CN109951282B - Pseudo-random sequence generation method based on segmented CUBIC chaotic mapping - Google Patents

Abstract

The invention relates to a method for generating a pseudo-random sequence based on segmented CUBIC chaotic mapping, belonging to the field of chaotic cryptography. The method specifically comprises the following steps: s1: randomly setting an initial value and an initial control parameter; s2: generating a pseudo-random number from the PCMFO; s3: iterating the PCMFO to obtain a temporary floating point number; s4: extracting four temporary values with the same digits from the temporary floating point number, and generating 4 output numbers as independent variables; s5: and paralleling four output numbers to obtain the pseudo-random sequence. The invention improves the complexity of the password and realizes good balance between efficiency and safety.

Description

Pseudo-random sequence generation method based on segmented CUBIC chaotic mapping

Technical Field

The invention belongs to the field of chaotic cryptography, and relates to a method for generating a pseudorandom sequence based on segmented CUBIC chaotic mapping.

Background

Chaotic systems for data security are roughly classified into two categories, one-dimensional chaotic systems and high-dimensional chaotic systems: the one-dimensional chaotic system has a simple structure and high calculation efficiency, but the sequence is not complex enough, and the prediction of chaotic tracks cannot be resisted; the high-dimensional chaotic system has the obvious advantages of complex track, long period and the like in a discrete environment, but the iterative high-dimensional chaotic system needs more calculation, and the iterative high-dimensional chaotic system usually brings efficiency problems to an algorithm based on the high-dimensional chaotic system. Therefore, how to construct a chaotic map with a good balance between complexity and efficiency becomes another key for designing chaotic-based ciphers.

The CUBIC map is a chaotic map proposed by May to describe the frequency of genes in genetics that contain one locus and two alleles. The CUBIC mapping is a one-dimensional discrete time dynamic system and has the characteristics of simple structure and complex dynamic behavior. The mathematical expression is as follows:

x_n+1＝f(x_n)＝ax_n ³+(1-a)x_n

wherein x_nE (-1, +1) is the state value, a is the control parameter. When a is equal to (3.3, 4)]The CUBIC map is in a chaotic state.

According to the analysis, the ergodicity of CUBIC mapping is general, the Lyapunov exponent is not large enough, and the value range of the control parameter a is small. Therefore, designing an encryption algorithm directly using the CUBIC mapping may present a security risk. To further improve the cryptographic properties, the present invention proposes an enhanced CUBIC mapping, or PCM.

Disclosure of Invention

In view of this, the present invention provides a method for generating a pseudorandom sequence based on segmented CUBIC chaotic mapping, which achieves a good balance between efficiency and security.

In order to achieve the purpose, the invention provides the following technical scheme:

a pseudo-random sequence generation method based on segmented CUBIC chaotic mapping specifically comprises the following steps:

s1: randomly setting an initial value x₀And an initial control parameter alpha₀Setting the number of segments N in the PCM to 128 to obtain good chaotic characteristics;

s2: generating a pseudo-random number using PCMFO; iterate PCMFO 1000 times to avoid the initial value influence;

s3: iterate once PCMFO if the current state value x₀If the value is less than 0,1 is added; then, a temporary floating point number t having a section of (0,1) is obtained₁；

S4: another temporary floating point number t is generated in the interval (0,1) according to the following formula₂：

t₂＝(t₁×L)mod 1

Wherein L is [5000,10000 ]]Is with an accuracy of 10^-5Mod is a modulo operation;

s5: from t₂After extracting its decimal point, 32 bits are divided into a₁、a₂、a₃And a₄Four parts, which are bits 1-8, bits 9-16, bits 17-24 and bits 24-32, respectively;

s6: 4 temporary values a₁、a₂、a₃And a₄4 output numbers of 8 bit length are generated as arguments, each k₁、k₂、k₃And k₄(ii) a The rules for generating these 4 numbers are:

k₁＝F(a₁,a₃,a₂)，k₂＝H(a₂,a₄,a₃)

k₃＝F(a₃,a₁,a₄)，k₄＝H(a₄,a₂,a₁)

wherein

Is the XOR operator;

s7: 4 output numbers from k₁Is juxtaposed to k₄As a 32-bit output sequence, if more pseudo-random sequences are needed, go to step S3 to generate the next 32-bit output sequence; otherwise, the above process is stopped.

Further, in step S2, the formula for generating the pseudo random number according to the PCMFO is:

x_n+1＝PCM(x_n,a_n)

wherein, PCM (x)_n,a_n) Representing segmented CUBIC chaotic mapping; logistic (x)_n+1)＝4x_n(1-x_n) It is a Logistic chaotic map.

Further, the calculation formula of the segmented CUBIC chaotic map is as follows:

wherein x is_nE (-1,1) is the state value, a E (0,4) is the control parameter, and N is the number of segments.

The invention has the beneficial effects that: the method of the invention improves the complexity of the password and realizes good balance between efficiency and safety.

Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.

Drawings

For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:

FIG. 1 is a flow chart of a pseudo-random sequence generation method according to the present invention;

FIG. 2 is a schematic structural diagram of a PCMFO according to the present invention;

FIG. 3 is a diagram illustrating the effect of PCM to CUBIC mapping;

FIG. 4 is a diagram of the corresponding PCM ergodic effect for different values of a and N;

fig. 5 is a diagram of the cycle division of PCM at different values of N when a is 4;

FIG. 6 is a plot of the Lyapunov exponent simulation of the PCM at fixed values of a, N, respectively;

FIG. 7 is a simulation diagram of probability density distribution of PCM with different values of a and N.

Detailed Description

The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.

Referring to fig. 1 to 2, fig. 1 is a method for generating a pseudo-random sequence by a pseudo-random number generator (PRNG) according to the present invention, which includes the following specific steps:

s1: randomly setting an initial value x₀And an initial control parameter alpha₀Setting the number of segments N in the PCM to 128 to obtain good chaotic characteristics;

s2: generating a pseudo-random number using PCMFO; iterate PCMFO 1000 times to avoid transient effects;

s3: iterate once PCMFO if the current state value x₀If the value is less than 0,1 is added; then, a temporary floating point number t having a section of (0,1) is obtained₁；

S4: another temporary floating point number t is generated in the interval (0,1) according to the following formula₂：

t₂＝(t₁*L)％1

Wherein L is [5000,10000 ]]Is with an accuracy of 10^-5Mod is a modulo operation;

s5: from t₂After extracting its decimal point, 32 bits are divided into a₁、a₂、a₃And a₄Four parts, which are bits 1-8, bits 9-16, bits 17-24 and bits 24-32, respectively;

s6: 4 temporary values a₁、a₂、a₃And a₄4 output numbers of 8 bit length are generated as arguments, each k₁、k₂、k₃And k₄(ii) a The rules for generating these 4 numbers are:

k₁＝F(a₁,a₃,a₂)，k₂＝H(a₂,a₄,a₃)

k₃＝F(a₃,a₁,a₄)，k₄＝H(a₄,a₂,a₁)

wherein

Is the XOR operator;

s7: 4 output numbers from k₁Is juxtaposed to k₄As a 32-bit output sequence, if more pseudo-random sequences are needed, go to step S3 to generate the next 32-bit output sequence; otherwise, the above process is stopped.

In step S2, as shown in fig. 2, the formula for generating pseudo random numbers from the PCMFO is:

x_n+1＝PCM(x_n,a_n)

wherein, Logistic (x)_n) Represents the Logistic mapping analysis, Logistic (x)_n+1)＝4x_n(1-x_n) The method is Logistic chaotic mapping, and the calculation formula is as follows:

example (b):

the invention relates to a method for generating a pseudorandom sequence based on segmented CUBIC chaotic mapping, which specifically comprises the following steps:

step 1: segmenting according to a segmented CUBIC chaotic mapping formula;

step 2: compare PCM to CUBIC mapping: PCM is a new mapping, independent of CUBIC mapping, as shown in fig. 3.

(1) Analyzing the ergodicity of the PCM at different values of a and N: as shown in FIG. 4, for different a and N, the state value of PCM can fill the whole range of [ -1,1], and the ergodicity is good. In fig. 4(a), a is 0.5, and N is 8; in fig. 4(b), a is 2 and N is 8; in fig. 4(c), a is 3 and N is 8; in fig. 4(d), a is 4, N is 8; in fig. 4(e), a is 0.5 and N is 32; in fig. 4(f), a is 2, N is 32; in fig. 4(g), a is 3, N is 32; in fig. 4(h), a is 4 and N is 32. As can be seen from the above figure, the CUBIC mapping is a chaotic full mapping only when a is 4.

(2) The cycle-doubling bifurcation diagram (plotted numerically) of the PCM at different values of N when a is 4 was analyzed, as shown in fig. 5: the larger chaotic mapping control parameter range provides a larger key space, improves the decoding difficulty and improves the safety.

In fig. 5(a), N is 2; in fig. 5(b), N ═ 3; in fig. 5(c), N ═ 12; in fig. 5(d), N ═ 13; in fig. 5(e), N is 64; in fig. 5(f), N ═ 65; as can be seen from the above figure, the control parameter range of the chaotic full map increases with the increase of N. Compared with non-full mapping, full mapping corresponds to stronger chaos intensity, and the range of iterative values is larger. Subsequent iteration values are not easy to approach the previous iteration values, so that the period of the chaotic digital sequence can be expanded, and the dynamic degradation of the chaotic sequence is improved.

As N increases, PCM gradually exhibits chaotic behavior throughout the parameter range. It is noted that when N is odd, there is a distinct periodic window that decreases as N increases, meaning that N is preferably set to an even number.

(3) Analyzing the Lyapunov exponent of the PCM at different values of a and N: the larger the positive LE value, the better sensitivity, divergence and unpredictability of the system, and the larger the range of positive LE values, the larger the key space.

a. When a is fixed, as shown in fig. 6(a) to (d), the lyapunov exponent of the PCM increases with an increase in N.

b. When N is fixed, as shown in fig. 6(e) to (h), the interval of the control parameter a corresponding to the positive lyapunov exponent increases as N increases, and the larger the value of the control parameter a, the larger the LE of the PCM.

(4) When the probability density distribution of the PCM is analyzed under different values a and N, as shown in fig. 7(a) to (h), the density probability of the PCM is not uniform, and the value of the control parameter a has a significant influence on the density probability distribution of the PCM.

Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (1)

1. A method for generating a pseudo-random sequence based on segmented CUBIC chaotic mapping is characterized by comprising the following steps:

s1: randomly setting an initial value x₀And an initial control parameter alpha₀The number of segments N is set to 128;

s2: generating a pseudo-random number by adopting a CUBIC chaotic map (PCMFO) with feedback operation;

the formula for generating pseudo-random numbers from PCMFO is:

x_n+1＝PCM(x_n,a_n)

wherein, PCM (x)_n,a_n) Representing segmented CUBIC chaotic mapping; logistic (x)_n+1)＝4x_n(1-x_n) Is Logistic chaotic mapping;

the calculation formula of the segmented CUBIC chaotic mapping is as follows:

wherein x is_nE (-1,1) is a state value, a E (0,4) is a control parameter, and N is a segment number;

s3: iterate once PCMFO if the current state value x₀If the value is less than 0,1 is added; then, a temporary floating point number t having a section of (0,1) is obtained₁；

S4: according to the formula in the interval (0,1) generating another temporary floating point number t₂：

t₂＝(t₁×L)mod 1

Wherein L is [5000,10000 ]]Is with an accuracy of 10^-5Mod is a modulo operation;

s5: from t₂After extracting its decimal point, 32 bits are divided into a₁、a₂、a₃And a₄Four parts, which are bits 1-8, bits 9-16, bits 17-24 and bits 24-32, respectively;

s6: 4 temporary values a₁、a₂、a₃And a₄4 output numbers of 8 bit length are generated as arguments, each k₁、k₂、k₃And k₄(ii) a The rules for generating these 4 numbers are:

k₁＝F(a₁,a₃,a₂)，k₂＝H(a₂,a₄,a₃)

k₃＝F(a₃,a₁,a₄)，k₄＝H(a₄,a₂,a₁)

wherein

Is the XOR operator;

s7: 4 output numbers from k₁Is juxtaposed to k₄As a 32-bit output sequence, if more pseudo-random sequences are needed, go to step S3 to generate the next 32-bit output sequence; otherwise, the above process is stopped.

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