking than RSA and ElGamal public key encryption of key encrypt method of the present invention has more complexity.

This key encrypt method can substitute RSA and ElGamal public key encryption, is used for any places that need information security such as secure communication, ecommerce, E-Government, network office.

Embodiment

Below by specific embodiment information encryption and decryption process of the present invention is further described.

Embodiment 1

In the application of reality, generally get P=n, and P to choose a big prime number.For convenience of explanation, get prime P=23 and Integer n=23 here, the computational fields of system is finite field F ₂₃With integer item Z ₂₃

The key that the user of need to be keep secret communication produces separately through the following steps is right, and the PKI of cipher key pair can be given the public key management center and preserve and publicity, and then the key of user A is as follows to production process:

1) picked at random integer $\mathrm{SK}=3\∈Z_{23}^{*}$ With $x=6\∈F_{23}^{*};$

2) calculate

K＝T _SK(x)(mod?P)＝T ₃(6)(mod23)

＝(4·6 ³-3·6)(mod23)

＝18

3) allow SK=3 as private key, PK={x=6, K=18} is as PKI;

Suppose that user B will be information $M=16\∈F_{23}^{*}$ Encrypt with key encrypt method and to be transferred to A, B at first obtains the PKI PK={x=6 of A by public key management center or alternate manner, K=18}, and user B encrypts M according to the following step then.

1) picked at random integer $R=8\∈Z_{23}^{*};$

2) calculate

K1＝T _R(x)(modP)＝T ₈(6)(mod23)

＝(128·6 ⁸-256·6 ⁶+160·6 ⁴-32·6 ²+1)(mod23)

＝5

3) calculate

K2＝T _R(K)(modP)＝T ₈(18)(mod23)

＝(128·18 ⁸-256·18 ⁶+160·18 ⁴-32·18 ²+1)(mod23)

＝2

4) calculate C1=MK2 (mod P)=162 (mod 23)=9;

5) form ciphertext C={C1=9, K1=5};

B sends ciphertext to A by common signal channel, and A receives cipher-text information C={C1=9, behind the K1=5}, utilizes the private key SK=3 of oneself to decipher recovering information M through the following steps.

1) calculates

K2＝T _SK(K1)(modP)＝T ₃(5)(mod23)

＝(4·5 ³-3·5)(mod23)

＝2

2) calculate M=C1 (K2) ^-1(modP)=9 (2) ^-1(mod23)=912 (mod23)=16, restore information M;

Embodiment 2

In the application implementation process of reality, because the data of calculating all bigger (binary number to last kilobit normally up to a hundred), so the polynomial-valued calculating of each Chebyshev in the foregoing description will adopt on the finite field in the previous technique scheme the polynomial-valued computing formula of Chebyshev to carry out.Because it has exponential form, therefore can utilize quick exponentiation algorithm to carry out.

Here get the T among the embodiment 1 ₈(6) (mod23) the polynomial-valued quick index calculation method of Chebyshev is described.

$\left[\begin{array}{c}{T}_7\left(6\right)\\ T_8\left(6\right)\end{array}\right]\≡{\left[\begin{array}{cc}0& 1\\ -1& 2\·6\end{array}\right]}^{8-1}\left[\begin{array}{c}1\\ 6\end{array}\right](\mathrm{mod}23)$

$\≡{\left[\begin{array}{cc}0& 1\\ -1& 2\·6\end{array}\right]}^1{\left[\begin{array}{cc}0& 1\\ -1& 2\·6\end{array}\right]}^2\left[\begin{array}{cc}0& 1\\ -1& 2\·6\end{array}\right]\left[\begin{array}{c}1\\ 6\end{array}\right]\left(\mathrm{mod}23\right)$

$\≡\left[\begin{array}{c}16\\ 5\end{array}\right]\left(\mathrm{mod}23\right)$

Can get T by top calculating ₈(6) (mod23)=5.

Claims (1)

1. key encrypt method that is used for information encryption, communicating pair is A and B, and FP is a finite field, and P is a prime number, and Zn is an integer item, and n is an integer, and B will be information $M\∈F_P^{*}$ Sending A in the mode of encrypting, realize by following process, at first is the right generation of key, secondly is ciphering process, is decrypting process at last, it is characterized in that,

Chebyshev multinomial on the finite field FP is defined as:

T _n(x)≡(2xT _n-1(x)-T _n-2(x))(modP) n≥2

N ∈ Z is wherein arranged _n, variable x ∈ F _P, and initial value T is arranged ₀(x) ≡ 1 (mod P), T ₁(x) ≡ x (mod P);

The generative process that key is right:

1) A picked at random integer $\mathrm{SK}\∈Z_n^{*}$ With $x\∈F_P^{*};$

2) A calculating K=T _SK(x) (mod P), wherein T _SK(x) (mod P) is above-mentioned Chebyshev multinomial;

3) A SK as private key, PK={x, K} is as PKI;

Ciphering process:

1) B is by the public key management center or directly obtained the PKI PK={x of A by A, K};

2) B obtains the PKI PK={x of A, behind the K}, and the picked at random integer $R\∈Z_n^{*};$

3) B utilizes the PKI element x calculating K 1=T of A _R(x) (mod P), wherein T _R(x) (mod P) is above-mentioned Chebyshev multinomial;

4) B utilizes the PKI element K calculating K 2=T of A _R(K) (mod P), wherein T _R(x) (mod P) is above-mentioned Chebyshev multinomial;

5) B calculates C1=MK2 (mod P);

6) B allows information encrypted ciphertext C={C1, K1}, and transmit ciphertext C and give A;

Decrypting process:

1) A receives information encrypted C={C1, behind the K1}, with the private key SK calculating K 2=T of oneself _SK(K1) (mod P), wherein T _SK(K1) (mod P) is above-mentioned Chebyshev multinomial;

2) A calculates M=C1 (K2) ^-1(mod P), reduction information encrypted M.