Coding and decoding method and system for decoding by combining detection codes with twisted RS (Reed-Solomon) code list

Abstract

The invention belongs to but is not limited to the technical field of communication coding, and particularly relates to a coding and decoding method and system for decoding a detection code combined twisted RS (Reed-Solomon) code list, comprising a coding method for embedding an algebraic operation detection mechanism (AMD code) into a twisted Reed-Solomon code structure, and a decoding system which is constructed based on the coding method and can support list decoding. According to the method, a twisted RS code with a disturbance item is constructed to enhance the structural security, a key-free AMD detection mechanism is fused, candidate results are subjected to tampering identification and screening in a list decoding stage, and a fused communication coding and decoding technology with high reliability, high security and high decoding efficiency is realized. The system can be widely applied to the fields of data storage, distributed communication, block chain base layer protocols, anti-pollution network coding and the like with high security requirements.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of communication coding, and in particular relates to a coding and decoding method and system for detecting code combined twisted RS code list decoding. BACKGROUND

[0002] In information transmission and data storage systems, error-correcting codes (ECC) are a class of core technologies, whose goal is to realize reliable restoration of the original information in the presence of noise or attacks in the channel. Error-correcting codes map k-dimensional information messages into n-dimensional code words (usually vectors over a finite field GF(q)) by introducing redundancy to enhance data recovery capability. One of the key indicators is the minimum Hamming distance d, which measures the minimum number of different bits between two legal code words. The classical unique decoding theorem points out that if the number of error bits is less than d / 2, the original information can be uniquely determined. However, in the face of more errors or targeted tampering by the enemy, the traditional unique decoding mechanism will fail.

[0003] To break through the limitations of the traditional decoding radius, list decoding technology has emerged. In this framework, the receiving end allows to output a finite size of code word candidate set, as long as it contains the correct message. Formally, if a class of linear codes has at most l code words within a radius of p n for any received vector, we call it (p, l)-list decodable. The rise of this theory has promoted the use of classic MDS codes including Reed-Solomon codes in a wider error scenario.

[0004] Among all linear codes, maximum distance separable codes (MDS codes) are the most ideal class. They achieve the Singleton bound, i.e., d = n-k+1, providing the maximum error correction or erasure recovery capability under the same redundancy length. Reed-Solomon (RS) codes are the most famous MDS codes, which are widely used in communication, distributed storage, optical discs, and other high-reliability scenarios. Due to its good structure and efficient decoding method, RS code is still one of the main coding schemes in the industry.

[0005] ​Although traditional RS codes have excellent error correction capability, their structure is single, and there are limitations in some security or flexibility requirements. In recent years, twisted Reed-Solomon codes (Twisted RS codes) have been proposed, which are an extended code class based on RS codes by introducing a disturbance structure. This class of codes constructs a more general evaluation code by adding a "twisted term" to the encoding polynomial. Its generator matrix is no longer a simple Vandermonde structure, but contains a nonlinear disturbance at a specific position. Formally, this coding method allows the introduction of disturbance terms in the information symbols, effectively increasing the degree of freedom of constructing MDS codes.

[0006] One of the advantages of twisted RS codes is that while retaining the advantages of RS codes, a large number of new MDS code families that are not equivalent to the classical RS code structure can be constructed. This has practical significance for preventing structural attacks such as structural identification attacks in cryptography. In addition, when the parameters are set reasonably, twisted RS codes can still be reduced to RS decoding problems through transformation, retaining the efficiency of existing decoding methods. Current research has constructed many MDS twisted RS codes with shorter code length, small Hull dimension, and good Schur square property.

[0007] However, compared to the mature decoding method research of traditional RS codes, the decoding method research results of twisted RS codes are still scarce. Due to the twisted term of twisted RS codes, many existing RS code decoding methods cannot be naturally extended to twisted RS codes. Although some decoding methods have been proposed for twisted RS codes, these algorithms have high complexity and the error correction capability is still not ideal.

[0008] In the coding model against active attackers, algebraic manipulation detection codes (AMD codes) provide a novel protection method. This class of codes is not aimed at error correction, but is committed to detecting algebraic tampering of ciphertexts by the enemy. The typical construction of AMD codes includes a randomized encoder and a deterministic decoder, whose goal is to discover any non-trivial tampering operation with high probability.

[0009] AMD codes were first formally defined by Cramer et al. Their main applications include robust secret sharing, fuzzy extractors, and keyless authentication mechanisms. Unlike authentication codes based on hash or signature, AMD codes do not rely on keys, making them particularly suitable for restricted environments or highly interactive systems.

[0010] He et al. published in 2025, The error-correcting pair for several classes of NMDS linear codes. This paper takes the twisted generalized Reed-Solomon (TGRS) code as a representative, and constructs an error-correcting pair that meets certain parameter constraints in the NMDS code family An algebraic decoding algorithm that can correct ≤ bit errors in polynomial time is presented. The detailed steps for instantiating error-correcting pairs using TGRS codes and performance analysis are provided, demonstrating that this class of codes has a unique decoding method that can be implemented while maintaining (near) MDS distance. However, this technology still has three limitations: first, the algorithm belongs to the unique decoding paradigm, and its error correction radius is limited to It is not available when facing higher error rates or deliberate tampering by an adversary; second, the decoding process needs to enumerate and verify sub-code pairs that meet multiple constraints, and the computational complexity rapidly expands with code length and parameter set, making it difficult to implement in real-time high-speed scenarios; third, the scheme does not introduce AMD codes or other algebraic manipulation detection mechanisms. When the decoder outputs a candidate set or is subjected to algebraic attacks, there is no means to determine the unique correct message, and it cannot meet the needs of high-security, strong anti-interference transmission-storage applications.

[0011] In summary, the existing technology has the following problems and deficiencies:

[0012] (1) Limited error correction radius, insufficient resistance to high error rates and malicious tampering: traditional unique decoding schemes (including the -error-correcting pair method described by He et al.) can only guarantee success when Once the number of errors or the number of tampered bits exceeds this threshold, the original information cannot be recovered or verified, and the safety margin is limited.

[0013] (2) Limited and complex decoding research for twisted RS codes: most existing decoding algorithms for twisted RS codes are directly modified from classical RS decoding or rely on enumerating / searching sub-code pairs. The algorithm complexity rapidly increases with code length and parameters, making it unsuitable for high-throughput or real-time scenarios.

[0014] (3) Lack of "post-decoding judgment" mechanism, unable to uniquely lock the correct message from the list: list decoding can expand the error correction radius, but existing schemes generally lack low-cost candidate screening methods. When the list size is > 1 or subjected to algebraic / structured attacks, the decoder cannot uniquely determine the legal code word, reducing system reliability and security.

[0015] (4) Without the function of algebraic manipulation detection (AMD), it is vulnerable to active attacks: existing literatures focus on error correction performance and ignore the detection ability of malicious "algebraic tampering"; in the environment without key or low trust, if there is no lightweight tampering detection mechanism such as AMD code, the system is difficult to defend against precise attacks on the coding structure. SUMMARY

[0016] In view of the problems in the prior art, the application provides a coding and decoding method and system of an algebraic manipulation detection code combined with twisted RS code list decoding.

[0017] The application is implemented as follows: a coding and decoding method of a detection code combined with twisted RS code list decoding, comprising:

[0018] Applying the to-be-transmitted information to the system AMD code to obtain a first code word;

[0019] Taking the first code word as a message vector, taking a twisted RS code with parameters [n, r+2] and performing twisted RS code encoding to generate a second code word;

[0020] Receiving a received vector of the second code word after passing through a channel, calling a twisted RS code list decoder and outputting a candidate code word list within a preset decoding radius;

[0021] Performing twisted RS structure checking on each candidate code word in the candidate code word list, and the code word meeting the twisted RS polynomial constraint enters subsequent processing;

[0022] Performing AMD code decoding on each candidate code word passing through the structure checking, and if only one decoding succeeds, the corresponding information is output; if the number of decoding successes is more than one or is zero, it is determined that the decoding fails.

[0023] Further, the AMD code encoding comprises:

[0024] Randomly selecting a non-zero element η in a finite field .

[0025] Constructing a polynomial f(x) with a to-be-transmitted information vector s1 to s r and a random element η.

[0026] The code word after AMD encoding is m=(m0,…,m r+1 )=(s,η,f(η).

[0027] Further, the twisted RS code encoding satisfies the following conditions:

[0028] Selecting an assignment point set {α1,α2,…,α n} belonging to a multiplicative commutative subgroup of a finite field and being different from each other and non-zero; the {α1,α2,…,αn}∈G,α i ≠0, and satisfies

[0029] The code word of the twisted RS code after encoding is c=(c1,…,c n )=(f(α1),…,f(α n )).

[0030] Further, the list decoding specifically includes:

[0031] Suppose that the received code word is r=(r1,…,r n ), input into the list decoder of the twisted RS code, and the size of the decoding list is the decoding list

[0032] Further, the structure check of the twisted RS code specifically includes:

[0033] Traverse For a candidate code word, such as m i =(m 1,1 ,…,m 1,r+3 ), judge whether m i,r+3 is equal to m i,1 *η, if yes, proceed to the next step of AMD decoding, otherwise decoding fails.

[0034] Further, the input of the traversing decoder, the decoding of the AMD code specifically includes:

[0035] If the candidate code word meets the decoding constraint of the AMD code, the decoding succeeds, and the candidate code word is the original information, if there are multiple candidate code words meeting the condition, the decoding is considered to fail, and the specific steps are as follows: suppose that m1=(m 1,1 ,…,m 1,r+3 ) is a candidate code word passing the structure check, calculate Judge whether f′ is equal to m 1,r+2 , if more than one candidate code word meets the condition, the decoding of the AMD code is considered to fail. Otherwise, the unique decoding result is taken as the output of the decoder.

[0036] Another object of the present application is to provide an encoding and decoding system of algebraic manipulation detection combined with twisted RS code list decoding, comprising:

[0037] An AMD code encoding module: encoding information using a systematic AMD code;

[0038] A twisted RS code encoding module: encoding the encoding result of the AMD code;

[0039] Twisted RS code list decoding module: list decoding the received code word and outputting a list containing possible decoding results;

[0040] Twisted RS code structure checking module: traversing the list decoding result, and performing structure checking on each candidate code word, and taking the code word satisfying the condition as the input of the AMD decoding module;

[0041] AMD code decoding module: performing AMD code decoding on each input code word, and if only one code word is successfully decoded, the decoding result is the original information, and otherwise, the decoding fails.

[0042] Further, the AMD code encoding module randomly selects a non-zero element r in a finite field Fq;

[0043] According to the information vector to be transmitted and the random element r, a message polynomial is constructed;

[0044] By evaluating the message polynomial at a set of predetermined evaluation points, the first code word is generated.

[0045] Further, the AMD code decoding module traverses For a candidate code word, such as m i =(m 1,1 ,…,m 1,r+3 ), it is judged whether m i,r+3 is equal to m i,1 *η, if yes, the next step of AMD decoding is performed, otherwise, the decoding fails.

[0046] Another object of the present application is to provide a computer readable storage medium having instructions stored thereon, which, when executed by a processor, cause the processor to operate according to the steps of the encoding and decoding method of detecting code combined with twisted RS code list decoding.

[0047] In combination with the above technical solutions and the technical problems solved, the technical solution to be protected by the present application has the following advantages and positive effects:

[0048] Firstly, the present application uses the way of algebraic manipulation detection code combined with twisted RS code list decoding, which avoids the decoding uncertainty caused by the twisted term of twisted RS code and improves the decoding radius of twisted RS code. The key advantages of this method include:

[0049] 1. Significantly enhances error detection and correction capability: the algorithm combines AMD code with twisted RS code list decoding mechanism, and breaks through the limitation of traditional unique decoding radius (half of the minimum distance d) under the probabilistic computing framework. Even in the case of transmission error number e>d / 2, the algorithm can still work effectively, and can correct up to The probability of effectively detecting errors and outputting unique decoding results is increased, thereby significantly enhancing the overall error detection and correction capability of the system. Figure 1 The number of decoding successes and failures is shown when the number of input code words is 1000, the domain size q=2 8 =256, the redundancy r=131, the AMD tag length l=5, the code length n=255, and the number of generated errors is within the error correction capability of the code words.

[0050] 2. Construction flexibility and structural non-equivalence: The twisted RS code breaks the linear structure constraint of the traditional RS code based on the Vandermonde matrix by introducing a nonlinear disturbance term (twist term), and can systematically construct a large-scale MDS code family with structural non-equivalence. This feature effectively avoids algebraic attacks (such as polynomial recovery attacks in secret sharing schemes and structural identification attacks in encryption algorithms) against standard RS structures, while supporting dynamic configuration of twist parameters according to security requirements to realize customizable encoding and decoding architecture.

[0051] 3. Anti-algebraic tampering and security enhancement mechanism: The fusion of the probabilistic encoding mechanism of the AMD code can detect any algebraic operation type tampering (such as malicious modification on the additive group) with a provable high probability under the threat model where the enemy fully knows the encoding structure. Its keyless feature supports stateless deployment, which is suitable for resource-constrained devices or high-concurrency communication scenarios; when encountering active attacks (data pollution, malicious injection), the system can trigger abnormal identification and block error propagation in real time, significantly improving system robustness.

[0052] Secondly, the present application uses a combination of AMD code and list decoding mechanism of twisted RS code, which not only enhances the error detection and correction capability but also has a wide range of application scenarios, achieving significant technological progress.

[0053] Thirdly, the expected income and commercial value of the technical solution of the present application after transformation are:

[0054] The information is placed in an extremely complex channel by using the combination of AMD code and RS list decoding, and the errors it faces may exceed half of the code distance. This encoding method allows the receiver to receive more than half of the code distance errors with a high probability, and to The probability of correctly obtaining unique original information can be enhanced, and the error detection and correction capabilities can be enhanced. The technology can be used in the fields of secure communication, sensitive information transmission and encryption.

[0055] The technical solution of the application fills the technical gap in the industry at home and abroad: at present, in the field of error correction coding and security mechanism fusion design, although RS code as the most classic MDS code has been widely used in communication, storage and other systems, its standard form still has limitations such as insufficient anti-interference ability and strong structure predictability when facing algebraic attack, malicious tampering and other scenes. Therefore, the academic circle proposes twisted RS code, which constructs a large-scale non-equivalent MDS code family by introducing a disturbance term. However, the twisted RS code generally has key technical bottlenecks such as complex structure, high computational overhead and difficulty in hardware implementation in the decoding process, especially the inability to directly compatible with the efficient decoder of the classic RS code, which limits its popularization in practical application.

[0056] On the other hand, although the existing tamper detection technology (such as authentication code or cryptography method) has certain security, it often relies on key management, communication interaction or introduces additional system burden, and is difficult to be directly applied to high-throughput error correction coding system. The algebraic operation detection code (AMD code) developed in recent years provides a lightweight solution for asymmetric tamper detection, but how to naturally integrate with the coding structure and improve its actual error correction ability in complex channels is still a key technical problem that has not been solved.

[0057] The application first proposes a list decoding scheme that deeply integrates twisted RS code structure and AMD code mechanism, which fundamentally solves the problem of difficult decoding of twisted RS code. The scheme not only bypasses the structural compatibility barrier caused by the twist term, realizes the reuse of the existing RS decoding framework, but also significantly improves the decoding performance and security strength by introducing list decoding and probability detection mechanism. At the same time, the method has low computational complexity, high decoding efficiency and is easy to implement in engineering, and there is no similar public technology at home and abroad that can meet the three requirements of “structural flexibility + decoding realizability + security detection ability” at the same time.

[0058] The technical scheme of the present application solves the technical problem that people have been eager to solve but have failed to obtain success: before this, due to the introduction of the twist term, the twisted RS code is no longer a simple extension of the traditional RS code in structure, and its decoding process cannot be directly reused in the decoding framework of the classic RS code. Although some decoding methods have been proposed to try to be compatible with the traditional RS decoder, these methods often have high computational complexity, close to the exhaustive brute-force decoding, and are not practical. In addition, some theoretical schemes rely on complex algebraic structures, which are difficult to efficiently implement in a hardware environment, and their supported decoding radius is limited, which limits the further improvement of decoding performance. The present application introduces the tamper detection mechanism of the AMD code, jointly designs with the twisted RS code, and adopts the list decoding strategy in the decoding stage, effectively bypassing the decoding obstacles brought by the twist term. This method not only can be compatible with the efficient list decoding framework of the existing RS code in structure, but also significantly enhances the error correction capability and tamper detection capability of the system. Compared with other twisted RS code decoding schemes, this scheme achieves a good balance between decoding efficiency, implementation complexity and security robustness, and is more suitable for engineering deployment and practical application.

[0059] Therefore, the present application fills the technical gap of the twisted RS code in the practical decoding mechanism, breaks through the fixed limit that the traditional error correction code lacks endogenous defense capability when facing active tampering, has distinct technical innovation and application leading nature, and has important academic and industrial value in the cross field of coding theory and security engineering. BRIEF DESCRIPTION OF DRAWINGS

[0060] Figure 1 is the number distribution of decoding success and failure when the input code word number is 1000, the domain size q=2 8 =256, the redundancy r=131, the AMD tag length l=5, the code length n=255, and the number of errors generated is within the error correction capability of the code word.

[0061] Figure 2 is the encoding and decoding system framework diagram of the algebraic manipulation detection code combined with the list decoding of the twisted RS code according to the embodiment of the present application. DETAILED DESCRIPTION

[0062] In order to make the purpose, technical scheme and advantages of the present application more clear and explicit, the present application will be further described in detail below in combination with embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application.

[0063] In existing communication and storage systems, facing channel interference, hardware defects or malicious tampering, codewords may not only suffer from random errors and burst errors, but also have the risk of being targetedly modified. Although traditional algebraic error-correcting codes such as RS codes have strong error-correcting ability, they are prone to decoding failure or misinterpretation in the case of errors beyond their error-correcting radius, especially in the presence of targeted tampering, the decoder may output an incorrect legal codeword, leading to the inability of the system to identify tampering risks at the application layer. This problem is particularly prominent in distributed storage, financial transaction data transmission, satellite communication and other scenarios that require high reliability and tamper detection.

[0064] To address the above challenges, the method introduces a cascade structure of system AMD (Algebraic Manipulation Detection) code and twisted RS (Twisted Reed-Solomon) code, combining tamper detection with strong error correction capability. First, the original information is encoded using the system AMD code, so that the codeword has the property of algebraic tamper detection, so that the data can be directly judged whether it has been maliciously modified during subsequent decoding. Then the output of the AMD code is used as the message vector of the twisted RS code, and the twisted RS encoding is performed by setting the [n, r+2] parameters, introducing additional structural constraints, improving the minimum distance and structural diversity of the codeword, and providing a basis for subsequent list decoding and structure screening.

[0065] At the receiving end, the twisted RS codeword transmitted through the noise channel will produce errors. The decoding process first outputs multiple candidate codewords within the preset radius by the twisted RS list decoder. The list decoder uses algebraic root-finding and polynomial interpolation methods to generate multiple potential correct solutions even when the unique decoding radius is exceeded, thereby expanding the error correction capability. This stage does not directly output information, but retains the candidate set for subsequent screening using structural information.

[0066] To ensure that only possible legal twisted RS codewords are retained in the candidate set, the method introduces a structure verification link. The verification process is based on the algebraic constraints of the twisted RS code generating polynomial and its twist parameters, and checks whether the candidate codeword satisfies the corresponding polynomial relationship one by one. The candidate that fails the verification is directly excluded, thereby excluding invalid or fake codewords due to the expansion of the decoding radius, and reducing the false positive probability of the AMD decoding stage.

[0067] In the code word set after structure screening, AMD code decoding operation is performed one by one, and whether tampering of the information domain occurs is judged by using the algebraic detection mechanism. AMD decoding can detect and reject the tampered result when the error is propagated to the information part by checking the specific check relationship. If only one candidate code word passes the AMD decoding and the check is successful, it is judged as the only correct solution and the original information is output. If the number of candidate codes passing the check is zero or more than one, it is considered that there is ambiguity or tampering that cannot be eliminated, and the decoding fails, avoiding the error information into the system.

[0068] The method realizes the fusion of error correction capability and tamper detection capability by introducing the AMD code detection mechanism and the twisted RS structure constraint in the traditional algebraic decoding framework. It not only improves the decoding success rate in a strong interference and malicious attack environment, but also significantly reduces the risk of misinterpretation. In the occasions such as satellite downlink, distributed storage node synchronization, blockchain consensus information exchange and the like which have very high requirements for data integrity and security, the method can significantly improve the fault tolerance and attack resistance of the system, while maintaining low decoding complexity and engineering realizability.

[0069] The application uses the algebraic manipulation detection code combined with the twisted RS code list decoding method to provide a more effective, flexible and secure twisted RS code decoding method, which specifically includes the following steps:

[0070] 1. Preprocessing stage: let n be a positive integer, take the twisted RS code with parameters [n, r+2], define the twisted polynomial as Take a random element Take information Wherein Indicates an affine space, and r indicates the length of the information. Let the multiplicative commutative group Take two different value point sets as {α1, α2, …, α n}∈G, α i ≠0, and satisfy Ensure that the obtained twisted RS code is an MDS code.

[0071] 2. AMD code encoding stage: encode the information s by using an approximately optimal AMD code construction method.

[0072] 3. The AMD code encoding result is taken as the coefficient of the RS code polynomial, a twist term is introduced, the RS code polynomial is further changed into a twisted RS code polynomial, the value point set is taken, and the twisted RS code encoding is performed.

[0073] 4. Twisted RS code list decoding stage at the receiving end: list decoding is performed on the received code word to obtain a candidate list.

[0074] 5. The receiving end twist RS code structure verification stage: first, the candidate list is subjected to structure verification, and the code word meeting the condition is input to the AMD code list decoder.

[0075] 6. The receiving end AMD code decoding stage: the candidate code word subjected to structure verification is traversed, and the information in the list is subjected to AMD code decoding.

[0076] Through the above steps, the decoding bottleneck caused by the twist term of the twist RS code can be avoided. The method not only can be compatible with the high-efficiency list decoding framework of the existing RS code in structure, but also can significantly enhance the error correction capability and tamper detection capability of the system. That is, when the number of errors exceeds half of the minimum distance, the unique original information can also be correctly obtained with a probability. The technology can be used in the fields of secure communication, sensitive information transmission and encryption.

[0077] The AMD code system code is encoded for the information m, and specifically includes the following steps.

[0078] Let n be a positive integer, take the twist RS code with parameters [n, r+2], define the twist polynomial as Take a random element Take the information Wherein represents an affine space, and r represents the length of the information. Let the multiplicative commutative group Take two mutually different assignment point sets as {α1, α2, …, α n}∈G, α i ≠0, and satisfy Ensure that the obtained twist RS code is an MDS code.

[0079] Calculate The AMD encoded code word is m=(m0, …, m r+1 )=(s, η, f(η)).

[0080] The RS code encoding of the AMD code encoding result is specifically as follows.

[0081] The twist RS code encoded code word is c=(c1, …, c n )=(f(α1), …, f(α n )).

[0082] The received code word is subjected to list decoding, and specifically includes the following steps.

[0083] Suppose that the received code word is r=(r1, …, r n ), which is input into the twist RS code list decoder, and a decoding list with a list size of is obtained.

[0084] The traversal above-mentioned decoding list, the information in the list is twisted RS code structure check, specifically including:

[0085] Traversal For candidate codeword, such as m i =(m 1,1 ,…,m 1,r+3 ), judge m i,r+3 Whether equal to m i,1 *η, if equal then the next step AMD decoding, otherwise decoding failure.

[0086] The traversal decoder input, AMD code decoding, specifically including:

[0087] If the candidate code word meets the decoding constraints of AMD, it is a decoding success, and the candidate code word is the original information, if there are multiple candidate code words that meet the condition, it is considered as decoding failure, the specific steps are: assuming m1=(m 1,1 ,…,m 1,r+3 ) is the candidate code word through the structure check, calculate Judge f′ whether equal to m 1,r+2 , if more than one candidate code word meets the condition, it is considered as AMD code decoding failure. Otherwise, the only decoding result is taken as the output of the decoder.

[0088] In high-reliability scenarios such as distributed storage arrays and deep space telemetry links, the unique decoding radius of the classic Reed-Solomon code is limited to When nodes are densely offline or channels are subject to interference peaks, the error rate often crosses this threshold, and engineers are forced to introduce additional copies or redundant links to maintain business continuity, resulting in a sharp rise in cost and delay. Existing industrial systems urgently need an encoding scheme that can maintain high recovery probability without increasing redundancy.

[0089] Twisted RS code provides the possibility of introducing structural diversity while maintaining MDS distance, which naturally raises the threshold of structural attacks based on pattern recognition. But the twist term destroys the recursive structure of traditional Berlekamp-Massey or Gao algorithm, directly leading to a significant increase in decoding cost; in hard real-time links, the processor often appears performance bottleneck due to multiple inverse and polynomial reorganization, this contradiction has long limited the industrialization of twisted RS code.

[0090] To recover the distorted data within the computational budget, the application adopts a list decoding procedure for twisted RS codes: the receiver first imposes soft decision constraints on a family of low-degree evaluation polynomials according to the received vector, and quickly merges a batch of candidate polynomials p(x) (degree no higher than r+2) through linear solving in finite field. The evaluation vectors corresponding to these polynomials collectively form a candidate codeword set, whose radius can approach d-1, greatly broadening the correctable interval while keeping the computational complexity at the polynomial level.

[0091] The candidate set often contains more than one legal vector, which must be further pruned. The inherent "tail term = head term * eta" structure of twisted RS codes plays a role here: the system only needs to verify whether the ratio of the last coordinate to the first coordinate of each candidate vector is equal to the sender's random element eta, and the vector that violates the ratio relationship is immediately excluded. This criterion is derived from the codeword generation formula itself and does not rely on additional redundancy, and its execution only involves one finite field multiplication and comparison, which can be embedded in the FPGA pipeline without affecting the throughput.

[0092] The vector set after structure screening is sent to the AMD decoder. Since the previous AMD encoding has embedded information in the "exponent-polynomial" check relationship, any adversary who fails to tamper with both the information word and the random element will trigger a check inconsistency during decoding and be rejected. On the contrary, only when a certain candidate codeword completely preserves the original exponent relationship will the decoder output the corresponding information. With this probability-1 algebraic manipulation detection mechanism, the multiple solution phenomenon that list decoding may produce is eliminated, and the system can still give a unique recovery result under high error conditions.

[0093] The present application mainly improves the following problems and defects of the prior art and realizes significant technical progress:

[0094] 1. Twisted RS codes are difficult to implement efficient decoding, restricting their practical promotion: Although twisted RS codes have theoretically constructed a large number of non-equivalent MDS codes, improving the diversity and security of the encoding system, the "twisted term" introduced by them destroys the traditional Vandermonde structure, making it impossible to directly reuse the decoders of classic RS codes. Most of the existing decoding methods are complex and difficult to implement in engineering, and even need brute force, which seriously affects their practicality. The present application introduces the AMD mechanism and list decoding strategy to bypass the structural barrier, realizes a feasible decoding path for twisted RS codes, and solves the long-standing decodability problem.

[0095] 2. Existing high-security error correction mechanisms are generally complex to implement and have low performance: Some studies have attempted to combine authentication mechanisms with error correction coding, but there are practical obstacles such as strong key dependence, difficulty in hardware implementation, and low decoding throughput, making them particularly difficult to implement in high-concurrency or edge devices. The solution proposed in this invention is not only highly compatible with existing RS systems in terms of decoding framework, but also has a simple design, low computational cost, and is suitable for rapid implementation in both hardware and software environments, possessing good engineering feasibility and promotion potential.

[0096] To address the problems existing in the prior art, the technical solution adopted in this invention is: a method of decoding using algebraic manipulation detection codes combined with twisted RS code lists.

[0097] like Figure 1 As shown, this embodiment relates to a probabilistic encoding and decoding scheme for increasing the unique decoding radius, which includes the following steps:

[0098] Step 1, Preprocessing stage: Let n∈N be a positive integer, take a twisted RS code with parameters [n,r+2], and define the twisted polynomial form. Take, select a random element Get information in Let represent the affine space, and r represent the length of the information. Let the multiplication commutative group be... Let the set of pairwise distinct assignment points be {α1, α2, ..., α...} n}∈G,α i ≠0, and satisfies Ensure that the obtained twisted RS code is an MDS code.

[0099] Step 2, AMD code encoding stage:

[0100] 2.1) Calculation

[0101] 2.2) The codeword after AMD encoding is m = (m0, ..., m r+1 )=(s,η,f(η))

[0102] Step 3, RS code encoding:

[0103] 3.1) Calculate the codeword after RS ​​code encoding as follows:

[0104] Step 4: Decode the RS code list at the receiving end:

[0105] Assume the received codeword is r = (r1, ..., r n The input is fed into the list decoder. The list decoding module for twisted RS codes mainly consists of two parts:

[0106] Interpolation module: Given twisted RS code, the received codeword is r = (r1, ..., rn Given n interpolation points (α) i ,r i ), and the corresponding minimum density m i The decoder is constructed using a bivariate polynomial with the minimum (1, r+2)-weighted degree.

[0107]

[0108] Through all interpolation points, i.e., Q(α) i ,r i ) = 0, where the minimum specificity m i This refers to the polynomial Q(x,y) passing through the point (α). i ,r i The number of times (α) i ,r i Let be the degree of the zeros of the polynomial Q(x,y). The weighted degree of the polynomial is:

[0109] degx i y j =i+(r+2)j

[0110] Factoring the polynomial module: The second step of decoding is to factor the bivariate polynomial with the smallest (1, r+2)-weighted degree obtained by inputting n interpolation points into the interpolation module into the form yp(x), where the degree of p(x) is ≤ r+2. All factors form an output list, where each polynomial p(x) could potentially be a polynomial used to construct the codeword. The resulting list has a size of [missing information]. Decoding list

[0111] Step 5, Verify the RS code structure at the receiving end: Traverse For candidate codewords, such as m i =(m 1,1 ,…,m 1,r+3 ), determine m i,r+3 Is it equal to m? i,1 If *η is equal, proceed to the next step of AMD decoding; otherwise, decoding fails.

[0112] Step 6: AMD code decoding at the receiving end: Traverse each candidate codeword that has passed the structure check and perform AMD code decoding. If the candidate codeword meets the AMD decoding constraints, the decoding is successful, and the candidate codeword is the original information. If multiple candidate codewords meet the conditions, the decoding is considered to have failed. The specific steps are as follows: Assume m1 = (m 1,1 ,…,m 1,r+3 For candidate codewords that pass the structure check, calculate... Determine if f′ is equal to m 1,r+2If more than one candidate codeword satisfies the condition, the AMD code decoding fails. Otherwise, the only decoding result is taken as the output of the decoder.

[0113] The present application systematically solves the core problems of twisted RS code in decoding efficiency, security and implementation complexity, and proposes an encoding technical solution with novel structure, feasible decoding and self-defense capability. Compared with the prior art, the solution has made significant progress in decodability, security robustness and engineering practicability, and has outstanding innovative value and broad application prospect.

[0114] Technical advantages:

[0115] 1. Breakthrough of twisted RS code decoding bottleneck, realization of efficient and practical decodable solution: the present application bypasses the algebraic obstacle brought by the twist term of twisted RS code from the structural level, and uses list decoding strategy to make the decoding process maintain low computational complexity while maintaining high error correction radius. Compared with the existing complex brute-force decoding method, the present solution is compatible with the existing RS code decoding architecture, easy to deploy on software and hardware platforms, and greatly reduces the engineering implementation threshold.

[0116] 2. Good theoretical support and application prospect: the present application is based on the two branches of twisted RS code and AMD code encoding theory, taking into account mathematical rigor and practical applicability, and has good theoretical promotion space and industry adaptability. Whether in academic research, industrial protocol formulation, or deployment in domestic self-controllable chips / communication equipment, it has broad development potential.

[0117] In order to prove that the above-mentioned embodiments have achieved some positive effects in the process of research and development or use, and indeed have great advantages compared with the prior art, a specific example of decoding success is shown below. Here, in order to facilitate the display, the twisted RS code is not guaranteed to be MDS, if you need to guarantee the MDS property, you need to take larger parameters:

[0118] Step 1, preprocessing stage: take the parameter of twisted RS code as Take a random element Take information Where represents the affine space.

[0119] Step 2, AMD code encoding stage:

[0120] 2.1) Calculate on

[0121] 2.2) The AMD encoded codeword is s=(s0,s1,s2,s3)=(m,η,f(η))=(10,7,2,11)

[0122] Step 3, twist RS code encoding:

[0123] 3.1) twist RS code information polynomial Take the assignment point set as

[0124] 3.2) calculate the twist RS code after encoding the code word as c=(c1, c2, …, c 15 )=(f(α1), …, f(α n ))=(10, 3, 2, 6, 7, 14, 6, 15, 5, 1, 6, 4, 13, 11, 11).

[0125] Step 4, twist RS code decoding at the receiving end:

[0126] 4.1) Assuming that the position (3, 7) is disturbed by noise, the code word received by the receiving end is r=(10, 3, 2, 7, 7, 14, 6, 14, 5, 1, 6, 4, 13, 11, 11), which is input to the list decoder, thereby obtaining a possible information {(10, 7, 2, 11, 7)}.

[0127] Step 5, twist RS code structure verification at the receiving end:

[0128] 4.1) Verify η*m0=2*10≡7=m r+2 , input the candidate code word to the AMD decoder Step 5, AMD code decoding at the receiving end:

[0129] 5.1) For (10, 7, 2, 11, 7), calculate m r+1 =2 4 +2*2 2 +7*2+10=11, so the candidate code word passes the AMD verification, thereby recovering the original information (10, 7).

[0130] The above is only a specific embodiment of the present application, but the protection scope of the present application is not limited thereto, any modification, equivalent replacement and improvement made by those skilled in the art within the technical range disclosed by the present application, as long as it is within the spirit and principles of the present application, should be covered within the protection scope of the present application.

Claims

1. A method for decoding a detection code combined with a twisted RS code list, characterized in that, include: The information to be transmitted is encoded using the system's AMD code to obtain the first codeword; Using the first codeword as the message vector, take the twisted RS code with parameters [n, r+2], perform twisted RS code encoding, and generate the second codeword; Receive the received vector after the second codeword passes through the channel, call the twisted RS code list decoder, and output a candidate codeword list within the preset decoding radius; Perform twisted RS structure verification on each candidate codeword in the candidate codeword list, and codewords that meet the twisted RS polynomial constraints are entered into subsequent processing. For each candidate codeword that passes the structure check, perform AMD code decoding. If only one codeword is successfully decoded, output the corresponding information. If more than one item is successfully decoded or zero items are successfully decoded, the decoding is considered a failure.

2. The encoding and decoding method for combining detection codes with twisted RS code list decoding as described in claim 1, characterized in that, The AMD code encoding includes: In a finite field Randomly select a non-zero element η from the data; The information vectors to be transmitted are s1 to s2. r And construct a polynomial f(x) using random element η. The AMD-encoded codeword is m = (m0, ..., m r+1 )=(s,η,f(η)).

3. The encoding and decoding method for combining detection codes with twisted RS code list decoding as described in claim 1, characterized in that, The twisted RS code encoding satisfies the following conditions: Select the set of assignment points {α1, α2, ..., α n } are distinct and non-zero commutative subgroups of multiplication in finite fields; The {α1, α2, ..., α n }∈G,α i ≠0, and satisfies The codeword after RS ​​encoding is c = (c1, ..., c n )=(f(α1),…,f(α n )).

4. The encoding and decoding method for combining detection codes with twisted RS code list decoding as described in claim 1, characterized in that, The list decoding specifically includes: Assume the received codeword is r = (r1, ..., r n The input is fed into a list decoder for the twisted RS code, resulting in a list of size . Decoding list 5. The encoding and decoding method for combining detection codes with twisted RS code list decoding as described in claim 1, characterized in that, The structural verification of the twisted RS code specifically includes: Traversal For candidate codewords, such as m i =(m 1,1 ,…,m 1,r+3 ), determine m i,r+3 Is it equal to m? i,1 If *η is equal, proceed to the next step of AMD decoding; otherwise, decoding fails.

6. The encoding and decoding method for combining detection codes with twisted RS code list decoding as described in claim 1, characterized in that, The process of traversing the input of the decoder to perform AMD code decoding specifically includes: If a candidate codeword meets the decoding constraints of AMD, the decoding is successful, and the candidate codeword is the original information. If multiple candidate codewords meet the conditions, the decoding is considered to have failed. The specific steps are as follows: Assume m1 = (m 1,1 ,…,m 1,r+3 For candidate codewords that pass the structure check, calculate... Determine f ′ Is it equal to m? 1,r+2 If more than one candidate codeword satisfies the condition, the AMD code decoding is considered to have failed; otherwise, the unique decoding result is used as the decoder's output.

7. A codec system for algebraic manipulation detection combined with twisted RS code list decoding as described in claim 1, characterized in that, include: AMD code encoding module: Encodes information using the system AMD code; Twisted RS code encoding module: Encodes the above AMD code encoding results into twisted RS codes; Twisted RS code list decoding module: decodes the received codewords in a list and outputs a list containing the possible decoding results; The twisted RS code structure verification module iterates through the results of the list decoding, performs structure verification on each candidate codeword, and uses the codewords that meet the conditions as input to the AMD decoding module. AMD code decoding module: Performs AMD code decoding on each input codeword. If only one codeword is successfully decoded, the decoding result is the original information; otherwise, the decoding fails.

8. The encoding and decoding system for algebraic manipulation detection combined with twisted RS code list decoding as described in claim 7, characterized in that, The AMD code encoding module randomly selects a non-zero element r in the finite field Fq; Construct a message polynomial based on the information vector to be transmitted and the random element r; The first codeword is generated by evaluating the message polynomial at a set of preset evaluation points.

9. The encoding and decoding system for algebraic manipulation detection combined with twisted RS code list decoding as described in claim 7, characterized in that, The AMD code decoding module traverses For candidate codewords, such as m i =(m 1,1 ,…,m 1,r+3 ), determine m i,r+3 Is it equal to m? i,1 If *η is equal, proceed to the next step of AMD decoding; otherwise, decoding fails.

10. A computer-readable storage medium, characterized in that, It stores instructions that, when executed by a processor, cause the processor to run according to the steps of the encoding and decoding method of the detection code combined with the twisted RS code list decoding as described in any one of claims 1 to 6.

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