Grid-based algebraic erasure construction method

Abstract

The invention relates to a lattice-based algebraic erasure construction method, and belongs to the technical field of information security. According to the method, firstly, abstract algebraic erasure core operation is strictly summarized into one half-per-unit group right action, and the abstract algebraic erasure core operation is improved for the first time on the basis of a non-homogeneous short integer solution problem in lattice cryptography. In order to further improve the safety, a plait group matrix with a more complex algebraic structure is proposed to replace a trivial replacement group, so that a post-quantum cryptography scheme of which the safety depends on double difficulty assumption of a lattice problem and a plait group problem is constructed.

Description

TECHNICAL FIELD

[0001] The application relates to a lattice-based algebraic erasure construction method and belongs to the technical field of information security. BACKGROUND

[0002] The security of the widely used public key cryptography system mainly depends on the calculation complexity of specific number theory problems, such as a large integer factorization problem and a discrete logarithm problem. However, with the rapid development of quantum computing technology, a computer capable of performing a quantum algorithm such as Shor algorithm will pose a fundamental threat to the existing public key cryptography system, making it invalid in the future. In response to this threat, the academic and industrial circles are committed to researching new cryptographic algorithms that can resist joint attacks of classical computers and quantum computers, which is referred to as post-quantum cryptography.

[0003] As an exploration to cope with quantum threats, algebraic erasure proposed by Iris Anshel, Michael Anshel, Dorian Goldfeld and Stephane Lemieux in 2005 provides a solution based on non-commutative algebraic problems. The technology is initially positioned as a lightweight cryptographic primitive suitable for computing resource-constrained environments, aiming to provide security protection for low-cost platforms such as Internet of Things devices. The core idea is that it acts on a monoid through a group, and applies a homomorphic mapping to effectively "erase" the details of the group action, thereby constructing a cryptographic problem that is difficult to reverse. Specific implementations based on this principle include the Colored Burau key agreement protocol and the Walnut DSA digital signature algorithm.

[0004] The security assumption of algebraic erasure is different from the number theory assumption based on integer factorization or discrete logarithm. Its security depends on the one-wayness of the "erasure" process, the hidden element assumption, the difficulty of the synchronous conjugate search problem and the secrecy of the key parameters. Although this scheme is not directly based on the number theory problem vulnerable to Shor algorithm attacks, its core algebraic problem has not been reduced to any recognized and fully analyzed post-quantum standard problem (such as lattice problems, coding problems, etc.). Therefore, the actual security of the algebraic erasure scheme under the quantum computing model is uncertain, and independent security analysis and evaluation need to be conducted for specific implementation schemes and their parameter selection.

[0005] Publications have made cryptographic analysis and attack evaluation on algebraic erasure and some of its specific instances. Research shows that algebraic erasure is extremely sensitive to parameter generation and instantiation. Under several public or random parameter instances, researchers have constructed attack methods that can recover shared keys or compromise security. Some practical attacks can be successfully implemented under conventional computing resources, indicating that some public parameter sets are difficult to achieve their claimed security strength in actual deployment.

[0006] Therefore, the existing algebraic erasure has the following defects: 1. The abstract framework of the method lacks a general proof that formalizes its security to a standard quantum-resistant problem, which makes its security foundation not clear enough; 2. It is extremely sensitive to parameters (including ring size, seed matrix, permutation selection, conjugate element length, etc.), and an error or weak parameter generation method may lead to a practical attack. In order to overcome the above defects, the present application proposes a lattice-based algebraic erasure construction method. SUMMARY

[0007] The purpose of the present application is to solve the problem that the existing algebraic erasure lacks a specific and provably secure post-quantum implementation scheme, and its security is difficult to be reduced to a recognized computational difficult problem, and proposes a lattice-based algebraic erasure construction method, which skillfully uses the inhomogeneous small integer solution (ISIS) problem in lattice cryptography to construct the algebraic erasure method. Specifically, the present application migrates the one-way function of the ISIS problem to the homomorphism in the algebraic erasure structure. Through this method, the security of the present scheme is successfully reduced to the difficulty of the ISIS problem. Since the ISIS problem is generally considered to be resistant to quantum computing, the present scheme also has the post-quantum security feature. In addition, the present application notes that the permutation group structure used in the original construction is relatively simple, which may have security risks. Therefore, the present application further proposes an enhanced scheme. The scheme uses a braid group matrix representation with a more complex algebraic structure to replace the original permutation group. In summary, the present application finally constructs a lattice-based algebraic erasure construction method, whose security depends on the double difficulty assumptions of the lattice problem (ISIS) and the braid group problem.

[0008] The purpose of the present application is achieved by the following technical solutions.

[0009] The lattice-based algebraic erasure construction method proposed by the present application includes the following steps:

[0010] Step 1: Define the algebraic erasure structure

[0011] The algebraic erasure structure is a six-tuple Let s m represents the result of s acting on m, where s∈S, m∈M, and the algebraic erasure structure satisfies the following conditions:

[0012] · M and N are two monoids. A and B are two sub-monoids of .

[0013] · S is a group. S acts on M on the left, but does not act on N.

[0014] • φ: M→N is a homomorphism.

[0015] • denotes the semi-direct product of M and S, m1, m2∈M, s1, s2∈S, and the binary operation in it satisfies:

[0016]

[0017] where: denotes the result of s1 acting on m2;

[0018] • N×S denotes the direct product of N and S, n1, n2∈N, and the binary operation in it satisfies:

[0019]

[0020] • The * operation is the core of the algebraic erasure, which is defined as follows:

[0021]

[0022]

[0023] The reason why the algebraic erasure structure can become a powerful tool for constructing cryptographic protocols is mainly due to its unique algebraic properties. The algebraic erasure satisfies the mixed associativity law, for any (n, s)∈N×S and has

[0024]

[0025] The mixed associativity law ensures that the continuous * operation on a series of generators is equivalent to first performing the product operation in and then performing the * operation once, so that the calculation in the algebraic erasure protocol can be iterated, greatly simplifying the calculation process of the protocol.

[0026] Step 2: Algebraic structure of the * operation - monoid right action

[0027] Let G be a group and X be a set. A right group action of G on X is a binary function X×G→X, which satisfies the unit element axiom and the consistency axiom.

[0028] The present application proposes that the * operation in the definition of the algebraic erasure formula (1) can actually be strictly regarded as a right action of a monoid on a set, that is, the set X to be acted on is regarded as N×S, and the monoid G to be acted on is regarded as (the internal operation of which is We define the right-acting mapping as the * operation in the algebraic erasure scheme. This perspective provides crucial insight into understanding the algebraic symmetry and structural rationality of the scheme. We will now prove that * is a valid monoid right-acting mapping by verifying that it satisfies two fundamental axioms.

[0029] (1) Identity element axiom: The identity element of an active monoid should keep the elements of the set unchanged. The unit element is (e M e S ), where e M and e S Let M and S be the identity elements, respectively. Apply it to any element (n, s) ∈ N×S:

[0030] (n, s) * (e) M e S )=(n(φ( s e M )), se S )

[0031] Under the standard definitions of group action and homomorphism, an action on the identity element remains an action on the identity element. Therefore, the axiom holds.

[0032] (2) Compatibility axiom: For any x∈X and g1, g2∈G, we always have Substituting the correspondence into the compatibility axiom of right-hand action, we obtain Substituting the variables x = (n, s), g1 = (m1, s1), and g2 = (m2, s2) further, the axiom is expressed as:

[0033]

[0034] This formula has the same structure as formula (2). Therefore, * is an algebraically valid monoid right action, and formula (2) is not an isolated algebraic trick; its purpose is to prove that the function * satisfies the compatibility axiom.

[0035] Step 3: Constructing an example of algebraic erasure based on the lattice problem

[0036] This invention proposes a lattice-based algebraic erasure construction method, specifically utilizing the one-way nature of the ISIS problem to construct a homomorphism φ for algebraic erasure.

[0037] The ISIS problem is defined as follows: Given a prime number modulo q, and a random matrix... ( (representing the n0×m0 order integer matrix group modulo q), the target vector ( Given a vector group of integers of order n0 modulo q and a norm bound β > 0, the ISIS problem requires finding a non-zero integer vector. ( Let m be a group of integer vectors of order m0, such that Am = n mod q and its norm satisfies ||m|| < β.

[0038] The algebraic erasure construction method based on the ISIS problem of the present invention is as follows:

[0039] Step 3.1 Construct monoids M and N.

[0040] The monoid M is defined as the additive group of integers. A monoid N is defined as an additive group modulo q of integers.

[0041] Step 3.2 Construct the homomorphism φ.

[0042] Choose a publicly available, randomly generated matrix. The parameters m0, n0, and q are configured such that m0≈O(n0log q), where Big O represents the computational complexity to ensure the safety of the preset.

[0043] A homomorphism φ is defined as M→N, which is determined by performing left multiplication of matrix A and subsequent reduction modulo q on any input vector m∈M, mathematically expressed as φ(m)=Am mod q. The homomorphism φ constitutes a valid monoid homomorphism due to the distributive law of matrix multiplication over vector addition. To ensure the one-wayness of this homomorphism φ for algebraic erasure, this invention imposes a cryptographic constraint on the input domain of the homomorphism φ, requiring that the selected preimage m∈M must be a short vector with a norm satisfying the condition ||m||<β. By imposing this norm constraint, the computation problem of the preimage of the homomorphism φ is transformed from a standard linear algebra problem into an instance of an ISIS problem. Therefore, the known quantum computational difficulty of the ISIS problem provides a security guarantee for the one-wayness of the homomorphism φ on the constrained set of short vector inputs.

[0044] Step 3.3 Construct group S and its role.

[0045] A group S is represented by a matrix of a permutation group acting on m coordinates, where S is a set of m0×m0 permutation matrices and forms a subgroup of GL(m0, Z). Elements P∈S in group S have left-handed actions on the monoid M. P m is defined as the matrix-vector multiplication operation. P The technical effect of m = Pm is to rearrange the components of vector m.

[0046] Based on the above definition, the core operation * for any element (n,P)∈N×S and It was identified as:

[0047] (n, P) * (m1, P1) = (n + φ( P m1), PP1)=(n+APm1,PP1)

[0048] In this construction, the algebraic erasure mechanism is implemented by performing a composite operation on the integer vector m1: first, the vector m1 is permuted using a private permutation matrix P; then, the result of the permutation is mapped from a high-dimensional space of m0 dimensions to a low-dimensional space of n0 dimensions using a public matrix A. This process constitutes a lossy information transformation, thereby achieving information erasure. Therefore, for an attacker who intercepts APm1, decomposing the result to reconstruct the original m1 and the permutation matrix P is computationally infeasible.

[0049] The security of this construction relies on the computational difficulty associated with the ISIS problem, providing a path to a concrete implementation of post-quantum cryptography through algebraic erasure. However, the permutation group chosen in the above construction has a relatively simple algebraic structure; the permutation operation is merely a rearrangement of vector coordinates. This highly structured operation may provide attackers (especially quantum attackers) with additional exploitable information. To construct a more robust scheme, this invention further proposes an improved method: constructing the group S using the matrix representation of the braid group.

[0050] Step 3.4. Redefine group S.

[0051] To enhance the robustness of the scheme, this invention redefines the group S. This definition is based on a selected faithful, m0-dimensional braided group matrix representation ρ, where ρ is a representation of a braided group. The homomorphic mapping to the general linear group GL(m0, Z), i.e.

[0052]

[0053] Group S was identified as braid group The image under the representation ρ, i.e. The elements of the group S constructed in this way are a unitary matrix of integers with a determinant of ±1. These elements are derived from the fundamental relations of braid groups, and their algebraic structure is more complex than that of permutation matrices.

[0054] Based on this redefined group S, the form of the core operation * is determined as follows:

[0055] (n, ρ(b))*(m1, ρ(b1))=(n+φ(ρ(b)m1), ρ(b)ρ(b1))

[0056] Where m1∈Z m Let ρ(b) ∈ GL(m, Z) be an integer vector, and let ρ(b) ∈ GL(m, Z) be an integer matrix. Utilizing the group homomorphism property of the aforementioned representation of ρ, i.e., ρ(b)ρ(b1) = ρ(b·b1) (where b·b1 represents the homomorphism in the braid group B) m (Multiplication operation in Chinese), the expression for the above operation can be further parsed as:

[0057] (n,ρ(b))*(m1,ρ(b1))=(n+A(ρ(b)m1)mod q,ρ(b·b1))

[0058] Thus, this invention completes the construction of a lattice-based algebraic erasure method. By rigorously reducing the underlying operations of algebraic erasure to a monoid right action and creatively combining the ISIS problem in lattice cryptography with the complex algebraic structure of braid groups, this invention successfully constructs a post-quantum cryptographic method whose security depends on the double-hard problem assumption. This method not only solves the problem of the lack of a clear security foundation in traditional algebraic erasure but also provides a concrete and reliable technical path for its practical application in the field of post-quantum information security.

[0059] Beneficial effects

[0060] 1. This invention proposes a secure algebraic erasure construction method. For the first time, this invention rigorously treats the core operation in algebraic erasure as an algebraically valid monoid right action, providing a solid algebraic foundation for iterative computation of cryptographic protocols. Based on this algebraic structure, this invention further introduces the ISIS problem from lattice cryptography, successfully constructing the various components of algebraic erasure using its one-way function properties. This results in an algebraic erasure scheme whose security depends on the computational difficulty of the ISIS problem and possesses post-quantum security potential.

[0061] 2. This invention significantly enhances the algebraic complexity of the scheme by elevating the constructed group S from a simple permutation group to a matrix representation of a braid group. When applied to public key generation in cryptography, the non-commutativity of the braid group makes the order of operations on private key elements crucial. The private key constitutes an ordered non-commutative sequence, greatly increasing the difficulty of cryptanalysis. Furthermore, the dense matrix representation of the braid group performs a thorough linear mixture on the private vector, rather than a simple coordinate permutation, achieving a more complex erasure effect, making it computationally more difficult to recover private information from public information.

[0062] 3. The security of this invention is built upon two heterogeneous computational problems, both considered resistant to quantum attacks. One is the lattice problem (ISIS) originating from a one-way function Am, and the other is the braid group problem (braid group conjugation problem, etc.) originating from factoring elements in a group S. This hybrid design combines the advantages of lattice-based cryptography and non-commutative algebraic cryptography, constructing a post-quantum cryptosystem with richer algebraic structures and more diverse security foundations. Attached Figure Description

[0063] Figure 1 A flowchart of the core operations of an algebraic erasure scheme based on the ISIS problem. Detailed Implementation

[0064] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0065] Example

[0066] Figure 1 This is a flowchart of the core algebraic erasure operation proposed in this invention. This operation is the basic module for constructing the cryptographic protocol described in this invention. It transforms the input state elements and private key through a series of defined algebraic steps to generate new state elements.

[0067] In a specific embodiment, the execution environment and components of this core operation* are defined as follows:

[0068] Monoids M and N: Define two monoids, where monoid M is the additive group of integers. The monoid N is an integer additive group.

[0069] • Group S: Define a group S that acts on a monoid M. In a preferred embodiment, group S is configured as a braid group. The faithful matrix represents the image of ρ, i.e.

[0070] • Homomorphism φ: Define a homomorphism φ(m) from M to N = Am mod q, where It is a publicly available, randomly generated matrix of system parameters. The one-wayness of this mapping is guaranteed by the computational difficulty of the ISIS problem.

[0071] Reference Figure 1 The specific execution flow of the core operation* is described from top to bottom as follows:

[0072] This operation receives two inputs: (n, P) and (m1, P1) at the top of the process. Here, n ∈ N, P ∈ S, m1 ∈ M, and P1 ∈ S. The vector m1 is constrained to be a short vector satisfying a specific norm condition.

[0073] The process enters the core computing module shown within the red dashed box and executes the following steps:

[0074] Step 1: Group Action. The matrix P from the input state elements and the vector m1 from the input private key are processed. Specifically, P performs a left-hand action on m1, defined as a matrix-vector multiplication, to compute intermediate results. P m1 = Pm1. This step applies a non-commutative, structurally complex linear transformation to vector m1 using matrix P.

[0075] Step 2: Homomorphic mapping (erasure). The intermediate results obtained in Step 1... P m1 is taken as input. It is processed through a predefined homomorphic mapping φ, i.e., left-multiplied by the public matrix A and reduced modulo q, to ​​calculate the result φ( P m1)=A( P m1)modq. This step, through a lossy mapping from a high-dimensional space to a low-dimensional space, achieves irreversible erasure of private information and is a key step in ensuring the security of the solution.

[0076] While performing step 2, two compound operations are executed in parallel:

[0077] • State composition: The output A of step 2 is combined. P The vector n from the input state element is added to m1 to generate a new state vector component n+A(Pm1).

[0078] • Group multiplication: Perform matrix multiplication on matrix P from the input state elements and matrix P1 from the input private key to generate a new state matrix component PP1.

[0079] Finally, the process reaches the bottom output layer. The outputs of the "State Composition" module and the "Group Multiplication" module are combined to obtain the final output of this core operation (n+A( P m1), PP1). The algebraic structure of the output result is consistent with the input state element, so it can be used as the input state element for the next iteration.

[0080] To clearly illustrate the technical solution of the present invention with a specific example, this embodiment will refer to... Figure 1 The process is shown, and a set of specific parameters and values ​​are used to explain each step.

[0081] 1. Instantiation of parameters and inputs.

[0082] In this embodiment, the basic parameters of the cryptographic system are set as follows:

[0083] Dimension and Modulus: Set the vector space dimension m0 = 3, n0 = 2, and choose a prime modulus q = 7.

[0084] Public matrix A: Randomly select a public matrix. Its value is:

[0085]

[0086] Instantiation of group S: Group S is constructed using the matrix representation ρ of the 3D braided group B3. Input state element (n, P): Assume the current state element contains vectors. The sum matrix P = ρ(b) ∈ S, where b is a braid in B3. The specific values ​​are:

[0087]

[0088] Input private key (m1, P1): Suppose the private key contains a short vector m1∈Z 3 The sum matrix P1 = ρ(b1) ∈ S, where b1 is a braid in B3. The specific values ​​are:

[0089]

[0090] 2. Numerical calculation process of core operations*.

[0091] in accordance with Figure 1 The calculation is performed by substituting the above values ​​into the top-down flow shown:

[0092] Step 1: Group Action

[0093] calculate P m1 = Pm1. This step uses the state matrix P to linearly mix the private vector m1:

[0094]

[0095] Step 2: Homomorphism

[0096] Calculate φ( P m1)=A( P m1)mod 7. This step maps the mixed vector to a low-dimensional space and performs modulo reduction, thus erasing information:

[0097]

[0098] Step 3: State Composition and Group Multiplication

[0099] State composition: Calculate n+A( P m1)(mod 7). Add the result of step 2 to the input state vector n:

[0100]

[0101] Group multiplication: Calculate PP1. In group S, this corresponds to the multiplication ρ(b·b1) in the braid group:

[0102]

[0103] Step 4: Output

[0104]

[0105] Figure 1 This invention clearly illustrates a specific implementation of the core algebraic erasure operation proposed in this paper using a data flow diagram. The diagram clarifies a deterministic computational process that receives two inputs (state elements and a private key) and produces one output (a new state element). Its core mechanism lies in performing a non-commutative algebraic transformation on the private information through a group action step, followed by irreversible information erasure through a homomorphic mapping step, and finally securely fusing the transformation result with the current state through "state composition" and "group multiplication." The entire process algebraically constitutes an effective monoid group right action, providing a solid foundation for constructing secure, iterative cryptographic protocols.

[0106] This invention proposes a secure algebraic erasure construction method. This method first rigorously reduces the abstract core operation of algebraic erasure to a monoid right action, and then, for the first time, improves upon it based on the non-homogeneous short integer solution problem in lattice cryptography. To further enhance security, this invention proposes replacing the trivial permutation group with a braid group matrix representation, which has a more complex algebraic structure, thereby constructing a post-quantum cryptographic scheme whose security depends on the dual difficulty assumptions of the lattice problem and the braid group problem.

Claims

1. A lattice-based algebraic erasure construction method, characterized in that, Includes the following steps: Step 1: Define the algebraic erasure structure The algebraic erasure structure is a six-tuple (M×S, N, φ, *, A, B), denoted as... s m represents the result of s acting on m, where s∈S, m∈M, and the algebraic erasure structure satisfies the following condition: ●M and N are two monoids, and A and B are two submonoids of M×S. ●S is a group that acts on M to the left, but not on N. ●φ: M→N is a homomorphism ●M×S represents the semi-direct product of M and S, where m1, m2∈M, s1, s2∈S, and the binary operations satisfy: in: This represents the result of s1 acting on m2. ●N×S represents the direct product of N and S, where n1, n2∈N, and the binary operations satisfy: ●*The operation is the core of algebraic erasure, defined as follows: *:(N×S)×(M×S)→N×S (1) The reason why algebraic erasure structures can be a powerful tool for constructing cryptographic protocols is mainly due to their unique algebraic properties. Algebraic erasure satisfies the mixed associative law, and for any (n, s) ∈ N×S and (m1, s1), (m2, s2) ∈ M×S, we have The hybrid associative law ensures that a series of * operations on a series of generators is equivalent to first performing a product operation in M×S and then performing a * operation, which allows the computation in the algebraic erasure protocol to be performed iteratively, greatly simplifying the computation process of the protocol. Step 2: *Algebraic structure of operations—right action of monoid groups Let G be a group and X be a set. A right group action of G on X is a bivariate function X×G→X, which satisfies the identity axiom and the compatibility axiom. The present invention proposes that the * operation in the definition of the algebraic erasure formula (1) can actually be strictly regarded as a right action of a monoid on a set, that is, the set X to be acted on is regarded as N×S, the monoid G to act on is regarded as M×S, and the mapping of the right action is defined as the * operation in the algebraic erasure scheme. Step 3: Constructing an example of algebraic erasure based on the lattice problem The ISIS problem is defined as follows: Given a prime number modulo q, and a random matrix... Let n0×m0 be the integer matrix group modulo q, and let the target vector be... Given a vector group of integers of order n0 modulo q, and a norm bound β > 0, the ISIS problem requires finding a non-zero integer vector. Let m represent the group of integer vectors of order m0 such that Am = n mod q and its norm satisfies ||m|| < β.

2. The lattice-based algebraic erasure construction method as described in claim 1, characterized in that, Step 2 verifies that the function satisfies two fundamental axioms, proving that * is an effective right action of a monoid: (1) Identity Axiom: The identity element of an action monoid should preserve the elements of the set. The identity element of M×S is (e M e S ), where e M and e S These are the identity elements of M and S, respectively. Applying them to any element (n, s) ∈ N×S: (n,s)*(e M ,e S )=(n(φ( s yes M )),see S ) Under the standard definitions of group action and homomorphism, an action on the identity element is still an action on the identity element; therefore, the axiom holds. (2) Compatibility axiom: For any x∈X and g1, g2∈G, we always have Substituting the correspondence into the compatibility axiom of right-hand action... get Substituting the variables x = (n, s), g1 = (m1, s1), and g2 = (m2, s2) further, the axiom is expressed as: This formula has the same structure as formula (2). Therefore, * is an algebraically valid monoid right action. Formula (2) is not an isolated algebraic trick. Its purpose is to prove that the function * satisfies the compatibility axiom.

3. The lattice-based algebraic erasure construction method as described in claim 2, characterized in that, Step 3 specifically involves: Step 3.1 Construct monoids M and N The monoid M is defined as the additive group of integers. A monoid N is defined as an additive group modulo q of integers. Step 3.2 Construct the homomorphism φ Choose a publicly available, randomly generated matrix. The parameters m0, n0, and q are configured to make m0≈O(n0logq), where Big O represents the computational complexity to ensure the safety of the preset. Define a homomorphism φ: M→N. This mapping is determined to be an operation of left multiplication of matrix A and subsequent reduction modulo q on any input vector m∈M. Its mathematical expression is φ(m)=Am mod q. The mapping φ constitutes an effective monoid homomorphism due to the distributive law of matrix multiplication over vector addition. In order to make the homomorphism φ have one-wayness to realize the algebraic erasure function, this invention imposes a cryptographic constraint on the input domain of the mapping φ, that is, the selected preimage m∈M must be a short vector and its norm must satisfy the condition ||m||<β. By imposing this norm constraint, the problem of solving the preimage of the homomorphism φ is transformed from a standard linear algebra problem into an instance of an ISIS problem. Therefore, the known resistance to quantum computation difficulty of the ISIS problem provides a security guarantee for the one-wayness of the homomorphism φ on the constrained set of short vector inputs. Step 3.3 Constructing group S and its role A group S is represented by a matrix of a permutation group acting on m coordinates, where S is a set of m0×m0 permutation matrices and forms a subgroup of GL(m0, Z), and the elements P∈S in S have left-handed action on the monoid M. P m is defined as the matrix-vector multiplication operation. P m = Pm, the technical effect of this action is to rearrange the components of vector m; Based on the above definition, the core operation * for any elements (n, P) ∈ N×S and (m1, P1) ∈ M×S is determined as follows: (n, P)*(m1, P1)=(n+φ( P m1), PP1)=(n+APm1, PP1) In this construction, the algebraic erasure mechanism is implemented by performing a composite operation on the integer vector m1: First, the vector m1 is permuted by a private permutation matrix P. Then, the result of the permutation is mapped from a high-dimensional space of m0 dimensions to a low-dimensional space of n0 dimensions by a public matrix A. This process constitutes a lossy information transformation, thereby achieving information erasure. Therefore, for an attacker who intercepts APm1, it is computationally infeasible to decompose the result to restore the original m1 and the permutation matrix P. The security of this construction relies on the computational difficulty associated with the ISIS problem, providing a path to a concrete implementation of post-quantum cryptography for algebraic erasure. However, the permutation group selected in the above construction has a relatively simple algebraic structure, and the permutation operation is only a rearrangement of vector coordinates. This highly structured operation may provide additional exploitable information for attackers, especially quantum attackers. To build a more robust scheme, the matrix representation of the braid group is used to construct the group S. Step 3.

4. Redefine group S The group S is redefined based on a chosen faithful, m0-dimensional braided group matrix representation ρ, where ρ is a representation of the braided group. The homomorphic mapping to the general linear group GL(m0, Z), i.e. Group S was identified as braid group The image under the representation ρ, i.e. The elements of the group S constructed in this way are a unitary matrix composed of integers, with a determinant of ±1. These elements are derived from the fundamental relations of braid groups, and their algebraic structure is more complex than that of permutation matrices. Based on this redefined group S, the form of the core operation * is determined as follows: (n, ρ(b)) * (m1, ρ(b1)) = (n + φ(ρ(b)m1), ρ(b)ρ(b1)) where m1 ∈ Zm is an integer vector, ρ(b) ∈ GL(m, Z) is an integer matrix, and using the group homomorphism property of ρ, i.e., ρ(b)ρ(b1) = ρ(b·b1), where b·b1 represents the group homomorphism of ρ in braid group B. m The multiplication operation in the above expression can be further parsed as: (n,ρ(b))*(m1,ρ(b1))=(n+A(ρ(b)m1)mod q,ρ(b·b1)).

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