WO2001069845A1

WO2001069845A1 - Chaos cipher system

Info

Publication number: WO2001069845A1
Application number: PCT/JP2001/000225
Authority: WO
Inventors: Shunji Kawamoto
Original assignee: Kansai Technology Licensing Organization Co., Ltd.
Priority date: 2000-03-10
Filing date: 2001-01-16
Publication date: 2001-09-20
Also published as: JP2001255817A; JP3976218B2

Abstract

A chaos generating device has a chaos function g(f(n)) which is a synthesis function created by combining a first function f and a second function g having a period T and generates a chaos time sequence by changing n of the chaos function g(f(n)) sequentially and deriving Xn in time sequence. To derive the (n+1)-th value Xn+1, the value f(n+1) calculated from the n-th value f(n) of the first function f and the first function f is divided by the period T of the second function g, and the remainder r is given to the second function g.

Description

Specification

Chaos encryption system

【Technical field】

The present invention relates to a chaos generation device and a chaos generation method used for, for example, an encryption technique.

[Background Art]

With the progress of computerization in recent years, the demand for information security has been increasing, and secure encryption technology that accurately transmits information to recipients and that prevents information from being read by anyone other than recipients is required. ing. Currently, cryptographic technologies that are in practical use, such as symmetric key cryptography such as DES (Data Encryption Standard) No.

Public key 喑 such as 喑), which was devised by Rivest, Shamir, and Adelman. However, these have many problems in terms of safety and processing speed. In recent years, chaos encryption methods have been proposed that have higher security strength than these encryption technologies, and have higher processing speeds for encryption and decryption.

Chaos changes from simple rules to complex and irregular, and chaos encryption encrypts using a chaos function that performs such irregular behavior. In other words, the chaos function is a function that derives unpredictable and irregular values in a time series, despite being expressed by a simple mathematical formula. Therefore, by encrypting information using the chaotic time series from this chaotic function, it is possible to perform unbreakable encryption.

For example, first, an initial value is set to a predetermined chaos function, and the output value is sequentially converted to a digital signal of “1” or “0” to generate a series of digital chaos time series. Then, a series of digital information to be transmitted is subjected to a logical operation using the generated digital force time series, and is sequentially encrypted. Similarly, on the receiving side, a digital chaos time series is generated from the same chaos function and initial value, and the received encryption is sequentially decrypted in the digital chaos time series. One of the characteristics of chaos is its initial sensitivity. This means that even if there is a slight difference in the initial value given to the chaotic map, the output value will differ greatly. In other words, if the initial value is slightly different, the generated chaotic time series will be significantly different. Therefore, by encrypting with the initial value as a key, it becomes very difficult for a third party to break the encryption.

In the above-mentioned DES encryption and RSA encryption, encryption is performed by performing complex arithmetic processing, so there is a problem that the processing speed of encryption and decryption is slow. Since the data is sequentially encrypted or decrypted using the chaotic time series generated from the chaotic functions, the processing speed can be significantly increased.

As described above, chaos has the property of being sensitive to the initial value, and the chaotic time series generated by a slight difference in the initial value is different, which makes it difficult for a third party to decipher. Conversely, if the initial value used as the key is slightly different between the encrypting side and the decrypting side, there is a problem that the decryption cannot be performed accurately.

Next, the change of the chaotic time series due to the change of the initial value will be described.

Logistic mapping, a well-known chaos mapping

Χ _{η + 1} = α X _n (1-Χ _π ) ... (1). In this logistic map, α = 4

In the case of _{_{Χ η + 1 = 4 Χ η}} (1 -Χ η) ■ ·· (2), exact chaotic solution

X _n = sin ² (c2 ⁿ ) (3) is obtained. A chaotic time series can be obtained by sequentially giving n (n = 1, 2, 3, ···) to this equation (3).

Next, in order to examine the change when the chaotic time series is calculated with double precision (rounding error _E = ^10-15 ), the initial value X in the above equation (3) is used. Figure 4 shows the chaotic time series when 0.1 and 0 · 100000000000001 are given. FIG 4 As can be seen from, appears to the extent that the influence of the rounding error is found even diagram (1 0 ²⁾ at about n = 4 5 th. This is the same even when iterative calculation of equation (1) is used.

Thus, in the chaos encryption method using the chaos mapping equation (1), the error _f = 1 (Even very small rounding errors in the order of T ¹⁵ which is integrated, several tens of times of errors increases rapidly with repeated calculations, reproduction faithful chaos is considered impossible. Further, such an error Is generated as the accuracy of electronic circuits and external noise, and it is impossible to reduce the error to zero.

Under such circumstances, it is said that practical application of chaos encryption technology is impossible. An object of the present invention is to provide a chaos generation device and a chaos generation method capable of faithfully reproducing a chaos time series without accumulating errors.

DISCLOSURE OF THE INVENTION

The present invention according to claim 1 has a chaotic function g (f (n)) which is a composite function of a first function f and a second function g having a period T, and the chaotic function g ( f (n)), a chaos generator that derives a unique value X _n in time series by sequentially changing n in the n + 1st value X _{n +1} The value f (n + 1) calculated from the value f (n) of the first function f and the first function f is divided by the period T of the second function g, and the remainder r A chaos generator characterized by deriving X _{n + 1} by giving it to a function g of.

The present invention according to claim 2 is characterized in that the first function f has a coefficient c and includes a predetermined integer k raised to the nth power (ck ⁿ ).

The first function f is f (n) = ck ⁿ in the case of claim 2, and f = c2 ^{n in} the example of the above equation (3). Also, the second function g that is a periodic function is g = sin ² (f (n)) in the example of Equation (3), and the chaos function that is a composite function of these functions is

X _n = sin ² (c 2 ")… (3) As described above, this equation (3) is a chaotic map

X _{n + 1} = 4 X _n (1— X _n )… It is an exact solution of (2). Therefore, the chaos function X _n = g (f (n)) is given sequentially as n = l, 2, 3, ... to derive X い₂ , Χ ₃ ··· and the chaotic time series be able to. At this time, first, η is given to the first function f to calculate f (n), and the calculated value f (n) is given to the second function g to calculate _Xn . Toes F (n) is given as n = l, 2, 3…, f (1), f (2), f (3)… are calculated, and these are sequentially given to the second function g. Derives X ₂ , X ₃ ….

Here, the first function f, when the f (n) = ck ^n, the value of the n times f is the f (n) = ck ^n, the value of the (n + 1) times the f f (n + 1) is f (n + 1) = kXf (n). In this way, the value f (n + 1) of the (n + 1) th first function f is obtained from the value f (n) of the nth first function f and the first function f. Can be Here, since the second function g is a periodic function, if two values given to the second function are exactly separated by the period T, the calculated X _n becomes the same value. From this, 値_π when the value r (n) of the first function given to the second function is divided by the period T and the remainder r is given to the second function g, and the value of the first function X _n when f (n) is given directly to the second function g is the same value.

The present invention utilizes the property of the periodic function g and the method of calculating f (n + 1) from f (n) described above. In other words, to calculate the the (n + 1) times the chi _{[pi + 1,} and a second η times the first function f value f (n) and the first function f (n + 1) th time Calculate the value f (n + 1) of f, divide this value f (n + 1) by the period T of the second function g, and give the remainder r to the second function g to obtain X _{n +} Calculate ₁ . In this way, X _n is calculated each time using the previous value. .

Here, for example, consider the case where the initial value includes an error f. The error increases when calculating the (n + 1) th value f (n + 1) from the nth first function f (n) and f. For example, if f (n) = ck ⁿ , the (n + 1) th value f (n + 1) is calculated by multiplying the n-th value f (n) by k times (k is an integer). If the value of contains the error ε, the error Ε is multiplied by k.

As described above, in the present invention, a remainder r obtained by dividing the (n + 1) th value of f f (n + 1) by the period T of the second function is given to the second function g to obtain Χπ _{+ 1} Is calculated. f

If the value k (ck ⁿ ) of (n + 1) becomes larger than the period T, f (n + 1) is divided by the period T, and only the remainder r is given to the second function g. This is prevented from being said. Therefore, even if an error occurs due to the accuracy or noise of the electronic circuit, the error is not integrated, and the chaotic time series can be faithfully derived.

The present invention according to claim 3 is characterized in that, when digital information is encrypted in the chaotic time series generated by the chaotic function, the period N of the chaotic function n is longer than the digital information to be encrypted. .

In the chaotic function as shown in the above equation (3), there is a period N for n. That is, X _n obtained from two different _n may have the same value, and in this case, the value of the subsequent chaotic time series X _n will also be the same. If such a chaotic function is used for decoding, the same chaotic time series will be used repeatedly, and the security will decrease. However, in the present invention, since the period N of the chaotic function is selected to be larger than the amount of information to be encrypted, the same chaotic time series is not used more than once in one encryption process. This can be equivalent to encrypting with chaos.

According to a fourth aspect of the present invention, digital information is sequentially encrypted or decrypted with a chaotic time series derived using the initial value set in the first function f as a key and this key and the chaotic function. It is characterized in that

The present invention according to claim 5 is characterized in that the digital information is obtained by converting a sentence, an image or a sound into a digital signal.

Text and images can be easily converted to digital information, and this digital information can be encrypted by performing arithmetic processing on chaotic time series.

When transmitting and receiving encrypted digital information, the transmitting side and receiving side have a common chaos function in advance. Then, on the transmitting side, an initial value is given to the chaos function to generate a chaos time series, and digital information is encrypted using the chaos time series. Then, it transmits the encrypted information and the initial value.

The receiving side generates the same chaos time series from the common chaos function and the received initial value, and decrypts the received encryption from the generated chaos time series.

The initial value sent as a key may include an error due to the accuracy of the circuit or noise at the time of transmission / reception, but by using the chaos generation algorithm of the present invention described above, This prevents the accumulation of errors when generating a chaotic time series. As a result, even if the initial value contains an error, the same chaotic time series as the transmitting side can be faithfully reproduced on the receiving side, and the encrypted information can be accurately restored.

As described above, the chaos generating device of the present invention is particularly suitably used as a device for encrypting or decrypting information such as text and images. For example, when the first function f is f (n) = ck ⁿ , the initial value as a key is a coefficient c.

The present invention according to claim 6 has a chaotic function g (f (n)) which is a composite function of a first function f and a second function g having a period T, and the chaotic function g ( f (n)) in the chaos generation method that derives a unique value X _n in a time series by sequentially changing _n of the n + 1st value Χ _{π + 1} The value f (n + 1) calculated from the value f (n) of the first function f and the first function f is divided by the period T of the second function g, and the remainder r is calculated as the second This is a chaos generation method characterized by deriving X _{n + 1} by giving it to a function g.

The present invention according to claim 7 has a chaotic function g (f (n)) which is a composite function of a first function and a second function g having a period T, and the chaotic function g (f (n)) in a pseudo-random number generator that sequentially derives a unique value X _n in a time series to generate a pseudo-random number.

To derive a first (n + 1) th value X _{n + 1,} the first n times the first function f value f (n) and the value f calculated from the first function f (n + 1), This pseudorandom number generator is characterized in that the second function g is divided by the period T of the second function g, and the remainder r is given to the second function g to derive ππ _{+ 1} .

The chaotic time series generated by the algorithm of the present invention can also be used as a pseudo-random number, and can be used where a random number is required, for example, in a computer game.

The present invention according to claim 8 has a chaotic function g (f (n)) which is a composite function of a first function f and a second function g having a period T, and the chaotic function g ( f (n)), chaos that derives a unique value X _n in time series by changing _n of A generator,

To derive a first (n + 1) th value X _{n + 1,} the n times of the first function f value f (n) and the value f calculated from the first function f (n + 1) , A pair of cryptographic devices having a chaos generator that divides the remainder by the period T of the second function g and gives the remainder r to the second function g to derive X _{n + 1} ,

The digital device encrypts digital information with the chaos time series derived from the chaos generator,

The chaos generation device of the other encryption device derives the same chaos time sequence as the chaos time sequence derived by the chaos generation device of the one encryption device, and decrypts the encrypted digital information. It is a cryptographic system.

[Brief description of the drawings]

The present invention, these and other objects, features and advantages will become more apparent from the following detailed description and drawings.

FIG. 1 is a block diagram showing a configuration of an encryption system 1 using the chaos generator of the present invention.

FIG. 2 is a diagram illustrating calculation results when c = (2049/200 11) π.

FIG. 3 is a diagram showing a period Ν when Μ is increased.

FIG. 4 is a diagram showing a calculation result of chaos by a conventional mapping in which rounding errors are accumulated.

BEST MODE FOR CARRYING OUT THE INVENTION

Hereinafter, preferred embodiments of a chaos generator, a chaos generation method, a pseudo-random number generator, and a cryptographic system according to the present invention will be described with reference to the accompanying drawings. FIG. 1 is a block diagram showing a configuration of a signal system 1 using a chaos generator according to one embodiment of the present invention. The cryptographic system 1 is composed of a pair of cryptographic devices 4 and 5 including chaos generators 2 and 3 realized by, for example, an integrated circuit.

For example, when sending digital information such as text or images, one of the encryption devices 4 encrypts the information using a certain key, and transmits the encrypted information to the other encryption device _5.c The key used for encryption is sent together with the encryption or passed separately. The other encryption device 5 decrypts the received encryption using the passed key.

Next, the encryption method will be described in more detail with reference to FIG. The encryption device 4 encrypts the plaintext 10. Plaintext 10 is digital information that represents a sentence with digital information consisting of “1” or “0”. The chaos generator 2 has a chaos function, and by setting an initial value to this chaos function, first, a chaos time series X _n

(Χ _{1 (} Χ ₂ , Χ ₃ ···). If the digital chaotic time series is output by digitizing it, the generated chaotic time series _{η η} is a chaotic function as shown in Fig. 2. In the case of (3), the value is in the range of 0 to Χ _η <1. Therefore, in order to convert the generated chaotic time series _{η η} to the digital chaotic time series _{π π} , for example, the chaos generator 2 By setting the reference value A (0 <A <1), the digital power is set to `` 1 '' when _Xn is greater than the set reference value A, and to `` 0 '' when _Xn is less than the reference value A. A time series is generated.

In this way, the chaos time series output from the chaos generator and the plaintext 10 are input to the arithmetic unit 11, where the arithmetic processing is sequentially performed to generate the ciphertext 12. In the present embodiment, the arithmetic unit 1 1 performs the arithmetic operation for an exclusive logical sum, exclusive OR of the plaintext 1 0 and chaos time-series chi _[pi Te convex to create a ciphertext 1 2. The created ciphertext 12 is transmitted, for example, via a public telephone line, or passed to the other encryption device 5 via a recording medium such as a floppy disk.

The other encryption device 5 also has the same chaos function as the chaos function of the chaos generation device 2 of one encryption device 4 and the reference value 、, and uses the key passed from one encryption device 4 to generate The same chaotic time series X _n as the cryptographic device 4 can be generated. Therefore, the received ciphertext 1 2 and the generated chaotic time series Χ _π are input to the arithmetic unit 13, where the ciphertext 12 and the chaotic time series _{π π} are exclusive-ORed for decryption. Thus, plaintext 10 is obtained.

In the above-described embodiment, the arithmetic unit 11 is configured to perform the exclusive OR operation. However, the present invention is not limited to this. For example, the arithmetic unit 11 adds on the encryption side, and the arithmetic unit 11 on the decryption side. The encryption may be decrypted by subtracting 1 in 3. The reference value A set in the chaos generator may be used as a key, and the reference value A may be changed for each encryption and decryption. Also, instead of one chaos function, a plurality of chaos functions are set for each of the chaos generators 2 and 3, and each time encryption is performed, the chaos function is changed, and which chaos function is used together with a key. You may pass it.

Next, an algorithm for generating the chaotic time series X _n in the chaotic generators 2 and 3 will be described.

In the present invention, the chaos function is a composite function g (f (n)) of the first function f and the second function g having the period T, and the first function f is f (n) = ck "(n is A positive integer, k is a prime number and c is an irrational number.

X _n = sin ² (c 2 ")… (3) Therefore, the first function f becomes f (n) = c 2 ⁿ and the second function g (n) = sin ² (f (n As described above, Equation (3) is a logistic map that is a chaotic map.

X _{n + 1} = 4 X _n (1 -X _n ) ... This is the exact solution of (2).

Next, the chaos time series generation algorithm of the present invention using the chaos function expressed by the equation (3) will be described. Where the coefficient c in equation (3) is

c = L / M-π ≠ p / 2 ^q · π… (4)

(L, M, p and q are positive integers)

And First, the chaos function of equation (3) is determined by determining the coefficient c in equation (3). Next, X is given by giving n = 0 to the determined equation (3). Are calculated, and then XX ₂ , X ₃ … are sequentially calculated according to the chaotic time series generation algorithm described below.

The chaotic time series generation algorithm calculates the (n + 1) th X _{n + 1} by calculating the value f (n) of the n-th first function f and the function f (n + 1) And the remainder r divided by the period T of the second function g is given to the second function g to calculate. Because it is ^π · 2 η, - in here the n-th of the first of the function f (n) is f (n) = L / M f (n + 1) = 2 · L / M-π · 2 ⁿ Since the period T of the second function g is π, the remainder r obtained by dividing f (n + 1) by π is

r = (2 · L / M · π · 2 ⁿ ) mod (π) · · · (5) Therefore, from equation (5) and the second function g, Χη _{+ 1} is

X _{n + 1} = sin ² {(2 · L / M · π · 2 ^η ) mo d (π)}-(6).

Therefore, equation (6) is

X _{n + 1} = sin ² (L _{n + 1} / M · π)… (7)

L _{n + 1} = (2 L _n ) mod (M)… (8)

Thus, in the chaotic time series generation algorithm of the present invention, to calculate X _{n + 1} , the value f (n) of the n-th first function f and the first function f Since (n + 1) is calculated and the remainder r obtained by dividing the result by the period T is given to the second function g to calculate _{Xn + 1} , no error is accumulated as described above. As a result, the chaotic time series used for encryption can be faithfully reproduced on the decryption side.

The chaotic function expressed by Eq. (3) has a period N with respect to n, and this period N depends on the coefficient c. As mentioned above, the coefficient c is

c = LZM · π ≠ p / 2 ^q · π (4)

(L, M, p and q are positive integers)

It is. Here, the period N can be increased by choosing L and M to be integers that are not divisible by each other. Note that although LZM is a rational number, c is an irrational number because π is an irrational number.

For example, Figure 2. = 2049 Ζ 200 1 1 · π is a diagram showing a chaotic time series when π, where _{N where} X ₀ = X N is 1 0005. That is, the cycle N is 1 0005. In this way, when the period N is very large and longer than the digital information to be encrypted, it is possible to perform encryption with virtually complete chaos without repeating the same chaotic time series in one encryption. it can.

Next, the relationship between the coefficient c and the period N will be described. Figure 3 shows L = 1, M = 2 s + 1

(s = 1, 2)

FIG. 9 is a diagram showing a calculation result of the period N in Equation (3) when M is increased. In Fig. 3, only the calculation results up to M and 500 are shown, but as M increases,

It can be seen that the period N can be increased by the relationship N = (M-1) / 2 ... (9). From this, it can be said that by appropriately selecting M, it is possible to generate a chaotic time series having a sufficient length according to the information to be encrypted. However, it has been confirmed by computer calculation that M must be a prime number (3, 5, 7, 11, 13, 17, 17, ...). Calculating a large period N takes several hours at a time, for example

M = 2 0 0 0 0 0 0 0 0 0 7, 7, 2 0 0 0 0 0 0 0 0 0 9 3 (both prime numbers)

N = 1 0 0 0 0 0 0 0 0 0 3 8, 1 0 0 0 0 0 0 0 0 0 4 6

Is obtained. Such a large cycle N can be regarded as an infinite cycle at the level of actual encryption. In other words, if the period N is longer than this, the calculation time increases rapidly. In the case of, the calculation takes about several months.

In the embodiment described above, the first function f is f = c2 ⁿ , the second function g that is a periodic function is g = sin ² (f (n)), and these composite functions are

X _n = sin ² (c2 ⁿ )… (3) is a chaotic function, but the present invention is not limited to this. For example, the first function f may be c 3 ^、, and the second function g may be Any periodic function such as cos (f (n)) or sin (f (n)) is acceptable. Further, the chaos function may be a combination of these _c example,

X _n = ∑ia _iC os ( _Cj ki ⁿ + β + ∑ _{j tt} jSin ( _Cj k + β ^ ■■ (10), so that the security can be further improved. In this case, the key α ,, aj, βi] 3 ”= real number, ki, kj = prime number, _Ci , _Cj = Lj π / M _{ , L _{i K} M (new Lj = odd number, Μ” = prime number). Μ ”to a large prime number Choosing 喑 will make it harder for you to choose.

In addition, such a technique can be applied not only to text encryption but also to digital audio information and digital image encryption. If digital image encryption technology is used, it can be applied to digital watermarking technology. For example,

X _m = cos (c 2 ^m )… (1 1)

X _n = sin (c 2 ⁿ ) (12)

(m, n = 0, 1, 2, 3, ...)

Then, two-dimensional (plane) chaos can be realized and applied to a security system.

In addition, the chaos generation device of the present invention can be applied to chaos control and the _like.c Also, the chaos time series generated by the chaos generation algorithm of the present invention can be used as pseudo-random numbers. It can also be applied to computer games and the like that require.

The present invention may be embodied in various other forms without departing from its spirit or essential characteristics. Therefore, the above-described embodiment is merely an example in every aspect, and the scope of the present invention is set forth in the appended claims, and is not limited by the specification.

Furthermore, all modifications and changes belonging to the equivalent scope of the claims are within the scope of the present invention.

[Industrial applicability]

As described above, the chaos generation device of the present invention can generate chaos without accumulating errors, and thus can faithfully reproduce a chaos time series. This enables the practical use of chaotic encryption technology.

As digital information to be encrypted, it can be applied to digitized text or digital audio-image. It is also possible to use the generated chaos as a pseudo-random number.

Claims

The scope of the claims

1. A chaotic function g (f (n)), which is a composite function of a first function king and a second function g having a period T, and n of the chaotic function g (f (n)) In the chaos generator that derives a unique value X _n in time series by sequentially changing the n + 1st value X _{n +1} , the n -th first function f Is divided by the period T of the second function g, and the remainder r is given to the second function g. Chaos generation characterized by deriving _{n + 1}

2. The chaos generator according to claim 1, wherein the first function f has a coefficient c and includes a predetermined integer k raised to the nth power (ck ⁿ ).

3. When encrypting digital information in the chaos time series generated by the chaos function, the period N of the chaos function with respect to n is longer than the digital information to be encrypted. Chaos generator.

4. A key is an initial value set in the first function f, and digital information is sequentially encrypted or decrypted in a chaotic time series derived using the key and the chaotic function. The chaos generator according to any one of claims 1 to 3.

5. The chaos generator according to claim 4, wherein the digital information is obtained by converting a sentence, an image, or a sound into a digital signal.

6. It has a chaotic function g (f (n)) which is a composite function of the first function f and the second function g having the period T, and n of this chaotic function g (f (n)) In the chaos generation method that derives the unique value X _n in time series by sequentially changing, the n + 1st value X _{n +1} is derived by calculating the n -th first function f The value f (n + 1) calculated from the value f (n) and the first function f is divided by the period T of the second function g, and the remainder r is given to the second function g to obtain X _{n +} A chaos generation method characterized in that ₁ is derived.

7. It has a chaotic function g (f (n)) which is a composite function of the first function f and the second function g having the period T, and n of this chaotic function g (f (n)) Change sequentially Thus, in a pseudo-random number generator that derives a unique value X _n in time series and generates a pseudo-random number,

For deriving a second eta + 1 th value chi _{eta + 1,} the eta times the first function f value f (n) and the value f calculated from the first function f (n + 1) A pseudo-random number generator that divides by the period T of the second function g and gives the remainder r to the second function g to derive ππ _{+ 1} .

8. It has a chaotic function g (f (n)) which is a composite function of the first function f and the second function g having the period T, and n of this chaotic function g (f (n)) Is a chaos generator that derives a unique value X _n in a time-series manner by sequentially changing, and derives an (n + 1) -th value Χ _{π + 1} by using an n-th first function Divide the value f (n + 1) calculated from the value f (n) of f and the first function f by the period T of the second function g, and give the remainder r to the second function g. It has a pair of encryption devices with a chaos generator that derives X _{n + I} ,

One of the encryption devices encrypts digital information with the chaos time series derived from the chaos generator,

A cipher characterized by deriving a chaos time series that is the same as the chaos time series derived by the chaos generating apparatus of the other cryptographic device, and decrypting the encrypted digital information. system.