Nonlinear feedback shift register and method of constructing the same

Abstract

The application provides a nonlinear feedback shift register and a construction method thereof, and the construction method comprises the following steps: S100, based on a cascade structure of Fibonacci type NFSR, an NFSR with a state diagram containing only four state circles is constructed; and S200, based on the NFSR with the state diagram containing only four state circles, a Galois type NFSR with a state diagram containing only one state circle and a maximum period is constructed. For the Galois type NFSR, the output sequence period is a factor of the circle length of the corresponding state circle, and how to ensure that the period is equal to the circle length is also a difficulty, and the application constructs the NFSR with only four state circles and the NFSR with only one state circle in the state diagram, and the periods of the NFSRs are equal to the corresponding circle lengths.

Description

Technical Field

[0001] This invention relates to a nonlinear feedback shift register, and more specifically to a nonlinear feedback shift register and its construction method. Background Technology

[0002] The use of nonlinear feedback shift registers (NFSRs) as the main component has become a mainstream design trend in stream cipher algorithms. However, due to a lack of effective tools, NFSR theory remains incomplete. Based on their implementation structure, NFSRs are generally classified into Fibonacci-type NFSRs and Galois-type NFSRs. If the state graph of an NFSR contains only cycles, it is called nonsingular. The cycle structure of a nonsingular NFSR refers to the number of state cycles in its state graph and the length of each cycle. Clearly, the cycle structure of an NFSR is a fundamental property and also a challenging aspect of NFSR design.

[0003] Currently, the loop structures of pure cyclic shift registers, complementary cyclic shift registers, and a class of symmetric NFSRs can be completely determined, but the loop structure of general NFSRs is difficult to predict. The upper bound on the number of loops in the state diagram of an n-level non-singular Fibonacci NFSR is:

[0004]

[0005] Here, d is a factor of n, and Φ(d) is the number of positive integers 1, 2, ..., d that are relatively prime to d.

[0006] The feedback function is The parity of the number of state cycles in the state diagram of an n-order Fibonacci type NFSR is the same as the parity of the Hamming weight of the Boolean function g. When the Boolean function g is a symmetric function, the Fibonacci type NFSR is called a symmetric NFSR. The state diagram of an n-order symmetric NFSR contains at least... There are several state cycles, but determining the length of each cycle is difficult.

[0007] The above are all research results on Fibonacci-type NFSRs. Compared with Fibonacci-type NFSRs, there are very few research results on the loop structure of Galois-type NFSRs. Summary of the Invention

[0008] The present invention aims to provide a Galois-type nonlinear feedback shift register and its construction method.

[0009] This invention provides a method for constructing a nonlinear feedback shift register, comprising:

[0010] S100, based on the cascaded structure of Fibonacci-type NFSR, constructs an NFSR with a state diagram containing only 4 state cycles;

[0011] S200, based on the NFSR whose state diagram contains only 4 state cycles, construct a Galois-type NFSR whose state diagram contains only 1 state cycle and whose period can reach the maximum.

[0012] Furthermore, the cascaded structure of the Fibonacci type NFSR is a cascaded structure of two Fibonacci type NFSRs.

[0013] Furthermore, the cascaded structure includes: a control-end nonlinear feedback shift register NFSR2 cascaded to a controlled-end nonlinear feedback shift register NFSR1, wherein the output sequence of the control-end nonlinear feedback shift register NFSR2 directly participates in the update of the tail register of the controlled-end nonlinear feedback shift register NFSR1 in the form of XOR, and at this time, the output sequence of the head register of the controlled-end nonlinear feedback shift register NFSR1 is the output of the cascaded structure.

[0014] In some embodiments, step S100 includes:

[0015] S101, Select the nonlinear feedback shift register NFSR1 of the controlled end. The nonlinear feedback shift register NFSR1 of the controlled end is an m-level non-singular second-largest period Fibonacci type NFSR, and its feedback function g satisfies g(1,1,…,1)=1;m≥2;

[0016] S102, Select the control-end nonlinear feedback shift register NFSR2. This control-end nonlinear feedback shift register NFSR2 is a level 1 Fibonacci type NFSR, and its feedback function is:

[0017] S103, the controlled-end nonlinear feedback shift register NFSR1 is cascaded to the control-end nonlinear feedback shift register NFSR2 to obtain an m+1 level NFSR, whose state diagram contains only 4 state circles.

[0018] Furthermore, the feedback function of the m+1 level NFSR is:

[0019]

[0020] Where g is a feedback function of an m-level non-singular sub-large periodic Fibonacci type NFSR and satisfies g(1,1,…,1)=1.

[0021] In some embodiments, step S200 includes:

[0022] S201, Select an NFSR constructed in step S100, denoted as NFSR3, and denote the four state cycles in its state diagram as C1, C2, C3, C4, where state cycle C1 consists of state a = [1 1…1] T The state cycle C2 is composed of states b = [0 1…1]. T The state structure is as follows: state c = [1 0 1…1] on state cycle C3. T The successor is d = [0 1 1…1 0]. T State e = [0 01…1] on state cycle C4 T The successor is f = [1 1 1…1 0] T ;

[0023] S202, swap the successors of states a and b to merge state loops C1 and C2 into a single state loop C of length 2. ′ 12 ;

[0024] S203, swapping the successors of state c and state e merges state loops C3 and C4 into a single loop of length 2. m+1 -2 state cycle C ′ 34 ;

[0025] S204, Exchange C ′ 12 State b and state cycle C ′ 34 The successor of state c will be the state circle C. ′ 12 and state circle C ′ 34 Merge into a single unit of length 2 m+1 State circle C ′ 1234 The m+1 level NFSR3 is equivalent to an m+1 level Galois type NFSR, whose state diagram is given by state cyclic C. ′ 1234 constitute.

[0026] Furthermore, the feedback function of NFSR3 at level m+1 is:

[0027]

[0028] Where h(X) = X3X4…X m+1 , g is a feedback function of an m-level nonsingular sub-large periodic Fibonacci type NFSR and satisfies g(1,1,…,1)=1.

[0029] Secondly, the present invention also provides a nonlinear feedback shift register, wherein the nonlinear feedback shift register is a nonlinear feedback shift register constructed by step S100 in the above-described method for constructing a nonlinear feedback shift register.

[0030] Thirdly, the present invention also provides a nonlinear feedback shift register, wherein the nonlinear feedback shift register is a nonlinear feedback shift register constructed by step S200 in the above-described method for constructing a nonlinear feedback shift register.

[0031] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0032] It should be noted that the circle structure of an NFSR is a fundamental property of NFSRs, reflecting the periodicity of the NFSR's output sequence. One of the security principles for designing NFSR-based stream cipher algorithms is that the NFSR used must have a large period. Analyzing the circle structure of a given NFSR is difficult; in particular, for Galois-type NFSRs, the period of its output sequence is a factor of the length of the corresponding state circle, and ensuring that the period equals the circle length is also a challenge. This invention constructs an NFSR with a state diagram containing only 4 state circles based on the cascaded structure of Fibonacci-type NFSRs; then, based on the NFSR with a state diagram containing only 4 state circles, it constructs a Galois-type NFSR with a state diagram containing only 1 state circle. The periods of both are equal to the corresponding circle lengths, and the period of the Galois-type NFSR can reach its maximum. Attached Figure Description

[0033] Figure 1 This is a flowchart illustrating the construction method of the nonlinear feedback shift register in an embodiment of the present invention.

[0034] Figure 2 This is a schematic diagram of the cascaded structure of a Fibonacci-type NFSR in an embodiment of the present invention.

[0035] Figure 3 This is a schematic diagram of the process of constructing a Galois-type NFSR with only one state cycle and the maximum period in an embodiment of the present invention. Detailed Implementation

[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0037] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0038] This invention provides a method for constructing a nonlinear feedback shift register, comprising:

[0039] S100, based on the cascaded structure of Fibonacci-type NFSR, constructs an NFSR with a state diagram containing only 4 state cycles;

[0040] S200, based on the NFSR whose state diagram contains only 4 state cycles, construct a Galois-type NFSR whose state diagram contains only 1 state cycle and whose period can reach the maximum.

[0041] In some embodiments, step S100 is implemented as follows:

[0042] like Figure 2 As shown, the cascaded structure of the Fibonacci type NFSR is a cascaded structure of two Fibonacci type NFSRs. Specifically, the control-end nonlinear feedback shift register NFSR2 is cascaded to the controlled-end nonlinear feedback shift register NFSR1. The output sequence of the control-end nonlinear feedback shift register NFSR2 directly participates in the update of the tail register of the controlled-end nonlinear feedback shift register NFSR1 in the form of XOR. At this time, the output sequence of the controlled-end head register is the output of this cascaded structure.

[0043] The cascaded structure based on the Fibonacci-type NFSR constructs an NFSR with a state diagram containing only 4 state cycles, specifically including:

[0044] S101, Select the nonlinear feedback shift register NFSR1 of the controlled end. The nonlinear feedback shift register NFSR1 of the controlled end is an m (m≥2) level non-singular second-largest period Fibonacci type NFSR, and its feedback function g satisfies g(1,1,…,1)=1;

[0045] S102, Select the control-end nonlinear feedback shift register NFSR2. This control-end nonlinear feedback shift register NFSR2 is a level 1 Fibonacci type NFSR, and its feedback function is:

[0046] S103, the controlled-end nonlinear feedback shift register NFSR1 is cascaded to the control-end nonlinear feedback shift register NFSR2 to obtain an m+1 level NFSR, whose state diagram contains only 4 state circles.

[0047] The following is a detailed explanation: Because the Fibonacci-type controlled-end nonlinear feedback shift register NFSR1 is a non-singular second-largest-cycle NFSR, it can output a sequence of all ones and a sequence with a period of 2. m -1 sequence. If the output sequence of the controlled nonlinear feedback shift register NFSR1 is an all-one sequence, that is, the input of the Fibonacci type second nonlinear feedback shift register NFSR2 is always one, then it can be directly calculated that the state diagram of this m+1 level NFSR has two states respectively. and The resulting loops C1 and C2 are of length 1. If the output of the controlled nonlinear feedback shift register NFSR1 has a period of 2... m Given a sequence of -1, according to reference [1], the period of the output sequence of this cascaded structure is α(2). m -1). Clearly, α = 1 or α = 2. Assuming α = 2, then the remainder is 2. m+1 -2 states will form a string of length 2 m+1 -2 loops. Adding loops C1 and C2, the state diagram of this cascaded structure consists of 3 state loops, which contradicts the statement in reference [2] that the state diagram of a cascaded structure consists of an even number of state loops. Therefore, α = 1, meaning the remaining 2 m+1 -2 states will form two pairs of length 2 m A circle of -1.

[0048] In summary, the m+1 level NFSR constructed according to this method contains only 4 state cycles, and the feedback function of the m+1 level NFSR is:

[0049]

[0050] Where g is a feedback function of an m-level non-singular sub-large periodic Fibonacci type NFSR and satisfies g(1,1,…,1)=1.

[0051] In some embodiments, step S200 is implemented as follows:

[0052] like Figure 3 As shown, based on the NFSR whose state diagram contains only 4 state cycles, a Galois-type NFSR with a state diagram containing only 1 state cycle and the maximum period is constructed, i.e., containing only 1 state cycle and a period of 2. m+1 The m+1 level NFSR specifically includes:

[0053] S201, Select an NFSR constructed in step S100, denoted as NFSR3, and denote the four state cycles in its state diagram as C1, C2, C3, C4, where state cycle C1 consists of state a = [1 1…1] T The state cycle C2 is composed of states b = [0 1…1]. T The state structure is as follows: state c = [1 0 1…1] on state cycle C3. T The successor is d = [0 1 1…1 0]. T State e = [0 01…1] on state cycle C4 T The successor is f = [1 1 1…1 0] T ;

[0054] S202, swap the successors of states a and b to merge state loops C1 and C2 into a single state loop C of length 2. ′ 12 ;

[0055] S203, swapping the successors of state c and state e merges state loops C3 and C4 into a single loop of length 2. m+1 -2 state cycle C ′ 34 ;

[0056] S204, Exchange C ′ 12 State b and state cycle C ′ 34 The successor of state c will be the state circle C. ′ 12 and state circle C ′ 34 Merge into a single unit of length 2 m+1 State circle C ′ 1234 The resulting m+1 level NFSR3 is equivalent to an m+1 level Galois type NFSR, and for any positive integer m≥6, we have Therefore, according to reference [3], the NFSR designed by the above steps S201 to S204 is a Galois-type NFSR with a maximum period, and its state diagram consists of state circle C. ′ 1234 constitute.

[0057] The feedback function of NFSR3 at level m+1 is:

[0058]

[0059] Where h(X) = X3X4…X m+1, g is a feedback function of an m-level nonsingular sub-large periodic Fibonacci type NFSR and satisfies g(1,1,…,1)=1.

[0060] Literature [1] Hu H, Gong G. Periods on two kinds of nonlinear feedback shiftregisters with time varying feedback functions. International Journal of Foundations of Computer Science, 2011, 22(6): 1317-1329. DOI: 10.1142 / S0129054111008738.

[0061] Literature [2] Wang Z, Zheng Q, Zhao XX, et al. Grain-like structures with minimal and maximal period sequences. Designs, Codes and Cryptography, 2021, 89: 679-693. DOI: 10.1007 / s10623-020-00839-3.

[0062] Literature [3] Yingyin Pan, Jianghua Zhong, Dongdai Lin. Generalized cyclejoining method and its application to the construction of long-period GaloisNFSRs, Design, Codes and Cryptography (2024). DOI: 10.1007 / s10623-024-01500-z.

[0063] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for constructing a nonlinear feedback shift register, characterized in that, include: S100, based on the cascaded structure of Fibonacci-type NFSR, constructs an NFSR with a state diagram containing only 4 state cycles; S200, based on the NFSR whose state diagram contains only 4 state cycles, construct a Galois-type NFSR whose state diagram contains only 1 state cycle and whose period can reach the maximum. Step S200 includes: S201, Select an NFSR constructed in step S100, denoted as NFSR3, and denote the four state cycles in its state diagram as follows: , , , Among them, the state circle From state Composition, state circle From state Composition, state circle upper state The successor to State Circle upper state The successor to ; S202, Exchange State and state The successor will be the state circle and state circle Merge into a state cycle of length 2 ; S203, Exchange state and state The successor will be the state circle and state circle Merge into a length of State Circle ; S204, Exchange medium state and state circle medium state The successor will be the state circle and state circle Merge into a long State Circle The m+1 level NFSR3 is equivalent to an m+1 level Galois type NFSR, whose state diagram is composed of state cycles. constitute.

2. The method for constructing a nonlinear feedback shift register according to claim 1, characterized in that, The cascaded structure of the Fibonacci type NFSR is a cascaded structure of two Fibonacci type NFSRs.

3. The method for constructing a nonlinear feedback shift register according to claim 2, characterized in that, The cascaded structure includes: a control-end nonlinear feedback shift register NFSR2 cascaded to a controlled-end nonlinear feedback shift register NFSR1, wherein the output sequence of the control-end nonlinear feedback shift register NFSR2 directly participates in the update of the tail register of the controlled-end nonlinear feedback shift register NFSR1 in the form of XOR, and at this time the output sequence of the head register of the controlled-end nonlinear feedback shift register NFSR1 is the output of the cascaded structure.

4. The method for constructing a nonlinear feedback shift register according to claim 3, characterized in that, Step S100 includes: S101, Select the controlled-end nonlinear feedback shift register NFSR1. This controlled-end nonlinear feedback shift register NFSR1 is an m-level non-singular second-largest period Fibonacci type NFSR, and its feedback function is... satisfy m≥2; S102, Select the control-end nonlinear feedback shift register NFSR2. This control-end nonlinear feedback shift register NFSR2 is a level 1 Fibonacci type NFSR, and its feedback function is: ; S103, the controlled-end nonlinear feedback shift register NFSR1 is cascaded to the control-end nonlinear feedback shift register NFSR2 to obtain an m+1 level NFSR, whose state diagram contains only 4 state circles.

5. The method for constructing a nonlinear feedback shift register according to claim 4, characterized in that, The feedback function of the m+1 level NFSR is: in, It is a feedback function of an m-order nonsingular sub-large periodic Fibonacci type NFSR and satisfies .

6. The method for constructing a nonlinear feedback shift register according to claim 4, characterized in that, The feedback function of NFSR3 at level m+1 is: in, It is a feedback function of an m-order nonsingular sub-large periodic Fibonacci type NFSR and satisfies .

7. A nonlinear feedback shift register, characterized in that, The nonlinear feedback shift register is a nonlinear feedback shift register constructed by step S100 in the construction method of the nonlinear feedback shift register as described in any one of claims 1-5.

8. A nonlinear feedback shift register, characterized in that, The nonlinear feedback shift register is a nonlinear feedback shift register constructed by step S200 in the construction method of the nonlinear feedback shift register as described in any one of claims 1-6.

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