CN100459487C - Chaotic cipher production method under limited precision - Google Patents

Abstract

The invention discloses a generating method for chaotic cipher in limited precise. The steps are: (1) initialization: generates in random or sets the initial value x1[i] (i=0; l=1, 2... n,) of the chaotic system, and sets with K kinds of variable carrying rules, k is equal or less than n; (2) the chaotic system output x j[i] (j=1, 2... n) correspondent to the carrying rule can be acquired with the K kinds of variable carrying rules according to the input xl [i]; (3) the outputs of different carrying rule x j [i] are converted into value with the same carrying rule x' j [i]; (4) the x' j[i] is coded according to the set coding rule and acquires x'' j[i]; (5) the n chaotic coding series x'' j [i] are arranged into M xi= M xi (I, j)=x'' j[i], xi=1, 2, ...; (6) xl [i+1]=x j[i] according to the set rule; (7) I=i+1, returns to the step (2), till the task is completed. The invention can reduce the correlation of series, increases the linear complexity, thus the chaotic series may has excellent performance.

Description

A Chaotic Cipher Generation Method with Finite Precision

technical field

The present invention belongs to the cipher generation technology in information security, specifically a chaotic cipher generation method under limited precision, which utilizes electronic computer technology, information coding technology and chaotic system to adopt multi-ary systems under the condition of limited precision. Computational techniques produce chaotic cipher sequences with excellent properties.

Background technique

With the continuous improvement of computer computing speed and the development of distributed processing technology, some original encryption algorithms have been cracked. At present, most of the encryption algorithms used in China are low-strength encryption algorithms that will be eliminated abroad. The resulting insecurity has become an important issue that hinders economic development and threatens national security.

In recent years, chaos has begun to be applied in the field of encrypted communication. Chaos is an external complex performance caused by internal randomness in a deterministic system, and it is a seemingly random non-random motion. Due to the characteristics of ergodicity, broadband, noise-like, sensitivity to initial conditions, fast-decaying autocorrelation and weak cross-correlation, chaotic signals provide rich mechanisms and methods for realizing secure communication.

However, the generation of digital chaotic sequences is realized on computers or other devices with limited precision. Therefore, any chaotic sequence generator can be described by a finite automaton. Under this condition, the digital chaotic sequence generated will inevitably show short period, strong correlation and small linear complexity. It is difficult to design A digital chaotic sequence that satisfies the requirements of cryptography.

In the literature "A New Idea of Using One-Dimensional PWLMap in Digital Secure Communications-Dual-Resolution Approach", Li-Hui Zhou et al. propose to use a dual-resolution method that increases the input resolution to make it much higher than the output resolution. In the discrete chaotic system, the above-mentioned problems are caused by the limitation of precision. Obviously, this method is very costly.

Contents of the invention

The object of the present invention is to overcome the above disadvantages and provide a digital chaotic cryptographic system, which effectively solves the above problem of characteristic degradation without increasing the calculation accuracy.

A kind of chaotic password generation method under the limited precision provided by the present invention, its steps are:

(1) Initialization: Randomly generate or set the initial value x _l [i] of the chaotic system, i=0; l=1, 2,..., n, n is the number of chaotic systems, and set k different progress system, and k≤n;

(2) According to the input x _l [i], adopt one of the k different base systems respectively, and obtain a chaotic system output x _l ′[i] corresponding to the base system;

(3) Convert the output x _l ′[i] of different bases into the set value x _l ″[i] of the same base system;

(4) Encode x _l ″[i] to obtain x _l ″′[i];

(5) After scrambling n chaotic coding sequences x _l ″'[i], output them as passwords;

(6) Judging whether the number of generated passwords reaches the required number, if so, end; otherwise, determine x _l [i+1]=x _l '[i] according to the set sorting rules, let i= i+1, jump back to step (2).

The present invention uses low-dimensional chaotic dynamical systems of different bases to perform iterative calculations, increases the period of the digital chaotic sequence without increasing the accuracy, reduces the correlation of the digital chaotic sequence, and increases the linear complexity of the digital chaotic sequence, thereby making the chaos Sequences have nice properties. Due to the use of different bases, the carry rules in the same chaotic mapping process are also different, and the rounding values are also different when doing rounding processing. The resulting differences make the same chaotic mapping under different bases The chaotic orbits are also different, so that the period of the digital chaotic sequence can be increased; since k chaotic sequence values are scrambled and output, the linear complexity is increased; k different bases are used, the carry rules and rounding size are all different, thus reducing the degree of cross-correlation of chaotic sequences.

Description of drawings

Fig. 1 is a flowchart of the present invention.

Detailed ways

In the technical solution adopted by the present invention, no matter it is the chaotic system of step (2), or the same base system set in step (3), and the encoding rule set in step (4), only the encryption and decryption parties are required It can be agreed in advance that the same method can be used, and it is not limited to a specific method. The implementation method given in step (5) has two meanings in essence: it can not only sort the n chaotic coding sequences, but output them after sorting the column x _j ″[i]; it can also output them directly according to the original sequence. Similar to Actually, the implementation method given in step (6) has two meanings in essence: the output x _j [i] sequences of n chaotic systems can be rearranged and then correspondingly used as the next input x _l [ i+1], at this time l and j are not necessarily equal; and the original output x _j [i] sequence can be directly fed back to the chaotic system as its next input x _l [i+1], at this time l and j are equal .

The present invention will be described in further detail below by way of enumeration.

Example one

The chaotic map in this method is a simple one-dimensional logistic iterative

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&Subset;</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1,1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

calculating $x_{i+1}=1-{2x}_i^2$ , because the input value of the next iteration is still 64 bits, but x _i ² is 128 bits, so x _i ² must be rounded. In this method, tail removal method is adopted. (Explanation of the rounding method).

However, in order to improve the encryption strength and iteration cycle, the present invention uses the "shuffling algorithm" to scramble the output values of the 8 chaotic maps after calculating the decimal numbers corresponding to 8 different bases, so that 8 chaotic sequences The values of are output in a non-fixed order.

Here, first introduce the shuffling algorithm:

Since the standard playing card originated in the United States in 1872, it has experienced more than a century of evolution and development, and has already been widely popularized around the world. It is not only popular in various folk entertainment activities (including gambling and divination), but also has appeared in the world-wide official competitive event-bridge. In order to ensure the fairness and credibility of these recreational activities and competitions, it is necessary to make the sequence of each playing card after shuffling have good "randomness". Inspired by this, the present invention proposes a "shuffling" algorithm for dynamically modifying the codebook.

1. Top card hits D _cb (q)

Taking q as a parameter, when p≥0, exchange the 1st and qth codewords in the sequence 0, 1, ... 7. The result sequence after completing the algorithm D _cb (pq) is:

q,...,q-1,0,q+1,,...,7

2. Card cutting algorithm T _cb (q)

With q as the boundary, the sequences 0, 1, ..., q-1, q, q+1, ..., 7 are crossed back and forth. The result sequence after completing the algorithm T _cb (p) is:

q+1,...,7,0,1,...,q-1,q

The rule adopted by the pseudo-random number generator is also one-dimensional logistic iteration: $m_{i+1}=1-{2m}_i^2$ The computation in the iterative process adopts binary.

In this algorithm, the shuffling algorithm adopted for this codebook transformation is determined according to the pseudo-random _mi . When m _i ≤ 0, use the top card insertion algorithm (q=[8*m _i ], -1<m _i <1), otherwise use the card cutting algorithm with q=1.

It must be pointed out that for convenience of description, the initial sequence is set as a sequence arranged in natural order, ie 0, 1, . . . , 7, without loss of generality. Obviously, the same treatment can be done for different initial sequences.

The algorithm process is as follows:

(1) Randomly generate the initial value m ₀ of the output control rule and the initial value x _l [0] of the chaotic system, (l=1, 2, ... 8), and select 8 bases 6, 7, 9, 11, 12 , 13, 14 and 15;

(2) x _j [i] = x _l [i], (j = 1, 2, ... 8);

(3) According to the chaotic mapping by x _j [i] $x_{i+1}=1-{2x}_i^2$ Use different base calculations to generate chaotic map output x _j [i+1] (j=1, 2, ... 8);

(4) Convert x _j [i+1] of different bases into decimal x' _j [i+1] (j=1, 2, ... 8);

(5) Encode x′ _j [i+1] according to [256*acos(-x _j ′[i+1])] ([] indicates rounding down) to obtain x _j ″[i+1];

(6) According to the chaotic mapping by m[i] $m_{i+1}=1-{2x}_i^2$ Use binary calculations to generate chaotic map output m[i+1], and select the corresponding shuffling algorithm according to the positive or negative of m[i+1] to scramble the order of x _j ″[i+1] and output M _ξ ≡ M _ξ (i,j)=x _j ″[i+1](j=1,2,...,n; ξ=1,2,...);

(7) Let x _l [i+1]=x _j [i+1], l=j, (l, j=1, 2,...8), i=i+1, jump back to step (2) , until the end of the task;

The cycle length of digital chaotic sequence using binary calculation is 29,685,894. See the table below for the periods of other bases:

base number cycle length base number cycle length Hexadecimal 92,219,771 Hexadecimal 161,223,070 Hexadecimal 89,744,731 Hexadecimal 71,878,812 9 base 24,744,402 Hexadecimal 284,595,078

Hexadecimal 23,354,844 Hexadecimal 544,406,163

From the above data, it can be seen that if a chaotic map calculation is performed only based on the binary in the computer or FPGA chip, its period is much smaller than that of other bases except that it is larger than the nine-base and eleven-base systems.

In this method, 8 chaotic maps are used. If these 8 chaotic maps and pseudo-random numbers are all generated using binary calculations, then the period of the final output digital chaotic sequence is: the minimum of 29685894 and Ta[i] (codebook transformation period) common multiple. The period of the final output digital chaotic sequence using this method is: 29685894, 92219771, 89744731, 24744402, 23354844, 161223070, 71878812, 284595078, 544406163 and the least common multiple of Ta[i], which is an astronomical number, far greater than Use only the period of the binary method.

Example two

The following method can also be used to increase the period and linear complexity of the digital chaotic sequence: use the method of cyclic shift to exchange the iterative values of different bases, and then perform iterative calculation. Its algorithm is described as follows:

(1) Randomly generate the initial value m ₀ of the output control rule and the initial value x _l [0] of the chaotic system, (l=1, 2, ... 8), and select 8 bases 12, 13, 14, 15, 28 , 29, 30 and 31;

(2) x _j [i]=x _l [i], j=l+1, (l=1, 2,...7), when l=8, j=1;

(3) According to the chaotic mapping by x _j [i] $x_{i+1}=1-{2x}_i^2$ The chaotic map output x _j [i+1] (j=1, 2, ... 8) is generated by using different base calculations respectively;

(4) Convert x _j [i+1] of different bases into decimal x' _j [i+1] (j=1, 2, ... 8);

(5) Encode x′ _j [i+1] according to [256*acos(-x _j ′[i+1])] ([] indicates rounding down) to obtain x _j ″[i+1];

(6) Pseudo-random number generator according to m[i] and piecewise linear chaotic mapping:

$m_{i+1}=\left\{\begin{array}{cc}m/p_1& 0\&le;m_i<p_1\\ (m_i-p_1)/(p_2-p_1)& p_1\&le;m_i<p_2\\ 1-\frac{m_i-p_2}{p_3-p_2}& p_2\&le;m_i<p_3\\ 1-\frac{m_i-p_3}{1-p_3}& p_3\&le;m_i<1\end{array}\right.$ where p ₁ =0.29, p ₂ =0.5, p ₃ =0.7

Generate m[i+1], and select the corresponding shuffling algorithm according to the positive or negative of m[i+1] to output M _ξ ≡ _{M ξ (} i, j)= x _j "[i+1] (j=1, 2,..., n; ξ=1, 2,...);

(7) Let x _l [i+1]=x _j [i+1], l=j, (l, j=1, 2,...8), i=i+1, jump back to step (2) , until the end of the task;

Explanation for step (2): at time i, the iterative output values of 8 bases are x ₁ [i], x ₂ [i]... x ₈ [i], at time i+1 in Example 1, 8 The chaotic iteration input values of different bases at time i+1 are: x ₁ [i], x ₂ [i]...x ₈ [i], that is, x _j [i+1]=x _j [i] (j=1, 2,...8), at the time i+1 in Example 2, the input values of 8 chaotic iterations of different bases are no longer: x _i [1], x _i [2]... x _i [8], but _xj [i+1]= _xl [i], j=(l+1) mod 8+1, (j,l=1, 2...8). The cyclic shift is only one of the methods, and other exchange methods can also be used.

Example three

Since the existing microprocessors and hardware arithmetic units are based on binary calculations, FPGA technology must be used to implement M-ary arithmetic operations when implementing M-ary digital chaotic mapping in Example 1. The design of the multiplier requires Lots of hardware resources. However, the existing binary-based hardware multipliers are quite mature in design method and quite optimized in algorithm. If the existing binary-based hardware multiplier and M-ary operation can be combined, then hardware resources can be reduced while reducing design difficulty.

Therefore, binary calculation can be used in the calculation of the chaotic map, and then the obtained result x _k+1 is converted into an M-ary number x′ _k+1 , and then the M-ary number x′ _k+1 is converted into a binary x″ _{k+ 1} .

Because the carry rule of the N-ary system is different from the binary system, the rounding precision and the binary system are also different during the rounding process, so under the premise of limited precision, x′ _k+1 ≠ x″ _k+1 , and x It can be seen that _k+1 -x″ _k+1 is a random disturbance. This improvement also increases the period of the sequence without increasing the numerical accuracy of the digitized chaos map. Finally, the 8 chaotic iterative values generated are scrambled and output sequentially, which can increase the linear complexity of the chaotic sequence and reduce the correlation of the chaotic sequence.

Its algorithm is described as follows:

(1) Randomly generate the initial value m ₀ of the output control rule and the initial value x _l [0] of the chaotic system, (l=1, 2, ... 8), and select 8 bases 60, 61, 62, 63, 124 , 125, 126 and 127;

(2) x _j [i] = x _l [i], (j = 1, 2, ... 8);

(3) According to piecewise linear chaotic mapping by x _j [i]:

$x_{i+1}=\left\{\begin{array}{cc}x/p_1& 0\&le;x_i<p_1\\ (x_i-p_1)/(p_2-p_1)& p_1\&le;x_i<p_2\\ 1-\frac{x_i-p_2}{p_3-p_2}& p_2\&le;x_i<p_3\\ 1-\frac{x_i-p_3}{1-p_3}& p_3\&le;x_i<1\end{array}\right.$ where p ₁ =0.29, p ₂ =0.5, p ₃ =0.7

Use binary calculation to generate chaotic map output x _j [i+1] (j=1, 2, ... 8), and then convert x _j [i+1] into corresponding base x _j ^k[j] [i+ 1], (j=1, 2,...8), k[1], k[2],...k[8] are the 8 bases selected in step (1) in turn, and then x _j ^{k[ j]} [i+1] is converted to the corresponding x _j [i+1] respectively;

(4) Convert x _j [i+1] of different bases into decimal x' _j [i+1] (j=1, 2, ... 8);

(5) Encode x′ _j [i+1] according to [256*acos(-x _j ′[i+1])] ([] indicates rounding down) to obtain x _j ″[i+1];

(6) The pseudo-random number generator is based on m[i] and $x_{i+1}=1-{2x}_i^2,$ And according to the positive or negative of m[i+1], select the corresponding shuffling algorithm to output M _ξ ≡ M _ξ (i, j)=x _j ″[i+1]( j = 1, 2, ..., n; ξ = 1, 2, ...);

(7) Let x _l [i+1]=x _j [i+1], l=j, (l, j=1, 2,...8), i=i+1, jump back to step (2) , until the end of the task;

The following two points need to be further clarified:

(1) Select chaotic map in the process of concrete realization of this method $x_{i+1}=1-{2x}_i^2$ and piecewise linear chaotic maps, the above method can also be used for other chaotic maps.

(2) In each specific realization of the method, 8 specific bases are adopted, and in fact, any base can be used. However, for the chaotic iterative formula, the chaotic properties of some bases are degraded, so it is recommended not to use these bases.

Example four

The following method can also be used to increase the period and linear complexity of the digital chaotic sequence: use the shuffling method to exchange the iterative values of different bases, and then perform iterative calculations. The algorithm is described as follows:

(1) Randomly generate the initial value m ₀ of the output control rule and the initial value x _l [0] of the chaotic system, (l=1, 2, ... 8), and select 8 bases 12, 13, 14, 15, 28 , 29, 30 and 31;

(2) by x _l [i] according to piecewise linear chaotic mapping:

$x_{i+1}=\left\{\begin{array}{cc}x/p_1& 0\&le;x_i<p_1\\ (x_i-p_1)/(p_2-p_1)& p_1\&le;x_i<p_2\\ 1-\frac{x_i-p_2}{p_3-p_2}& p_2\&le;x_i<p_3\\ 1-\frac{x_i-p_3}{1-p_3}& p_3\&le;x_i<1\end{array}\right.$ where p ₁ =0.29, p ₂ =0.5, p ₃ =0.7

Use binary calculation to generate chaotic map output x _j [i] (j = 1, 2, ... 8)

(3) Convert x _j [i] of different bases into decimal x' _j [i] (j=1, 2, ... 8);

(4) Encode x′ _j [i] according to [256*acos(-x _j ′[i])] ([] indicates rounding down) to obtain x _j ″[i];

(5) output x _j ″[i] sequentially;

(6) Pseudo-random number generator according to m[i] and piecewise linear chaotic mapping:

$m_{i+1}=\left\{\begin{array}{cc}m/p_1& 0\&le;m_i<p_1\\ (m_i-p_1)/(p_2-p_1)& p_1\&le;m_i<p_2\\ 1-\frac{m_i-p_2}{p_3-p_2}& p_2\&le;m_i<p_3\\ 1-\frac{m_i-p_3}{1-p_3}& p_3\&le;m_i<1\end{array}\right.$ where p ₁ =0.29, p ₂ =0.5, p ₃ =0.7

Generate m[i+1], and select the corresponding shuffling algorithm according to the positive or negative of m[i+1], and then assign x _l [i+1] to x l [i+1] after shuffling the order of x _j [i]: x _l [i+ 1]=x _j [i], j is the codebook address before the shuffling algorithm, and l is the codebook address after the shuffling algorithm;

(7) i=i+1, jump back to step (2), until the end of the task.

Claims (3)

1. A chaotic password generation method under a limited precision, the steps of which are: (1) Initialization: Randomly generate or set the initial value x _l [i] of the chaotic system, i=0; l=1, 2,..., n, n is the number of chaotic systems, and set k different progress system, and k≤n; (2) According to the input x _l [i], respectively adopt one of the set k different bases to obtain a chaotic system output x′ _l [i] corresponding to the base; (3) Convert the output x′ _l [i] of different bases into the set value x″ _l [i] of the same base; (4) encode x″ _l [i] to obtain x″′ _l [i]; (5) After n chaotic coding sequences x″' _l [i] are scrambled, they are output as passwords; (6) Judging whether the number of passwords generated reaches the required number, if reached, end; otherwise, determine x _l [i+1]=x' _l [i] according to the set sorting rules, make i= i+1, jump back to step (2). 2. The method according to claim 1, characterized in that: step (5) is realized in the following manner: according to the data characteristics of x "' _l [i], n chaotic coded sequences x "' _l according to the shuffling algorithm [i] Output after scrambling. 3. The method according to claim 1 or 2, characterized in that the chaotic system in step (2) is a one-dimensional logistic iterative chaotic mapping system or a piecewise linear chaotic mapping system.

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