Separable code construction method and system

Abstract

The invention discloses a separable code construction method and system. The method comprises the following steps: constructing an association relationship between an anti-framing code and a separable code; q is a positive integer relatively prime to 6, infinity is an element, a definition code C of # imgabs0 # is on Zq-1 U {infinity}, and C = C1 U C2; a, b belongs to C and are different code words, I (a, b) = 2, when and only when there is an element x belongs to Zq-1, a and b belong to {(infinity, x, x, x) T, (x, infinity, x, x) T, (x, x, infinity, x) T}, and T is a transposed symbol; n and t are positive integers, q is larger than or equal to 2, n is larger than or equal to 2, t is larger than or equal to 2, a definition code C is a q-element # imgabs1 #-separable code, the length of the definition code C is n, and C is smaller than or equal to (t-1) qn / (t-1); and when M is less than or equal to 2q2 for any 3-sc (4, M, q), C = C1 union C2 = C1 + C2 = (q-1) (q-2) + 4 (q-1) = q2 + q-2 code words are constructed, and order optimization is achieved. The construction of the # imgabs2 # with the code length of 4 is a brand-new construction, the construction comprises q2 + q-2 code words, and the number of the code words can reach the optimal order.

Description

Technical Field

[0001] The present invention belongs to the technical field of separable code construction, and in particular, relates to a separable code construction method, system and system. Background Art

[0002] Multimedia fingerprint is a new copyright protection technology developed in recent years. It achieves the purpose of copyright protection and pirate tracking by embedding a unique "fingerprint" in each distributed multimedia content. When pirated content is found, the fingerprint information in the pirated content can be extracted and the tracking algorithm of the multimedia fingerprint code can be applied to identify the user involved in the piracy.

[0003] After searching the existing construction methods, it was found that there is currently no technical solution that provides a separable code construction with three anti-collusion users and a code length of 4. Summary of the invention

[0004] The present invention provides a separable code construction method and system, and proposes a method for constructing a separable code according to the relationship between the anti-fraud code and the separable code. The construction method with a code length of 4 fills the gap in the construction of separable codes with a code length of 4 for anti-collusion users t=3.

[0005] In a first aspect, the present invention provides a separable code construction method for constructing a separable code with t=3 anti-collusion users and a code length of 4, the method comprising:

[0006] Constructing the correlation between anti-fraud codes and separable codes;

[0007] Let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2;

[0008] Let a, b∈C be different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol;

[0009] Let n and t be positive integers such that q ≥ 2, n ≥ 2 and t ≥ 2, and define the code C to be q-ary -separable code and its length is n, then

[0010] In MM≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords to achieve the optimal order.

[0011] In a second aspect, the present invention provides a separable code construction system for constructing a separable code with t=3 anti-collusion users and a code length of 4, the system comprising:

[0012] A construction module configured to construct an association relationship between the anti-fraud code and the separable code;

[0013] The first definition module is configured to let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2;

[0014] Assume that the module is configured so that a, b∈C are different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol;

[0015] The second definition module is configured to let n and t be positive integers and make q≥2, n≥2 and t≥2, and the definition code C is q-element -separable code and its length is n, then

[0016] Construction modules, configured to be in M≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords to achieve the optimal order.

[0017] According to a third aspect, an electronic device is provided, comprising: at least one processor, and a memory communicatively connected to the at least one processor, wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor so that the at least one processor can perform the steps of the separable code construction method of any embodiment of the present invention.

[0018] In a fourth aspect, the present invention further provides a computer-readable storage medium having a computer program stored thereon, wherein when the program instructions are executed by a processor, the processor is caused to execute the steps of the separable code construction method of any embodiment of the present invention.

[0019] In summary, the separable code construction method and system of the present application proposes a code length of 4 The construction is a new construction, which contains q 2 +q-2 codewords, the number of codewords can reach the optimal order. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0021] Figure 1 A flowchart of a separable code construction method provided by one embodiment of the present invention;

[0022] Figure 2 A structural block diagram of a separable code construction system provided by an embodiment of the present invention;

[0023] Figure 3 It is a schematic diagram of the structure of an electronic device provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0024] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0025] See also Figure 1 , which shows a flow chart of a separable code construction method of the present application.

[0026] like Figure 1 As shown, the separable code construction method specifically includes the following steps:

[0027] Step S101, constructing an association relationship between the anti-false accusation code and the separable code.

[0028] In this step, the separable code and the anti-fraud code are defined respectively, and the separable code is defined as follows:

[0029] Assumptions is an (n, M, q) code and t ≥ 2 is an integer, n, M, q are positive integers;

[0030] If for any subset Make and have holds, that is, there is at least one coordinate i, 1≤i≤n, such that Then it is called is a - Separable codes, or

[0031] The anti-fraud code is defined as:

[0032] Assumptions is a (n, M, q) code and t ≥ 2 is an integer;

[0033] If for any subset Make have That is, for any There is at least one coordinate i, 1≤i≤n, such that Then it is called is a t-framework-proof code, or t-FPC(n,M,q).

[0034] It should be noted that the associated relationships include:

[0035] (n, M, q) code yes If and only if are (t-1)-FPC(n,M,q) and t-SC(n,M,q);

[0036] t-FPC(n,M,q) is also

[0037] (4, M, q) code yes If and only if the preset condition is met, the preset condition is:

[0038] is 2-FPC(4,M,q);

[0039] c does not contain the codeword set Δ i and its conjugation class, i∈{1,2,3,4,5,6,7,8,9}.

[0040] Step S102, let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2.

[0041] In this step, the expression of C1 is:

[0042] C1={(x,x+i,x+2i,x+3i)T |x∈Z q-1 , i∈Z q-1 \{0}},

[0043] The expression of C2 is:

[0044] C2={(∞,x,x,x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T |x∈Z q-1},

[0045] Where i is any positive integer, x∈Z q-1 , i∈Z q-1 \{0}, Z q-1 The domain of positive integers.

[0046] Step S103, let a, b∈C be different code words, then I(a, b)=2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol.

[0047] Step S104, let n and t be positive integers such that q≥2, n≥2 and t≥2, and define the code C to be q-element -separable code and its length is n, then

[0048] Step S105, when M≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 code words to achieve the optimal order, where M is a positive integer.

[0049] In summary, the method of this application proposes a code length of 4 The construction is a new construction, which contains q 2 +q-2 codewords, the number of codewords can reach the optimal order.

[0050] In a specific embodiment, let n, M, q be positive integers, and Q = {0, 1, ..., q-1}. A set of size M It is called (n, M, q) code and each is called a codeword. and any 1≤i≤n, The i-th dimension coordinate set of is defined as: For any subset of codewords definition The following code is

[0051] The definitions of separable codes (SC) and frame-proof codes (FPC) are also given.

[0052] Definition 1: Assumption is a (n, M, q) code and t ≥ 2 is an integer. If for any subset Make and have holds, that is, there is at least one coordinate i, 1≤i≤n, such that Then it is called is a - Separable codes, or

[0053] Definition 2: Assumption is a (n, M, q) code and t ≥ 2 is an integer. If for any subset Make have That is, for any There is at least one coordinate i, 1≤i≤n, such that Then it is called is a t-framework-proof code, or t-FPC(n,M,q).

[0054] The relationship between frameproof code (FPC) and separable code is as follows:

[0055] 1. (n, M, q) code yes If and only if They are (t-1)-FPC(n,M,q) and t-SC(n,M,q).

[0056] 2. t-FPC(n,M,q) is also

[0057] (4, M, q) code yes If and only if it satisfies the following two conditions:

[0058] 1) is 2-FPC(4,M,q);

[0059] 2) Does not contain the following codeword set Δ i and its conjugation class, i∈{1,2,3,4,5,6,7,8,9}.

[0060]

[0061] Where a≠b, c≠d, x≠y,

[0062]

[0063] Among them, a≠b≠c, w≠q, f≠e, x≠y.

[0064]

[0065] Among them, a1≠b1≠c1, a2≠b2≠c2 and a3≠b3≠c3.

[0066]

[0067] where a1≠b1≠c1, a2≠b2≠c2, a3≠b3≠c3 and x≠y

[0068]

[0069] Among them, a1≠b1≠c1, a2≠b2≠c2, a3≠b3≠c3 and a4≠b4≠c4.

[0070] The proof of the equivalence condition is as follows. The relationship between frameproof code (FPC) and separable code shows that the necessity of the equivalence condition is established. As for its sufficiency, the proof by contradiction is used. Assume that both 1) and 2) of the equivalence condition are established, but C is not Then there are two different subsets A, Such that 1≤|A|, |B|≤3 and desc(A)=desc(B).

[0071] First, prove |A|=|B|=3. Without loss of generality, assume 1≤|A|≤2. From the equivalent condition 1), we know that C is 2-FPC(4, M, q). We discuss each case.

[0072] When |B|=1, A=B must be true, which is a contradiction. This situation does not hold.

[0073] When |B| = 2 or 3, then there exists a codeword c∈B such that However This contradicts the fact that C is 2-FPC(4, M, q), so this situation does not hold.

[0074] According to the above discussion, we know that |A|=|B|=3. So assuming

[0075]

[0076] Where a, b and c are all different, and d, e and f are all different. And because desc(A) = desc(B), the following equation holds:

[0077] {a1, b1, c1} = {d1, e1, f1}, {a2, b2, c2} = {d2, e2, f2},

[0078] {a3, b3, c3} = {d3, e3, f3}, {a4, b4, c4} = {d4, e4, f4}.

[0079] Then we will prove three cases: |A∩B|=2, 1 or 0.

[0080] (I) |A∩B|=2. Without loss of generality, assuming b=e, c=f and a≠d, it is equivalent to A={a, b, c}, B={b, c, d}. The codeword matrix of A∪B is listed as follows:

[0081]

[0082] where a, b, c and d are all different.

[0083] Since C is 2-FPC(4, M, q), we have and

[0084] because Without loss of generality, assume Since {a1, b1, c1} = {b1, c1, d1}, then d1 = c1 and a1 = b1 hold. Then the codeword matrix of A∪B is listed as follows:

[0085]

[0086] because Without loss of generality, assume Since {a2, b2, c2} = {b2, c2, d2}, then d2 = a2. Then the codeword matrix of A∪B is as follows:

[0087]

[0088] Also because Without loss of generality, assume Since {a3, b3, c3} = {b3, c3, d3}, then d3 = b3 and a3 = c3 hold. Then the codeword matrix of A∪B is listed as follows:

[0089]

[0090] Finally, let's discuss {a4, b4, c4} = {b4, c4, d4}. There are two cases:

[0091] a4=d4. Then A∪B is the conjugation class of Δ1, which is a contradiction. This situation does not exist.

[0092] a4≠d4. Since {a4, b4, c4}={b4, c4, d4}, then a4=b4 and c4=d4, or a4=c4 and b4=d4. If a4=b4 and c4=d4, then A∪B is the conjugated class of Δ2, which is a contradiction and does not exist. If a4=c4 and b4=d4, then A∪B is the conjugated class of Δ3, which is a contradiction and does not exist.

[0093] (II) |A∩B|=1. Without loss of generality, assuming c=f, then A={a,b,c},B={c,d,e}. Then the codeword matrix of A∪B is listed as follows:

[0094]

[0095] where a, b, c, d and e are all different.

[0096] Since C is 2-FPC, we have and

[0097] because Without loss of generality, assume Since {a1, b1, c1} = {b1, c1, d1}, then d1 = c1 and a1 = b1 hold. Then the codeword matrix of A∪B is listed as follows:

[0098]

[0099] because Without loss of generality, assume Since {a2, b2, c2} = {d2, c2, e2}, then e2 = c2 and a2 = b2 = d2 hold. Then the codeword matrix of A∪B is listed as follows:

[0100]

[0101] because Without loss of generality, assume Since {a3, b3, c3} = {b3, c3, d3}, then d3 = a3 and e3 = b3 hold. Then the codeword matrix of A∪B is listed as follows:

[0102]

[0103] Also because Without loss of generality, assume Since {a4, b4, c4} = {d4, c4, e4}, then d4 = b4 and e4 = a4 hold. Then the codeword matrix of A∪B is listed as follows:

[0104]

[0105] Obviously, b∈desc({d,e}) holds. This contradicts the definition of 2-FPC, so this situation does not exist.

[0106] (III) |A∩B|=0. Similarly, since C is a 2-FPC, we have: and

[0107] because Without loss of generality, assume Since {a1, b1, c1} = {b1, c1, d1}, then d1 = c1 and a1 = b1. Then,

[0108]

[0109] because Without loss of generality, assume Since {a2, b2, c2} = {d2, e2, f2}, d2 = b2. Then,

[0110]

[0111] because Without loss of generality, assume Since {a3, b3, c3} = {d3, e3, f3}, d3 = a3. Then,

[0112]

[0113] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0114] if It holds true, because {a1, b1, c1}={d1, e1, f1}, then c1=e1.

[0115] if It holds true, because {a2, b2, c2} = {d2, e2, f2}, then c2 = e2.

[0116] if It is true, because {a3, b3, c3} = {d3, e3, f3}, then c3 = e3.

[0117] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0118] if It holds true, because {a1, b1, c1}={d1, e1, f1}, then b1=e1.

[0119] if It holds true, because {a2, b2, c2} = {d2, e2, f2}, then b2 = e2.

[0120] if It is true, because {a3, b3, c3} = {d3, e3, f3}, then b3 = e3.

[0121] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0122] if It holds true, because {a1, b1, c1}={d1, e1, f1}, then a1=e1.

[0123] if It holds true, because {a2, b2, c2} = {d2, e2, f2}, then a2 = e2.

[0124] if It is true, because {a3, b3, c3} = {d3, e3, f3}, then a3 = e3.

[0125] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0126] (4.1) If It holds true, because {a1, b1, c1}={d1, e1, f1}, then c1=f1.

[0127] (4.2) If It holds true, because {a2, b2, c2} = {d2, e2, f2}, then c2 = f2.

[0128] (4.3) If It is true, because {a3, b3, c3} = {d3, e3, f3}, then c3 = f3.

[0129] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0130] (5.1) If It holds true, because {a1, b1, c1} = {d1, e1, f1}, then b1 = f1.

[0131] (5.2) If It holds true, because {a2, b2, c2} = {d2, e2, f2}, then b2 = f2.

[0132] (5.3) If It is true, because {a3, b3, c3} = {d3, e3, f3}, then b3 = f3.

[0133] because Then there is Then there must be an index 1≤i≤3 such that Established.

[0134] (6.1) If It holds true, because {a1, b1, c1}={d1, e1, f1}, then a1=f1.

[0135] (6.2) If It holds true, because {a2, b2, c2} = {d2, e2, f2}, then a2 = f2.

[0136] (6.3) If It is true, because {a3, b3, c3} = {d3, e3, f3}, then a3 = f3.

[0137] It is easy to check that for any i∈{1,2,3} or any i∈{4,5,6} and for any j∈{1,2,3}, once situation (ij) occurs, then situation (ij) cannot occur for any i′∈{1,2,3}\{i} or i′∈{4,5,6}\{i}. So there are 3×2×1×3×2×1=36 situations to consider.

[0138] {(1.3)，(2.2)，(3.1)，(4.2)，(5.1)，(6.3)}，{(1.1)，(2.3)，(3.2)，(4.2)，(5.1)，(6.3)}，

[0139] {(1.1)，(2.3)，(3.2)，(4.2)，(5.3)，(6.1)}，{(1.1)，(2.3)，(3.2)，(4.3)，(5.1)，(6.2)}，

[0140] {(1.1)，(2.3)，(3.2)，(4.3)，(5.2)，(6.1)}，{(1.2)，(2.1)，(3.3)，(4.1)，(5.3)，(6.2)}，

[0141] {(1.2)，(2.1)，(3.3)，(4.2)，(5.3)，(6.1)}，{(1.2)，(2.1)，(3.3)，(4.3)，(5.1)，(6.2)}，

[0142] {(1.2)，(2.1)，(3.3)，(4.3)，(5.2)，(6.1)}，{(1.2)，(2.3)，(3.1)，(4.1)，(5.3)，(6.2)}，

[0143] {(1.2)，(2.3)，(3.1)，(4.2)，(5.1)，(6.3)}，{(1.2)，(2.3)，(3.1)，(4.3)，(5.2)，(6.1)}，

[0144] {(1.3)，(2.1)，(3.2)，(4.1)，(5.3)，(6.2)}，{(1.3)，(2.1)，(3.2)，(4.2)，(5.1)，(6.3)}，

[0145] {(1.3)，(2.1)，(3.2)，(4.3)，(5.2)，(6.1)}，{(1.3)，(2.2)，(3.1)，(4.1)，(5.3)，(6.2)}，

[0146] {(1.3)，(2.2)，(3.1)，(4.2)，(5.3)，(6.1)}，{(1.3)，(2.2)，(3.1)，(4.3)，(5.1)，(6.2)}，

[0147] {(1.1), (2.2), (3.3), (4.1), (5.2), (6.3)}, {(1.1), (2.2), (3.3), (4.1), (5.3), (6.2)},

[0148] {(1.1), (2.2), (3.3), (4.2), (5.1), (6.3)}, {(1.1), (2.2), (3.3), (4.2), (5.3), (6.1)},

[0149] {(1.1), (2.2), (3.3), (4.3), (5.1), (6.2)}, {(1.1), (2.2), (3.3), (4.3), (5.2), (6.1)},

[0150] {(1.1), (2.3), (3.2), (4.1), (5.2), (6.3)}, {(1.1), (2.3), (3.2), (4.1), (5.3), (6.2)},

[0151] {(1.2), (2.1), (3.3), (4.1), (5.2), (6.3)}, {(1.2), (2.1), (3.3), (4.2), (5.1), (6.3)},

[0152] {(1.2), (2.3), (3.1), (4.1), (5.2), (6.3)}, {(1.2), (2.3), (3.1), (4.2), (5.3), (6.1)},

[0153] {(1.3), (2.1), (3.2), (4.1), (5.2), (6.3)}, {(1.3), (2.1), (3.2), (4.3), (5.1), (6.2)},

[0154] {(1.3), (2.2), (3.1), (4.1), (5.2), (6.3)}, {(1.3), (2.2), (3.1), (4.3), (5.2), (6.1)},

[0155] {(1.2), (2.3), (3.1), (4.3), (5.1), (6.2)}, {(1.3), (2.1), (3.2), (4.2), (5.3), (6.1)}.

[0156] 1) For the first 18 cases, take the first case {(1.3), (2.2), (3.1), (4.2), (5.1), (6.3)} as an example to prove. Then we have:

[0157]

[0158] in and Since {a2, b2, c2} = {d2, e2, f2} = {b2, c2}, then a2∈{b2, c2}. And because It can be seen that a2=c2, and similarly b3=c3. Therefore

[0159]

[0160] Then, we know that a4=d4. Using the proof by contradiction, since {a4, b4, c4}={d4, e4, f4}, assuming a4≠d4, a4∈{e4, f4}. Therefore, a∈{e, f}, which contradicts the definition of 2-FPC. Similarly, we can get b4=d4. Therefore, according to

[0161] {a4, b4, c4} = {d4, e4, f4}, then c4 = e4 = f4.

[0162]

[0163] in b2≠c2, a3≠c3 and a4≠c4. When a1=b1, then b∈desc({d,e}), a contradiction. So a1, b1, c1 are all different. Obviously, this matrix is ​​equivalent to Δ4, a contradiction, so this situation does not hold. Similarly, it can be proved that the remaining 17 cases of A∪B are all conjugate classes of Δ4, so the other cases do not hold.

[0164] 2) For the next 16 cases, we take the cases {(1.1), (2.2), (3.3), (4.1), (5.2), (6.3)} as examples. Then we have:

[0165]

[0166] Because {a4, b4, c4} = {d4, e4, f4}, then d4∈{a4, b4, c4}. If d4=a4, then d∈desc({a, e}), a contradiction. Similarly, when d4=b4, d∈desc({b, e}) and when d4=c4, d∈desc({c, e}). So this case does not hold, and the same logic can be used to prove that the other 15 cases do not hold either.

[0167] 3) The last two cases can be classified into the same category. Take the cases {(1.3), (2.1), (3.2), (4.2), (5.3), (6.1)} as examples to prove that:

[0168]

[0169] Since {a4, b4, c4} = {d4, e4, f4}, it can be divided into the following 25 cases for discussion.

[0170] 3-1) a4 = b4 = c4 = d4 = e4 = f4;

[0171] 3-2) a4 = b4 = d4 = e4, c4 = f4 and a4 ≠ c4;

[0172] 3-3) a4 = b4 = d4 = f4, c4 = e4 and a4 ≠ c4;

[0173] 3-4) a4 = b4 = e4 = f4, c4 = d4 and a4 ≠ c4;

[0174] 3-5) a4 = c4 = d4 = e4, b4 = f4 and a4 ≠ b4;

[0175] 3-6) a4 = c4 = d4 = f4, b4 = e4 and a4 ≠ b4;

[0176] 3-7) a4 = c4 = e4 = f4, b4 = d4 and a4 ≠ b4;

[0177] 3-8) b4 = c4 = d4 = e4, a4 = f4 and a4 ≠ c4;

[0178] 3-9) b4 = c4 = d4 = f4, a4 = e4 and a4 ≠ c4;

[0179] 3-10) b4 = c4 = e4 = f4, a4 = d4 and a4 ≠ c4;

[0180] 3-11) a4 = b4 = d4, c4 = e4 = f4 and a4 ≠ c4;

[0181] 3-12) a4 = b4 = e4, c4 = d4 = f4 and a4 ≠ c4;

[0182] 3-13) a4 = b4 = f4, c4 = d4 = e4 and a4 ≠ c4;

[0183] 3-14) a4 = c4 = d4, b4 = e4 = f4 and a4 ≠ b4,;

[0184] 3-15) a4 = c4 = e4, b4 = d4 = f4 and a4 ≠ b4;

[0185] 3-16) a4 = c4 = f4, b4 = d4 = e4 and a4 ≠ b4;

[0186] 3-17) b4 = c4 = d4, a4 = e4 = f4 and a4 ≠ c4;

[0187] 3-18) b4=c4=e4,a4=d4=f4 and a4≠c4;

[0188] 3-19) b4=c4=f4,a4=d4=e4 and a4≠c4;

[0189] 3-20) a4, b4, c4 are all different and d4=a4, e4=b4, f4=c4;

[0190] 3-21) a4, b4, c4 are all different and d4=a4, e4=c4, f4=b4;

[0191] 3-22) a4, b4, c4 are all different and d4=b4, e4=a4, f4=c4;

[0192] 3-23) a4, b4, c4 are all different and d4=b4, e4=c4, f4=a4;

[0193] 3-24) a4, b4, c4 are all different and d4=c4, e4=a4, f4=b4;

[0194] 3-25)a4, b4, c4 are all different and d4=c4, e4=b4, f4=a4.

[0195] For case 3-1), the matrix is ​​as follows:

[0196]

[0197] where |{a i , b i , c i}|=3 holds for any i∈{1,2,3}. Obviously, this matrix is ​​equivalent to Δ5, which is a contradiction.

[0198] For cases 3-2) to 3-10), we take case 3-2) as an example. Then we have

[0199]

[0200] where |{a i , b i , c i}|=3 holds for any i∈{1,2,3} and a4≠c4. Obviously, this matrix is ​​equivalent to Δ6, which is a contradiction, so this situation does not hold. Similarly, it can be proved that A∪B in the remaining 8 cases are all conjugate classes of Δ6, so the other cases do not hold either.

[0201] For cases 3-11) to 3-19), we take case 3-11) as an example. Then we have

[0202]

[0203] where |{a i , b i , c i}|=3 holds for any i∈{1,2,3} and a4≠c4. Obviously, this matrix is ​​equivalent to Δ7, which is a contradiction, so this situation does not hold. Similarly, it can be proved that A∪B in the remaining 8 cases are all conjugate classes of Δ7, so the other cases do not hold either.

[0204] For cases 3-20), 3-23), and 3-24), we take case 3-20) as an example. Then we have

[0205]

[0206] where |{a i , b i , c i}|=3 holds for any i∈{1,2,3} and a4≠c4. Obviously, this matrix is ​​equivalent to Δ8, which is a contradiction, so this situation does not hold. Similarly, it can be proved that A∪B in the remaining two cases are conjugate classes of Δ8, so the other cases do not hold either.

[0207] For cases 3-21), 3-22), and 3-25), we take case 3-21) as an example. Then we have

[0208]

[0209] where |{a i , b i , c i}|=3 holds for any i∈{1,2,3} and a4≠c4. Obviously, this matrix is ​​equivalent to Δ9, which is a contradiction, so this situation does not hold. Similarly, it can be proved that A∪B in the remaining two cases are conjugate classes of Δ9, so the other cases do not hold either.

[0210] The proof of the equivalent condition is completed. Next, we will propose And the code length is 4.

[0211] Let q be a positive integer coprime to 6, ∞ is an element and Then, define code C in Z q-1 ∪{∞}, C=C1∪C2, where:

[0212] C1={(x,x+i,x+2i,x+3i) T |x∈Z q-1 , i∈Z q-1 \{0}},

[0213] C2={(∞,x,x,x) T, (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T |x∈Z q-1}.

[0214] Prove that it is a

[0215] First, to prove that it is 2-FPC, we need to have the following conclusion: let a, b∈C be different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}. Proof by contradiction.

[0216] Suppose there exists an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}Obviously I(a, b)=2.

[0217] On the contrary, assuming I(a, b) = 2, it can be divided into the following three cases.

[0218] 2-1) a∈C1, b∈C1. This situation does not satisfy the assumption. Obviously, the minimum Hamming moment of subcode C1 is 3.

[0219] 2-2) a∈c1, b∈C2. Then there exist x, y∈Z q-1 $and$i∈Z q-1 \{0}, such that a=(x,x+i,x+2i,x+3i) and b∈{(∞,y,y,y) T , (y,∞,y,y) T , (y, y, ∞, y) T , (y, y, y, ∞) T}. With b = (∞, y, y, y) T For example, since i≠0, I(a, b)≤1, which contradicts the assumption, so this subcase does not exist. Similarly, we can get other subcases b∈{(y,∞,y,y) T , (y, y, ∞, y) T , (y, y, y, ∞) T} does not exist either.

[0220] 2-3) a∈C2, b∈C2. Then there exist x, y∈Z q-1 , so that:

[0221] a∈{(∞,x,x,x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T},

[0222] And b∈{(∞,y,y,y) T , (y,∞,y,y) T , (y, y, ∞, y) T , (y, y, y, ∞) T}. Assuming x≠y, we can directly check that I(a,b)≤1, which is a contradiction, so x=y. That is, a,b∈{(∞,x,x,x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}.

[0223] Then it is proved that the structure is 2-FPC. Using the proof by contradiction, assuming that C is not 2-FPC, there are three different codewords a, b, c∈C, such that a∈desc({b, c}). Therefore, I(a, b)=1 and I(a, c)=3, or I(a, b)=2 and I(a, c)=2. According to the structures of C1 and C2, I(a, c)=3 does not exist, so only I(a, b)=2 and I(a, c)=2 need to be considered. From the conclusions proved above, it can be seen that there exists an element x∈Z q-1 , so that a, b, c ∈ {(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}. Obviously, the condition a∈desc({b,c}) does not hold, so the proof is obtained.

[0224] Next, we prove that C does not contain any of the nine forbidden configurations Δ1, Δ2, Δ3, Δ4, Δ5, Δ6, Δ7, Δ8, and Δ9.

[0225] Assume that code C contains the forbidden configuration Δ1 as follows:

[0226]

[0227] Where a≠b, c≠d, Obviously I(a, d) = 2. From the above conclusion, we can know that there exists an element x∈Z q-1, so that a, d∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}. Then {a, d} = {(∞, x, x, x) T , (x, ∞, x, x) T}, let a(3) = d(3) = e and a(4) = d(4) = h. Without loss of generality, assume a = (∞, x, x, x) T and d = (x, ∞, x, x) T , then e=h=x. Then a=(∞,e,e,e) T , d = (e, ∞, e, e) T Therefore, b=(∞,∞,f,m), b(1)=b(2)=∞. It is not difficult to see that Contradiction. So code C does not contain the forbidden configuration Δ1. According to the same idea of ​​proof by contradiction, we can also get that the other eight forbidden configurations are not in it, so code C is a

[0228] According to the theorem: Let q, n and t be positive integers such that q ≥ 2, n ≥ 2 and t ≥ 2. Let C be a q-element -separable code and its length is n, then

[0229] Then we can get M≤2q 2 For any And the construction is |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords can achieve the optimal order.

[0230] See also Figure 2 , which shows a structural block diagram of a separable code construction system of the present application.

[0231] like Figure 2 As shown, the separable code construction system 200 includes a construction module 210 , a first definition module 220 , an assumption module 230 , a second definition module 240 and a construction module 250 .

[0232] The construction module 210 is configured to construct an association relationship between the anti-fraud code and the separable code; the first definition module 220 is configured to set q to be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C = C1 ∪ C2; Assume that module 230 is configured so that a, b∈C are different codewords, then I(a, b) = 2 if and only if there is an element x∈Zq-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol; the second definition module 240 is configured to let n and t be positive integers and make q≥2, n≥2 and t≥2, and the definition code C is q-element -separable code and its length is n, then Construction module 250, configured to 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords to achieve the optimal order.

[0233] It should be understood that Figure 2 Modules and references documented in Figure 1 Therefore, the operations and features described above for the method and the corresponding technical effects are also applicable to Figure 2 The modules in it will not be described in detail here.

[0234] In some other embodiments, the embodiments of the present invention further provide a computer-readable storage medium having a computer program stored thereon, wherein when the program instructions are executed by a processor, the processor executes the separable code construction method in any of the above method embodiments;

[0235] As an implementation mode, the computer-readable storage medium of the present invention stores computer-executable instructions, and the computer-executable instructions are configured as follows:

[0236] Constructing the correlation between anti-fraud codes and separable codes;

[0237] Let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2;

[0238] Let a, b∈C be different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol;

[0239] Let n and t be positive integers such that q ≥ 2, n ≥ 2 and t ≥ 2, and define the code C to be q-ary -separable code and its length is n, then

[0240] In M≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords to achieve the optimal order.

[0241] The computer-readable storage medium may include a program storage area and a data storage area, wherein the program storage area may store an operating system, an application required for at least one function; the data storage area may store data created according to the use of the separable code construction system, etc. In addition, the computer-readable storage medium may include a high-speed random access memory, and may also include a memory, such as at least one disk storage device, a flash memory device, or other non-volatile solid-state storage device. In some embodiments, the computer-readable storage medium may optionally include a memory remotely arranged relative to the processor, and these remote memories may be connected to the separable code construction system via a network. Examples of the above-mentioned network include, but are not limited to, the Internet, an intranet, a local area network, a mobile communication network, and combinations thereof.

[0242] Figure 3 is a schematic diagram of the structure of an electronic device provided by an embodiment of the present invention, such as Figure 3 As shown, the device includes: a processor 310 and a memory 320. The electronic device may also include: an input device 330 and an output device 340. The processor 310, the memory 320, the input device 330 and the output device 340 may be connected via a bus or other means. Figure 3 In the example, the connection through the bus is taken. The memory 320 is the above-mentioned computer-readable storage medium. The processor 310 executes various functional applications and data processing of the server by running the non-volatile software programs, instructions and modules stored in the memory 320, that is, the separable code construction method of the above-mentioned method embodiment is realized. The input device 330 can receive input digital or character information, and generate key signal input related to user settings and function control of the separable code construction system. The output device 340 may include display devices such as display screens.

[0243] The electronic device can execute the method provided by the embodiment of the present invention, and has the functional modules and beneficial effects corresponding to the execution method. For technical details not described in detail in this embodiment, please refer to the method provided by the embodiment of the present invention.

[0244] As an implementation mode, the electronic device is applied to a separable code construction system and is used for a client, and includes: at least one processor; and a memory connected to the at least one processor in communication; wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor so that the at least one processor can:

[0245] Constructing the correlation between anti-fraud codes and separable codes;

[0246] Let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2;

[0247] Let a, b∈C be different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞x,x,x,x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T}, where T is the transposition symbol;

[0248] Let n and t be positive integers such that q ≥ 2, n ≥ 2 and t ≥ 2, and define the code C to be q-ary -separable code and its length is n, then

[0249] In m≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 codewords to achieve the optimal order.

[0250] Through the description of the above implementation methods, those skilled in the art can clearly understand that each implementation method can be implemented by means of software plus a necessary general hardware platform, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solution is essentially or the part that contributes to the prior art can be embodied in the form of a software product, and the computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, a disk, an optical disk, etc., including a number of instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods of each embodiment or some parts of the embodiment.

[0251] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A separable code construction method for constructing a separable code with t=3 anti-collusion users and a code length of 4, characterized in that: The method comprises: Constructing the correlation between anti-fraud codes and separable codes; Let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2; Let a, b∈C be different codewords, then I(a, b) = 2 if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T }, where T is the transposition symbol; Let n and t be positive integers such that q ≥ 2, n ≥ 2 and t ≥ 2, and define the code C to be q-ary -separable code and its length is n, then In M≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 code words to achieve the optimal order, where M is a positive integer.

2. A separable code construction method according to claim 1, characterized in that: Before constructing the association relationship between the anti-false accusation code and the separable code, the method further includes: The separable code and the anti-fraud code are defined respectively, and the separable code is defined as: Assumptions is an (n, M, q) code and t ≥ 2 is an integer, n, M, q are positive integers; If for any subset Make and have holds, that is, there is at least one coordinate i, 1≤i≤n, such that Then it is called is a - Separable code, or The anti-fraud code is defined as: Assumptions is a (n, M, q) code and t ≥ 2 is an integer; If for any subset Make have That is, for any There is at least one coordinate i, 1≤i≤n, such that Then it is called is a ,-framework-proof code, or t-FPC(n,M,q).

3. A separable code construction method according to claim 2, characterized in that: The association relationship includes: (n, M, q) code yes If and only if are (t-1)-FPC(n,M,q) and t-SC(n,M,q); t-FPC(n,M,q) is also (4, M, q) code yes If and only if the preset condition is met, the preset condition is: is 2-FPC(4,M,q); The codeword set Δi and its conjugate class, i∈{1, 2, 3, 4, 5, 6, 7, 8, 9}, are not included.

4. A separable code construction method according to claim 1, characterized in that: in, The expression of C1 is: C1={(x, x+i, x+2i, x+3i)T|x∈Z q-1 ,i∈Z q-1 \{0}}, The expression of C2 is: C2={(∞,x,x,x) 2 ,(x,∞,x,x) 2 ,(x,x,∞,x) 2 ,(x,x,x,∞) 2 |x∈Z q-1 }, Where i is any positive integer, x∈Z q-1 , i∈Z q-1 \{0}, Z q-1 The domain of positive integers.

5. A separable code construction system for separable code construction with t=3 anti-collusion users and a code length of 4, characterized in that: The system comprises: A construction module configured to construct an association relationship between the anti-fraud code and the separable code; The first definition module is configured to let q be a positive integer coprime to 6, ∞ is an element and Definition code C in Z q-1 ∪{∞}, C=C1∪C2; Assume that the module is configured so that a, b∈C are different codewords, then I(a, b) = if and only if there is an element x∈Z q-1 , so that a, b∈{(∞, x, x, x) T , (x, ∞, x, x) T , (x, x, ∞, x) T , (x, x, x, ∞) T }, where T is the transposition symbol; The second definition module is configured to let n and t be positive integers and make q≥2, n≥2 and t≥2, and the definition code C is q-element -separable code and its length is n, then Construction modules, configured to be in M≤2q 2 For any When |C|=|C1∪C2|=|C1|+|C2|=(q-1)(q-2)+4(q-1)=q 2 +q-2 code words to achieve the optimal order, where M is a positive integer.

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