CN102624523A - 1D-ICMIC-based encryption and decryption method by utilizing synchronous chaotic stream cipher - Google Patents

Abstract

The invention relates to a 1D-ICMIC (One-dimensional-Iterative Chaotic Map With Infinite Collapses)-based encryption and decryption method by utilizing a synchronous chaotic stream cipher. The encrypting method comprises the following steps: 1, generating a key stream sequence Kn plus 1 by an encrypting end through utilizing the synchronous chaotic stream cipher on the basis of 1D-ICMIC; and 2, after encrypting a plaintext sequence Mn plus 1 by utilizing the key stream sequence Kn plus 1, obtaining a mod value and generating a ciphertext sequence Cn plus 1, wherein cn plus 1 equals to mod (mn plus 1 plus kn plus 1, 1). The invention provides the encryption and decryption method by utilizing the synchronous chaotic stream cipher on the basis of 1D-ICMIC, which uses composite mapping formed by one-dimensional piecewise linear mapping (1D-PLM) and 1D-ICMIC as an encrypting function and has the advantages of simpleness, high efficiency and high safety.

Description

Encipher-decipher method based on the synchronous chaos stream cipher of 1D-ICMIC

Technical field

The invention belongs to field of information security technology, relate to a kind of synchronous chaos stream cipher encrypting and decrypting method based on 1D-ICMIC.

Background technology

Along with the develop rapidly of the communication technology and internet, information security issue becomes the focus of social concerns day by day, and this has promoted the research of novel private communication technology greatly.The discovery of chaos phenomenon and the good cryptography character that is had thereof are for cryptography and communication security provide new thinking, the visual field and method.In chaos phenomenon, as long as initial condition is slightly different, its result is just far from each other, is difficult to prediction, but in some cases, but reflect that the Mathematical Modeling of this type phenomenon is very simple, even the one-dimensional nonlinear iteration function just can demonstrate this chaotic characteristic.

Chaos stream cipher and some other stream cipher arithmetics (Zhou hong based on one dimension piecewise linear maps (1D-PLM); Ling Xieting.Generating chaotic secure sequences with desired statistical properties and high security [J]; Int J Bifurcation and Chaos; 1997,7 (1): 205-213.Zhou Hong, Luo Jie, Ling Xieting, the Design Theory of chaos nonlinear feedback keying sequence and limited precision realize [J], electronic letters, vol, 1997,25 (10): 58-60.Sang Tao, Wang Ruli, Yan Yixun, the Design Theory [J] of one type of novel chaos feedback cipher sequence, electronic letters, vol, 1999,27 (7): 47-50.) chaotic maps be the limited folding mapping in the finite interval.In order to improve the fail safe of cryptographic algorithm, need to increase the folding times in the finite interval.For the simple effective method of limited folding mapping is the compound number of times that increases mapping, but this is a cost to reduce ageing.Have unlimited folding chaotic maps (Iterative Chaotic Maps with Infinite Collapses, ICMIC) (Qiu Yuehong, He Chen in a kind of simply constructed finite interval; All great writings, the unlimited folding chaotic maps [J] in a kind of finite interval, high-tech communication; 2002,12 (9): 12-15, Qiu Yuehong; He Chen, all great writings, a kind of unlimited folding chaotic maps machine and quantized sequences [J] thereof; Shanghai Communications University's journal, 2002,36 (12): 1788-1790) have than limited folding mappings such as one dimension piecewise linear maps complex dynamic characteristic more; Can obtain enough complicated phase space structure with less compound number of times, thereby be the desirable member of structure cryptographic algorithm.Have complex dynamic characteristic and ideal uniform distribution character though one dimensional infinite folds chaotic maps (1D-ICMIC), have only a parameter, may be not enough when reality is used.And one dimension piecewise linear maps (1D-PLM) has good parameter autgmentability and statistical property.In conjunction with the advantage of two kinds of mappings, through shining upon compound mode, the infinitely folding easily chaotic maps of the parameter space that just can construct expansion.

Summary of the invention

In order to solve the above-mentioned technical problem that exists in the background technology; The Compound Mappings that the invention provides the folding mapping of a kind of one dimension piecewise linear maps (1D-PLM) and one dimensional infinite (1D-ICMIC) formation is an encryption function, has the synchronous chaos stream cipher encrypting and decrypting method based on 1D-ICMIC simple, efficient, high safety.

Technical solution of the present invention is: the invention provides a kind of synchronous chaos stream cipher encrypting method based on 1D-ICMIC, its special character is: said synchronous chaos stream cipher encrypting method based on 1D-ICMIC may further comprise the steps:

1) generates key stream sequence K by encrypting end _N+1

2) utilize key stream sequence K _N+1To plaintext sequence M _N+1Get the mod value after encrypting, produce the ciphertext sequence C _N+1, wherein:

c _n+1＝mod(m _n+1+k _n+1，1)。

Above-mentioned steps 1) the synchronous chaos stream cipher that is based on 1D-ICMIC generates key stream sequence K _N+1

A kind of decryption method of the synchronous chaos stream cipher based on 1D-ICMIC, its special character is: the decryption method of said synchronous chaos stream cipher based on 1D-ICMIC may further comprise the steps:

1) by decrypting end generating solution decryption key sequence K _N+1

2) to the ciphertext sequence C _N+1Utilize decruption key sequence K _N+1Decipher, recover to obtain expressly sequence M _N+1

Above-mentioned steps 1) be to utilize the motor synchronizing chaos stream cipher of 1D-ICMIC to generate key stream sequence K _N+1

Advantage of the present invention is: the synchronous chaos stream cipher encrypting and decrypting method based on 1D-ICMIC provided by the present invention combines the unlimited aliasing effect that is easy to autgmentability and one dimension piecewise linear maps of the key space of one dimension piecewise linear maps, folds the chaos stream cipher algorithm that shines upon a kind of novel feedback-feed forward architecture of formation with one dimension piecewise linear maps and one dimensional infinite.The unlimited aliasing character that this encipher-decipher method is had by means of the folding mapping of one dimensional infinite; Make that the key stream sequence is extremely responsive to the change of key parameter, on phase space, approach even distribution, the sequence adjacent element approaches statistics independently; And between drive sequences and key stream sequence dependence unusual complicated and both to approach statistics independent, have the advantage of simple, efficient, high safety etc.

Description of drawings

Fig. 1 is the principle schematic of encipher-decipher method provided by the present invention.

Embodiment

The invention provides a kind of synchronous chaos stream cipher encrypting method based on 1D-ICMIC, this encryption method may further comprise the steps:

1) generates key stream sequence K by encrypting the synchronous chaos stream cipher of end group in 1D-ICMIC _N+1

2) utilize key stream sequence K _N+1To plaintext sequence M _N+1Get the mod value after encrypting, produce the ciphertext sequence C _N+1, wherein:

c _n+1＝mod(m _n+1+k _n+1，1)。

A kind of decryption method of the synchronous chaos stream cipher based on 1D-ICMIC, this decryption method may further comprise the steps:

1) utilizes the motor synchronizing chaos stream cipher generating solution decryption key sequence K of 1D-ICMIC by decrypting end _N+1

2) to the ciphertext sequence C _N+1Utilize decruption key sequence K _N+1Decipher, recover to obtain expressly sequence M _N+1

The one dimension piecewise linear maps is:

$x_{n+1}=\mathrm{\Φ}(x_n,\mathrm{\Θ})=\left\{\begin{array}{cc}(x_n-{\mathrm{\θ}}_i)/({\mathrm{\θ}}_{i+1}-{\mathrm{\θ}}_i)& x_n\∈[{\mathrm{\θ}}_i,{\mathrm{\θ}}_{i+1})\\ 0& x_n=0.5\\ \mathrm{\Φ}(1-x_n,\mathrm{\Θ})& x_n\∈\left(\mathrm{0.5,1}\right)\end{array}\right.---\left(1\right)$

Wherein, x ∈ [0,1); 0=θ ₀＜θ ₁＜Λ＜θ _i＜Λ＜θ _N+1=0.5, i=0,1 ... N, N>=1; Θ=[θ ₁, θ ₂.., θ _N]

The folding chaotic maps of one dimensional infinite is:

$x_{n+1}=\mathrm{\&Psi;}(x_n,a)=\left\{\begin{array}{cc}\mathrm{mod}(a/(1/2-x_n),1)& x_n<1/2\\ 0& x_n=1/2\\ \mathrm{mod}(a/(x_n-1/2),1)& x_n>1/2\end{array}\right.---\left(2\right)$

In the formula, x _n∈ [0,1), and a ∈ [1, ∞), n=0,1,2,3..., mod (, 1) and be mould 1 function.

The composite chaotic that is made up of mapping (1) and mapping (2) is mapped as:

x _n+1＝Λ(x _n，Θ，a)＝Ψ(Φ(x _n，Θ)，a) (3)

As the drive sequences generator, the chaos sequence of generation is as the drive sequences of feedforward function with Compound Mappings (3):

x _n+1＝Λ(x _n，Θ，a) (4)

With the mapping (2) repeatedly Compound Mappings be feedforward function, the key stream sequence

that obtains encrypting usefulness is:

k _n+1＝Ψ ^q(x _n+1，a) (5)

Encryption function is:

c _n＝mod(m _n+k _n，1) (6)

Corresponding decryption function is:

m _n＝mod(c _n-k _n，1) (7)

Ψ ^q(x, a)=Ψ (Ψ (..., a), a) be Ψ (, a) to q the Compound Mappings of variable x.That

in formula (6), (7) and

are respectively is interval [0,1) go up expressly sequence, key stream sequence and ciphertext sequence.The parameter Θ of mapping (3) is a seed key, and parameter a and q are the non-key parameter of adjustment aliasing effect.As the reason of key be not: because the ambiguity of formula (4) with the initial value of formula (4); Make under the identical situation of parameter Θ; Some initial values inequality obtain identical drive sequences

thereby obtain identical key stream sequence

like this; These initial values are of equal value; The number N of parameter Θ is big more; The initial value of equivalence is many more; And the concrete numerical value of these equivalent key depends on parameter Θ; Like this, just there is stronger correlation between key parameter, thereby effective key space is reduced greatly.

Key stream sequence has following character: when a levels off to when just infinite, key stream sequence

levels off to even distribution and have the two-value auto-correlation function.

The above-mentioned character of sequence just becomes the necessary condition of safe key stream sequence.And adequate condition should be:

(1) key stream sequence

is independent with drive sequences

statistics; Promptly can't obtain the information of drive sequences

, also just can't and then confirm parameter Θ as key from key stream sequence

.

(2) key stream sequence

is the independent same distribution sequence, other element or the reconstruct whole sequence that promptly can't come forecasting sequence through a fragment of sequence .

(3) key stream sequence is extremely responsive to the change of key parameter Θ.

If the lyapunov index of mapping Ψ () is λ, then as a enough greatly the time, Compound Mappings Ψ ^q(x, lyapunov index a) are q λ, and the promptly compound phase space complexity that increased has improved the aliasing effect.A or q are big more, Compound Mappings Ψ ^q(x, phase space complexity a) is high more, mapping output is sensitiveer to the variation of mapping input, statistics dependence between the two more a little less than.As q or a when being infinitely great, mapping Ψ ^q(x, output a), input statistics are independently.But the size of a receives the restriction of actual realization precision, and the efficient of the bigger then algorithm of q is low more.Therefore, under given realization precision, need to confirm the upper limit a of a _MaxLower limit q with q _Min, under the prerequisite that guarantees safe enough property, make algorithm have best real-time.

In following analysis, to suppose to adopt floating-point operation, operational precision is P, key parameter Θ is P _ΘPosition precision fixed point decimal, expressly sequence

Drive sequences

With the key stream sequence

Be P _kThe fractional fixed point of position precision.

Because the translocation of modular function; Precision is that last

at least ( expression is not more than the maximum integer of x) behind the decimal point of the fixed-point number y in the interval (0.1) that obtains after through mapping (2) of floating number x in the interval (0.1) of P is zero; Thereby the number of significant digit of y to become

a big more, the number of significant digit behind the decimal point of y is few more.

Fixed-point number y in the interval (0.1) that number of significant digit is

behind the decimal point is during as the input of mapping (2); Owing to ask the stretching action of computing reciprocal in the mapping (2); Make intermediate object program x ∈ (0; 0.5); Or

x ∈ (0.5; 1) is the floating number of P position precision; The interior fixed-point number in interval (0.1) of t obtains

under function m od (2at, 1) effect position precision.

Therefore, Compound Mappings Ψ ^q(x, a) precision of the number of gained still does

When reality realizes, can be according to realizing precision P and key stream sequence

Required precision P _kConfirm the upper limit a of parameter a _Max:

$a_{\max }=2^{P-P_k-1}---\left(8\right)$

After a upper limit of having confirmed, need to confirm that compound number of times q is to obtain satisfied aliasing effect.Though compound number of times q is big more, the aliasing effect is good more, and correspondingly operand is also big more, cryptographic algorithm ageing also poor more.Under limited precision realizes, because there is minimum compound number of times q in the quantization effect of limited realization precision _Min, make compound number of times q>=q _MinThe limited precision key stream sequence and the compound number of times of gained are q _MinThe time gained limited precision key stream sequence on statistics, be undistinguishable, therefore, only need compound q _MinThe inferior aliasing effect that can obtain satisfaction.Discuss below and how to confirm q _Min

Compound Mappings y=Ψ ^q(x, output y a) to the sensitivity of input x is:

${\mathrm{\&xi;}}_{\mathrm{\&Psi;}}=\left|\frac{\mathrm{dy}}{\mathrm{dx}}\right|=\left|\frac{d{\mathrm{\&Psi;}}^q(x,a)}{\mathrm{dx}}\right|\&GreaterEqual;{\left(4a\right)}^q---\left(9\right)$

When input x and output y are P _kPosition precision fixed point number, x changes

Promptly

If the change of caused output y:

|Δy|＝ξ _Ψ·|Δx|≥(4a) ^q|Δx|≥1 (10)

Then two inputs of deviation for are positioned in the different non-linear segmentation of formula (5).Like this, the information about x among the output y is zero, and promptly x and y statistics is independent.

Therefore; When inequality (10) was set up, drive sequences

and corresponding key stream sequence

statistics were independent.In addition, when inequality (10) is set up, if drive sequences

Adjacent element deviation more than or equal to

The corresponding key stream sequence of gained then

Adjacent element statistics independent, and chaotic maps x _N+1=Λ (x _n, Θ, the drive sequences that a) is produced

The deviation of adjacent element

Therefore, key stream sequence

Adjacent element statistics independent.Consider chaotic maps x _N+1=Λ (x _n, Θ a) is Markov mapping, then key stream sequence Be the even distribution series of statistical independent.

Therefore, can obtain the key stream sequence that obtains to have desirable cryptography characteristic by inequality (10)

Required minimum compound number of times q _Min:

In the formula,

expression is not more than the maximum integer of x.

At last, according to the key stream sequence

The extremely responsive requirement of the change of key parameter Θ is confirmed the precision P of key parameter Θ _ΘFormulation to this requirement is exactly:

Arbitrary element θ as Θ _i(i=1,2 ... N) change Become

The time, correspondingly, satisfy following formula if cause the change of formula (4) output:

$|x_{n+1}^{\&prime;}-x_{n+1}|=|\mathrm{\&Lambda;}(x_n,\mathrm{\&Theta;},a)-\mathrm{\&Lambda;}(x_n,{\mathrm{\&Theta;}}^{\&prime;},a)|$ (12)

$\&GreaterEqual;\min \left(\right|\mathrm{\&Lambda;}(x_n,\mathrm{\&Theta;},a)-\mathrm{\&Lambda;}(x_n,{\mathrm{\&Theta;}}^{\&prime;},a)\left|\right)\&GreaterEqual;2^{-P_k}$

Then, can know x by formula (10) _N+1And x ' _N+1The element k of pairing key stream sequence _N+1And k ' _N+1Uncorrelated.

(x, output y Θ) is to parameter θ to get y=Φ by formula (1) _iThe sensitivity that changes:

${\mathrm{\&xi;}}_{\mathrm{\&Phi;}}=\left|\frac{\mathrm{d\&Phi;}(x,\mathrm{\&Theta;})}{d{\mathrm{\&theta;}}_i}\right|=\frac{x-{\mathrm{\&theta;}}_{i-1}}{{({\mathrm{\&theta;}}_i-{\mathrm{\&theta;}}_{i-1})}^2}$ i＝1，2，...N

Then as x ≠ θ _I-1The time,

${\mathrm{\&xi;}}_{\mathrm{\&Phi;}}=\frac{x-{\mathrm{\&theta;}}_{i-1}}{{({\mathrm{\&theta;}}_i-{\mathrm{\&theta;}}_{i-1})}^2}\&GreaterEqual;2^{-P_k+2}---\left(13\right)$

Convolution (9) and formula (13) can get:

$|\mathrm{\&Lambda;}(x_n,\mathrm{\&Theta;},a)-\mathrm{\&Lambda;}(x_n,{\mathrm{\&Theta;}}^{\&prime;},a)|$

$=|{\mathrm{\&xi;}}_{\mathrm{\&Psi;}}\&CenterDot;{\mathrm{\&xi;}}_{\mathrm{\&Phi;}}\&CenterDot;\mathrm{\&Delta;}{\mathrm{\&theta;}}_i|$ (14)

$\&GreaterEqual;4a\&CenterDot;2^{-P_k+2}\&CenterDot;2^{-P_{\mathrm{\&Theta;}}}$

$=a\&CenterDot;2^{-P_{\mathrm{\&Theta;}}-P_k+4}$

Convolution (12), (14) can get:

${a2}^{-P_k-P_{\mathrm{\&Theta;}}+4}\&GreaterEqual;2^{-P_k}$

Promptly have:

P _Θ≤log ₂a+4 (15)

When the given maximum of a modus ponens (8), can get by formula (15):

P _Θ≤P-P _k+3 (16)

So, at P and P _kAfter confirming, maximum key entropy H (K) ≈ N (P-P of this cryptographic algorithm _k+ 3).The main computing of this cryptographic algorithm is a floating-point multiplication, needs q altogether _Min+ 2 floating-point multiplications.If adopt the double-precision floating point computing, then the number of significant digit P=52 of mantissa.If obtain P _kThe key stream sequence of=32 precision

Can get corresponding a by formula (8) _Max=2 ¹⁹, can obtain minimum compound number of times q by formula (11) _Min=2.Can obtain the maximal accuracy P of key parameter Θ by formula (16) _{Θ max}=23, corresponding key entropy H (K) ≈ 23N, N is the element number of parameter Θ.

In order to understand scheme of the present invention better, do further to describe below in conjunction with accompanying drawing and specific embodiment.

The present invention is based on the Compound Mappings that the folding mapping of one dimension piecewise linear maps and one dimensional infinite constitutes, schematic diagram is as shown in Figure 1.Wherein, the key generator of encryption and decryption end constitutes by formula (4), (5), and the key and the initial condition of two key generators are identical, thereby the key sequence that is produced is synchronous fully.Encrypt end and encrypt with formula (6), obtain the ciphertext sequence, send it to decrypting end with the key sequence of key generator.Decrypting end is used with the synchronous key sequence of transmitting terminal the ciphertext sequence is deciphered with formula (7), obtains expressly sequence.And the third party who does not know key can't decrypting ciphertext because of producing and encrypt the synchronous key sequence of end.

Expressly sequence is the SIN function sequence on interval (0,1)

The key sequence that is produced with formula (4), (5)

It is encrypted obtain the ciphertext sequence

Under the prerequisite that knows for sure as the parameter of key, decrypting end can accurately be duplicated key sequence

And then the plaintext sequence after obtaining deciphering

And if only know approximation as the parameter of key, then can not accurately duplicate key sequence

Also just can not be from the ciphertext sequence

Accurately recover expressly sequence

Initial value x ₀And during Θ '=[0.2645757], with the sequence of gained

To the ciphertext sequence

The sequence of deciphering gained

Claims (2)

1. encryption method based on the synchronous chaos stream cipher of 1D-ICMIC, it is characterized in that: said synchronous chaos stream cipher encrypting method based on 1D-ICMIC may further comprise the steps:

1) generates key stream sequence K by encrypting the synchronous chaos stream cipher of end group in 1D-ICMIC _N+1

2) utilize key stream sequence K _N+1To plaintext sequence M _N+1Get the mod value after encrypting, produce the ciphertext sequence C _N+1, wherein:

c _n+1＝mod(m _n+1+k _n+1，1)。

2. decryption method based on the synchronous chaos stream cipher of 1D-ICMIC is characterized in that: the decryption method of said synchronous chaos stream cipher based on 1D-ICMIC may further comprise the steps:

1) utilizes the motor synchronizing chaos stream cipher generating solution decryption key sequence K of 1D-ICMIC by decrypting end _N+1

2) to the ciphertext sequence C _N+1Utilize decruption key sequence K _N+1Decipher, recover to obtain expressly sequence M _N+1

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