Electronic encryption device and method

• US 5,857,025 A
• Filed: 09/09/1996
• Issued: 01/05/1999
• Est. Priority Date: 09/09/1996
• Status: Expired due to Fees

• Current Assignee: Intelligent Security Systems Incorporated
• Original Assignee: Intelligent Security Systems Incorporated
• Inventors: Anderson, Robert, Schultz, George R., Heptig, John P.
• Primary Examiner(s): Tarcza, Thomas H.
• Assistant Examiner(s): Laufer, Pinchus M.
• Application Number:
  US08/711,770
• Time in Patent Office: 848 Days
• Field of Search: 380/28, 380/46, 380/48, 364/717.01, 364/717.05
• US Class Current: 380/28
• CPC Class Codes: H04L 9/001 using chaotic signals
• CPC Class Codes: H04L 9/003 for power analysis, e.g. differential power analysis [DPA] or simple power analysis [SPA]
• CPC Class Codes: H04L 9/16 the keys or algorithms being changed during operation
• CPC Class Codes: H04L 9/3226 using a predetermined code, e.g. password, passphrase or PIN network architectures or network communication protocols for supporting authentication of entities using passwords in a packet data network H04L63/083

Abstract

A device and method is provided using a split processor based architecture to provide physical security of critical information by implementing a required protocol before releasing critical information. The critical information is used as a key to a cryptographic algorithm. A chaotic cryptographic algorithm is also provided including a defined key space employing six parameters of a Lorenzian system of equations.

Claims

1. A data encryption and decryption system comprising:

a. means for selecting a set of parameters from a predetermined valid parameter space;

b. means for inserting the parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and x, y, z, σ

, r and b are parameters;

c. means for iterating the chaotic equations to derive x'"'"', y'"'"' and z'"'"' iterate streams;

d. means for deriving a cipher number from the iterate streams; and

e. means for xoring the cipher number with the data.

2. The system of claim 1, wherein the predetermined valid parameter space further comprises one parameter from each of the following ranges:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

3. A data encryption and decryption system comprising:

a. means for selecting a set of parameters from a predetermined valid parameter space;

b. means for inserting the parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and x, y, z, σ

, r and b are parameters;

c. means for iterating the chaotic equations to derive x'"'"', y'"'"' and z'"'"' iterate streams;

d. means for deriving a cipher number from the iterate streams;

e. means for xoring the cipher number with the data;

f. means for iterating the chaotic equations a predetermined number of times; and

g. means for appending the predetermined number to encrypted data.

4. A data encryption and decryption system comprising:

a. storage means containing multiple sets of valid parameters from a predetermined valid parameter space;

b. means for selecting a set of parameters from the storage means;

c. means for inserting the selected parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and x, y, z, σ

, r and b are parameters;

d. means for iterating the chaotic equations to derive x'"'"', y'"'"' and z'"'"' iterate streams;

e. means for deriving a cipher number from the iterate streams;

f. means for xoring the cipher number with the data; and

g. means for storing the numeric results.

5. The system of claim 4, wherein the predetermined valid parameter space further comprises one parameter from each of the following ranges:

x=about -30 to about 50

y=about -30 to about 50

z=about -30 to about 50

σ

=about 0.006 to about 0.028

r=about 24 to about 40

b=about 1 to about 8

where x, y and z cannot simultaneously be zero.

6. The system of claim 4, wherein the predetermined valid parameter space further comprises one parameter from each of the following ranges:

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

7. The data encryption/decryption system of claim 4, wherein the means for deriving a cipher number from the iterate streams further comprises a means for selecting a predetermined byte of a predetermined iterate stream for the cipher number.

8. A data encryption and decryption system comprising:

a. means for selecting a set of parameters from a predetermined valid parameter space;

b. means for inserting the parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and x, y, z, σ

, r and b are parameters;

c. means for iterating the chaotic equations to derive x'"'"', y'"'"' and z'"'"' iterate streams;

d. means for deriving a cipher number from the iterate streams;

e. means for xoring the cipher number with the data;

f. means for iterating the chaotic equations a predetermined number of times;

g. means for encoding the predetermined number; and

h. means for appending the encoded number to encrypted data.

9. A data encryption system comprising:

a. a predetermined set of valid initial conditions;

b. a protocol responsive memory programmed to store and allow access to the initial conditions in response to a protocol;

c. a processor connected to the protocol responsive memory, the processor programmed to;

(1) execute the protocol;

(2) receive and store the initial conditions returned by the protocol responsive memory;

(3) insert the initial conditions into and iterate a set of chaotic equations to arrive at iterated results; and

(4) create encrypted data by combining data with the iterated results.

10. The data encryption systems of claim 9, further comprising a transmitter connected to the processor for transmitting the encrypted data.

11. The data encryption system of claim 9, further comprising a second memory connected to the processor to store the encrypted data.

12. The system of claim 9, wherein the set of chaotic equations are of the form:

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are iterated results of initial conditions x, y, z, σ

, r, and b.

13. The system of claim 12, wherein the predetermined valid initial conditions are in the range:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

14. The system of claim 9, wherein the processor encrypts data by xoring data to be encrypted after a preselected number of iterations of the chaotic equations.

15. The data encryption/decryption system of claim 9, wherein the processor connected to the protocol responsive memory is further programmed to create encrypted data by combining data with a preselected byte of the iterative results.

16. A data encryption system comprising:

a. a predetermined set of valid initial conditions;

b. a protocol responsive memory programmed to store and allow access to the initial conditions in response to a protocol;

c. a processor connected to the protocol responsive memory, the processor programmed to;

(1) execute the protocol;

(2) receive and store the initial conditions returned by the protocol responsive memory;

(3) insert the initial conditions into and iterate a set of chaotic equations to arrive at iterated results;

(4) create encrypted data by combining data with the iterated results;

d. a memory partition for storing a read password; and

e. a comparator, connected to the memory partition and to the processor, for executing the protocol.

17. The system of claim 16, wherein the protocol comprises:

a. a check password generated by the processor and transmitted to the comparator;

b. a comparison of the check password to the read password by the comparator;

c. transmission of the initial conditions by the protocol responsive memory to the processor if the comparison results in a match between the check password and the read password.

18. A data encryption system comprising:

a. a predetermined set of valid initial conditions;

b. a protocol responsive memory programmed to store and allow access to the initial conditions in response to a protocol;

c. a processor connected to the protocol responsive memory, the processor programmed to;

(1) execute the protocol;

(2) receive and store the initial conditions returned by the protocol responsive memory;

(3) insert the initial conditions into and iterate a set of chaotic equations to arrive at iterated results;

(4) create encrypted data by combining data with the iterated results;

d. generation of an address by the processor;

e. transmission of the address by the processor to the protocol responsive memory; and

f. transmission of the initial conditions by the protocol responsive memory to the processor upon receipt of the address.

19. A data encryption system comprising:

a. a predetermined set of valid initial conditions;

b. a protocol responsive memory programmed to store and allow access to the initial conditions in response to a protocol;

c. a processor connected to the protocol responsive memory, the processor programmed to;

(1) execute the protocol;

(2) receive and store the initial conditions returned by the protocol responsive memory;

(3) insert the initial conditions into and iterate a set of chaotic equations to arrive at iterated results;

(4) create encrypted data by combining data with the iterated results; and

d. a memory connected to the processor to store and allow access to the iterated results.

20. The date encryption/decryption system of claim 19, wherein the processor connected to the protocol responsive memory is further programmed to create encrypted data by combining data with a preselected byte of the iterative results.

21. An encryption system comprising:

a. a first memory for holding data to be encrypted;

b. a second memory for holding encrypted data;

c. a slave processor having a memory for storage of initial conditions and a memory for storage of a password;

d. the slave processor programmed to release the initial conditions upon receipt of the password;

e. a master processor connected to the first and second memories and the slave processor, programmed to;

(1) send the password to the slave processor;

(2) store the initial conditions returned by the slave processor;

(3) iteratively apply the initial conditions to equations of the following type;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are iterated results of the equations given initial conditions x, y, z, σ

, r, and b;

(4) read data from the first memory;

(5) combine the data with the iterated results to create encrypted data; and

(6) store the encrypted data in the second memory.

22. The encryption system of claim 21, further comprising:

a. a third memory connected to the processor for holding the iterated results of the equations; and

b. the master processor being further programmed to insert the iterated results into the equations.

23. The encryption system of claim 21, wherein the initial conditions are in the range:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

24. The encryption system of claim 21, wherein the master processor is a member of the MOTOROLA 68000 family of microprocessors.

25. The encryption system of claim 21, wherein the master processor is a member of the INTEL IX86 family of microprocessors.

26. The encryption system of claim 21, wherein the slave processor is the XICOR X76041.

27. The encryption system of claim 21, wherein the processor is programmed to iterate the equations a predetermined number of times before combining the data with the iterated results.

28. The encryption system of claim 21, wherein the initial conditions are in the range:

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

29. A data encryption system comprising:

a. a first memory for holding data to be encrypted;

b. a second memory for holding encrypted data;

c. a third memory for holding a master processor password;

d. an input device;

e. a slave processor having a memory for storage of a serial number, a read password and a set of initial conditions;

f. the slave processor programmed to release the serial number when polled and to release the initial conditions only upon receipt of the read password;

g. a master processor connected to the first, second and third memories and the input device and removably connected to the slave processor;

h. the master processor programmed to;

(1) poll the slave processor and store the serial number returned by the slave processor in temporary memory;

(2) access the input device and store identification data input from the input device in temporary memory;

(3) access the third memory, retrieve the master processor password and store the master processor password in temporary memory;

(4) xor the serial number, the identification number and the master processor password and store the result in temporary memory as the read password;

(5) transmit the read password to the slave processor;

(6) read the initial conditions returned by the slave processor;

(7) apply the initial conditions to a set of chaotic equations;

(8) iterate the chaotic equations and store iterated results;

(9) read the data to be encrypted from the first memory;

(10) xor the data to be encrypted with the iterated results and store the result.

30. The system of claim 29, wherein the chaotic equations are of the form:

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and where x, y, z, σ

, b and r are initial conditions.

31. The system of claim 29, wherein the initial conditions comprise one parameter from each of the following ranges:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

32. The system of claim 29, where the input device is a keyboard.

33. The system of claim 29, where the input device is a card reader.

34. The system of claim 29, where the input device identifies a physical trait of the user to be identified.

35. The system of claim 29, wherein the identification data is a personal identification number.

36. The system of claim 29, wherein the slave processor is a XICOR X76041.

37. The system of claim 29, wherein the master processor is a member of the INTEL IX86 family of microprocessors.

38. The system of claim 29, wherein the master processor is a member of the MOTOROLA 68000 family of microprocessors.

39. The system of claim 29, wherein the master processor is further programmed to iterate the chaotic equations a preselected number of times before xoring the data to be encrypted with the iterated results.

40. The system of claim 29, wherein the master processor is further programmed to insert the stored iterated results into the chaotic equations before iterating the chaotic equations.

41. The system of claim 29, further comprising:

a. a nonvolatile memory means for storing the iterated results; and

b. the master processor further programmed to;

(1) poll the nonvolatile memory means and determine if iterated results have been stored there; and

(2) insert the iterated results into the chaotic equations before iterating chaotic equations if iterated results have been found in the nonvolatile memory.

42. The system of claim 29, wherein the initial conditions comprise one parameter from each of the following ranges:

x=-30 to 50

y=-30 to 50

z=-30 to 50

43. σ

=0. 006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

45. The method of claim 43, further comprising the step of:

a. iterating the chaotic equations a predetermined number of times before creating the cipher stream.

46. The method of claim 43, further comprising the steps of:

a. storing the numeric results; and

b. inserting the numeric results in the chaotic equations before iterating the chaotic equations.

47. The method of claim 43, wherein the step of selecting a set of parameters further comprises:

a. selecting a set of parameters from the following valid parameter space;

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

48. The method of claim 43, wherein the step of selecting a set of parameters further comprises:

a. selecting a set of parameters from the following valid parameter space;

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

50. The method of claim 48, further comprising the step of:

a. iterating the chaotic equations a preselected number of times before xoring the cipher number with the data.

51. The method of claim 48, wherein the predetermined seed space further comprises one value from each of the following ranges:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

52. The method of claim 48, further comprising the steps of:

a. storing the numeric results; and

b. inserting the numeric results into the chaotic equations before iterating the chaotic equations.

53. The method of claim 48, wherein the predetermined seed space further comprises one value from each of the following ranges:

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

55. The data decryption system of claim 53, further comprising a transmitter connected to the processor for transmitting the decrypted data.

56. The data decryption system of claim 53, further comprising a second memory connected to the processor to store the decrypted data.

57. The system of claim 53, wherein the set of chaotic equations are of the form:

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are iterated results of initial conditions x, y, z, σ

, r, and b.

58. The system of claim 53, wherein the predetermined valid initial conditions are in the range:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

60. The system of claim 58, wherein the processor is programmed to iterate the equations a predetermined number of times before combining the data with the iterated results.

61. The system of claim 58, wherein the processor is programmed to iterate the equations a predetermined number of time before xoring the data with the iterate results.

63. The system of claim 61, wherein the chaotic equations are of the form:

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are numeric results and where x, y, z, σ

, b and r are initial conditions.

64. The system of claim 61, wherein the initial conditions comprise one parameter from each of the following ranges:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

65. The system of claim 61, wherein the master processor is further programmed to iterate the chaotic equations a preselected number of times before xoring the data to be decrypted with the iterated results.

66. The system of claim 61, wherein the initial conditions comprise one parameter from each of the following ranges:

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

68. The method of claim 66, further comprising the step of iterating the chaotic equation a predetermined number of times before creating the cipher stream.

69. The method of claim 66, wherein the step of selecting a set of parameters further comprises:

a. selecting a set of parameters from the following valid parameter space;

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

71. The method of claim 69, further comprising the step of:

a. iterating the chaotic equations a number of times before xoring the cipher number with the data.

72. The method of claim 69, wherein the predetermined seed space further comprises one value from each of the following ranges:

x=about -30 to about 50, excluding 0

y=about -30 to about 50, excluding 0

z=about -30 to about 50, excluding 0

σ

=about 0.006 to about 0.028

b=about 24 to about 40

r=about 1 to about 8.

73. The method of claim 69, wherein the predetermined seed space further comprises one value from each of the following ranges:

x=-30 to 50

y=-30 to 50

z=-30 to 50

σ

=0.006 to 0.028

r=24 to 40

b=1 to 8

where x, y and z cannot simultaneously be zero.

44. A method of encrypting data comprising of the steps of:

a. selecting a set of parameters from a predetermined valid parameter space;

b. inserting the parameters into the chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x, y, z, σ

, r and b are parameters and x'"'"', y'"'"' and z'"'"' are numeric results;

c. iterating the chaotic equations to derive a x'"'"' iterate stream, a y'"'"' iterate stream and a z'"'"' iterate stream;

d. creating a cipher stream by choosing a segment of one or more iterate streams; and

encrypting data by combining the cipher stream with the data.

49. A method of encrypting data comprising the steps of:

a. selecting a set of parameters from a predetermined seed space;

b. inserting the parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x, y, z, σ

, b and r are parameters and x'"'"', y'"'"' and z'"'"' are numeric results;

c. iterating the chaotic equations to derive are x'"'"' iterate stream, a y'"'"' iterate stream and a z'"'"' iterate stream;

d. xoring the x'"'"' iterate stream, the y'"'"' iterate stream and the z'"'"' iterate stream to obtain a cipher number;

e. xoring the cipher number with the data to create encrypted data.

54. A data decryption system comprising:

a. a predetermined set of valid initial conditions;

b. a protocol responsive memory programmed to store and allow access to the initial conditions in response to a protocol;

c. a processor connected to the protocol responsive memory, the processor programmed to;

(1) execute the protocol;

(2) receive and store the initial conditions returned by the protocol responsive memory;

(3) insert the initial conditions into and iterate a set of chaotic equations to arrive at iterated results; and

(4) create decrypted data by xoring the data with the iterated results.

59. An encryption system comprising:

a. a first memory for holding data to be decrypted;

b. a second memory for holding decrypted data;

c. a slave processor having a memory for storage of initial conditions and a memory storage of a password;

d. the slave processor programmed to release the initial conditions upon receipt of the password;

e. a master processor connected to the first and second memories and the slave processor, programmed to;

(1) send the password to the slave processor;

(2) store the initial conditions returned by the slave processor;

(3) iteratively apply the initial conditions to equations of the following type;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x'"'"', y'"'"' and z'"'"' are iterated results of the equations given initial conditions x, y, z, σ

, r, and b;

(4) read data from the first memory;

(5) combine the data with the iterated results to create decrypted data; and

(6) store the decrypted data in the second memory.

62. A data decryption system comprising:

a. a first memory for holding data to be decrypted;

b. a second memory for holding decrypted data;

c. a third memory for holding a master processor password;

d. an input device;

e. a slave processor having a memory for storage of a serial number, a read password and a set of initial conditions;

f. the slave processor programmed to release the serial number when polled and to release the initial conditions only upon receipt of the read password;

g. a master processor connected to the first, second and third memories and the input device and removably connected to the slave processor;

h. the master processor programmed to;

(1) poll the slave processor and store the serial number returned by the slave processor in temporary memory;

(2) access the input device and store identification data input from the input device in temporary memory;

(3) access the third memory, retrieve the master processor password and store the master processor password in temporary memory;

(4) xor the serial number, the identification number and the master processor password and store the result in temporary memory as the read password;

(5) transmit the read password to the slave processor;

(6) read the initial conditions returned by the slave processor;

(7) apply the initial conditions to a set of chaotic equations;

(8) iterate the chaotic equations and store iterated results;

(9) read the data to be decrypted from the first memory;

(10) xor the data to be decrypted with the iterated results and store the results.

67. A method of decrypting data comprising of the steps of:

a. selecting a set of parameters from a predetermined valid parameter space;

b. inserting the parameters into the chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x, y, z, σ

, r and b are parameters and x'"'"', y'"'"' and z'"'"' are numeric results;

c. iterating the chaotic equations to derive a x'"'"' iterate stream, a y'"'"' iterate and a z'"'"' iterate stream;

d. creating a cipher stream by choosing a segment of one or more iterate streams; and

e. decrypting data by combining the cipher stream with the data.

70. A method of encrypting and decrypting data comprising the steps of:

a. selecting a set of parameters from a predetermined seed space;

b. inserting the parameters into chaotic equations of the form;

space="preserve" listing-type="equation">x'"'"'=-σ

x+σ

y

space="preserve" listing-type="equation">y'"'"'=rx-y-xz

space="preserve" listing-type="equation">z'"'"'=-bz+xy

where x, y, z, σ

, b and r are parameters and x'"'"', y'"'"' and z'"'"' are numeric results;

c. iterating the chaotic equations to derive are x'"'"' iterate stream, a y'"'"' iterate stream and a z'"'"' iterate stream;

d. xoring the x'"'"' iterate stream, the y'"'"' iterate stream and the z'"'"' iterate stream to obtain a cipher number;

e. encrypting data by xoring the cipher number with the unencrypted data; and

f. decrypting data by xoring the cipher number with the encrypted data.