JP2012247575A - Information vector coding device, information vector decoding device, information vector coding method, information vector decoding method and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an information vector coding device, etc which handles desired linear codes while ensuring high confidential protection performance with a simple configuration.SOLUTION: The information vector coding device inputs a generator matrix G of a desired [l+n, k] linear code C and an MDS-rank e of a dual code Cof C to create a partial matrix [A B], and then converts the partial matrix [A B] into a row reduced echelon matrix [AQB]. The information vector coding device also performs a row change of the row reduced echelon matrix [AQB] to output a systematic generator matrix Gof a conversion code C. The information vector coding device encodes an information vector and a k-l-th random number vector using the systematic generator matrix Gof the conversion code C, a l-th to output n code word symbols.

Description

  The present invention relates to an information vector coding apparatus, an information vector restoration apparatus, an information vector coding method, an information vector restoration method, and a program which are intended for an arbitrary linear code and have strong secret protection characteristics with a simple configuration.

  In recent years, a method of constructing a secret sharing method based on an existing error correction code (referred to as “secret sharing code”) has attracted attention (see, for example, Non-Patent Documents 1 to 5). The secret sharing code can make use of the characteristics of the original existing error correction code, and can use the encoder and decoder as they are to generate shared information and restore the secret information, or it can be distributed by error correction processing. It is possible to detect and correct information falsification.

  As shown in Non-Patent Document 1, McEliece et al. Pointed out for the first time that the threshold method based on Shamir's polynomial interpolation (see Non-Patent Document 6) can be configured with Reed-Solomon codes. Furthermore, as shown in Non-Patent Document 3, Piepryzk et al. Point out that the threshold method can be composed of a maximum distance separation code (MDS code).

  Here, the MDS code achieves the Singleton limit and has an ideal property as an algebraic code, but there is a restriction that the maximum code length (maximum number of distributed information) is limited by the size of a finite field. Have.

  Also, the secret sharing method having an access structure other than the threshold type cannot be simply configured using an MDS code. Therefore, as shown in Non-Patent Document 4, Massey extended the result of McEliece et al. To a configuration with a general linear code. Furthermore, as shown in Non-Patent Document 5, dela Cruz et al. Have extended the Massey configuration for one element of secret information to a configuration for a vector composed of a plurality of elements.

  On the other hand, in the conventional threshold method based on polynomial interpolation shown in Non-Patent Document 6, a method of encoding a plurality of information (vectors) is called a “ramp type threshold method”. The ramp-type threshold method can reduce the communication bandwidth and storage usage by processing multiple pieces of information together, but the information vector itself from the number of pieces of distributed information less than the threshold Even if it cannot be decoded, there is a possibility that a part of the vector element can be uniquely decoded. A method in which the vector itself cannot be decrypted (a method in which no one of the elements can be definitely decrypted) is called a “ramp threshold method having strong secret protection characteristics”. As shown in Non-Patent Document 2, Nishiara et al. Showed a configuration method of a strong ramp-type threshold method that can be realized by polynomial interpolation.

R.J. McEliece and DV Sarvaate, “On sharing secrets and Reed-Solomon codes,” Commun. ACM, vol. 24, no. 9, pp. 583-584, 1981. M. Nishiara and K. Takizawa, "Strongly secure secret sharing scheme with ramp threshold based on Shamir's polynomial interpolation scheme," IEICE Transactions on Fundamentals, vol. J92-A, no. 12, pp. 1009-1013,2009. J. Pieprzyk and X.-M. Zhang, “Ideal threshold schemes from MDS codes,” in Proceeding of ICISC 2002, Lecture Notes in Computer Science, p3, p3, p3. J. L. Massey, “Some applications of coding theory in cryptography,” in Codes and Ciphers. Cryptography and Coding IV, pp. 33-47, 1995. R. dela Cruz, A. Meyer, and P. Sole, “An extension of Massesy scheme for secret sharing,” in Proceeding of IEEE Information Theory 10, 20 p. A. Shamir, “How to share a secret,” Commun. ACM, vol. 22, no. 11, pp. 612-613, 1979.

  However, the secret sharing code proposed so far is a specific code (Reed-Solomon (RS) code or Maximum Distance Separable (MDS) code including the RS code) although the secret protection characteristic is guaranteed. There is a problem that it is a special configuration based on the code, a configuration based on an arbitrary linear code, but a strong secret protection characteristic cannot be guaranteed, and the configuration of dela Cruz et al. However, the method does not have strong secret protection characteristics, for example, even if you do not want to decrypt any element of the information vector, some of the elements can be uniquely decrypted. There is a problem that it may become.

  Therefore, the present invention has been made in view of the above-described problems, and is an information vector encoding device, an information vector restoration device, an information vector code, which is intended for an arbitrary linear code and has a strong secret protection characteristic with a simple configuration. It is an object of the present invention to provide a conversion method, an information vector restoration method, and a program.

  The present invention proposes the following matters in order to solve the above problems. In addition, in order to make an understanding easy, although the code | symbol corresponding to embodiment of this invention is attached | subjected and demonstrated, it is not limited to this.

(1) The present invention is an information vector encoding device that receives an l-dimensional information vector and a kl-dimensional random number vector and outputs n codeword symbols by an encoder. any [l + n, k] and the generator matrix G of the linear code C, the partial matrix selector constituting the partial matrix [a B] and MDS-rank e of dual codes C ^⊥ of C as an input (e.g., FIG. 2 And a row basic deformer (for example, corresponding to the row basic deformer 220 in FIG. 2) for converting the partial matrix [A B] into a _reduced staircase matrix [A _RRE QB]. , A code comprising a column exchanger (for example, equivalent to the column exchanger 230 in FIG. 2) that performs column exchange of the reduced staircase matrix [A _RRE QB] and outputs a systematic generation matrix G _t of the conversion code C _t systematic generator matrix G of the conversion code C _t obtained from the conversion device L dimension information and vector and k-l-dimensional random vector encodes proposes information vector coding apparatus which outputs a n number of codeword symbols by.

According to the present invention, an encoder forms a submatrix [AB] with an arbitrary [l + n, k] linear code C generator matrix G and an MDS-rank e of C dual code C ^{入 力} as inputs. A submatrix selector that performs a column exchange between the submatrix [A B] and a row basic deformer that converts the submatrix [A B] into an _irreducible staircase matrix [A _RRE QB], and a conversion code C and l-dimensional information vector and k-l-dimensional random vector encoded by systematic generator matrix G _t of the transformation code C _t obtained from the code conversion device consisting of a column exchanger for outputting a systematic generator matrix G _t of _t N codeword symbols are output. Therefore, it has a strong secret protection characteristic that any element of the information vector cannot be deterministically decoded from any d1 ( ^C⊥ ) -2 codeword symbols. Here, d1 (C ^⊥ ) represents the first generalized Hamming weight of the code C ^⊥ .

(2) The present invention inputs k + e codeword symbols out of n codeword symbols generated by the information vector encoding device of (1), and converts them into an l-dimensional information vector by a decoder. an information vector restoring device for decoding the k-l-dimensional random vector, the decoder, the generator matrix G of an arbitrary [l + n, k] linear code C, and MDS-rank e of dual codes C ^⊥ of C To the submatrix selector (for example, equivalent to the submatrix A selector 210 in FIG. 2), and the submatrix [A B] to the _reduced staircase matrix [A _RRE QB]. row basic deformation for converting (e.g., corresponding to a row basic deformation 220 in FIG. 2) and performs column swapping of echelon matrix [a _{RRE QB],} outputs the systematic generator matrix G _t of the transformation code C _t Column exchanger (eg, column exchanger 23 in FIG. 2) To the systematic generator matrix G _t of the transformation code C _t obtained from the code conversion apparatus consisting of a considerable) proposes information vector restoring device for decoding the l-dimensional information vector and k-l-dimensional random vector.

According to the present invention, decoder constitute any [l + n, k] and the generator matrix G of linear code C, the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as an input The submatrix selector, the row basic deformer for converting the submatrix [A B] to the _reduced staircase matrix [A _RRE QB], and column exchange of the _reduced staircase matrix [A _RRE QB] are performed, and the conversion code C _t the systematic generator matrix G _t obtained from the code conversion device consisting of a column exchanger outputs the converted code C _t systematic generator matrix G _t, decodes the l-dimensional information vector and k-l-dimensional random vector. Therefore, the transformed matrix G _t is obtained by performing basic transformation of the C generation matrix G with the parameter e. By using the code C ′ generated by G _t , an information vector can be decoded from any k + e codeword symbols.

  (3) The present invention has a regular matrix Q multiplier (for example, equivalent to the regular matrix Q multiplier 410 in FIG. 6) that receives an l-dimensional information vector and a kl-dimensional random number vector as input and performs linear transformation using the regular matrix Q. ) And an arbitrary [l + n, k] linear code C encoder (for example, the code in FIG. 6) that inputs a k-dimensional vector linearly converted in the regular matrix Q multiplier and outputs n codeword symbols. And an information vector encoding device characterized by comprising the following:

  According to the present invention, the regular matrix Q multiplier receives the l-dimensional information vector and the kl-dimensional random number vector as inputs, and performs linear transformation using the regular matrix Q. An encoder of an arbitrary [l + n, k] linear code C inputs a k-dimensional vector linearly converted in a regular matrix Q multiplier, and outputs n codeword symbols. Therefore, strong secret protection characteristics can be obtained by preprocessing of the existing C encoder without converting the code itself.

(4) The present invention inputs k + e codeword symbols out of n codeword symbols generated by the information vector encoding device of (3), and inputs them into an l-dimensional information vector and k−1 dimensions. An information vector restoration device for decoding into a random number vector, which receives the k + e codeword symbols and outputs a k-dimensional vector (for example, equivalent to the decoder 510 in FIG. 8), the k-dimensional vector is multiplied by the inverse matrix ^{Q -1} of the regular matrix Q, the inverse matrix ^{Q -1} multiplier for decoding the l-dimensional information vector and k-l-dimensional random vector (e.g., the inverse matrix ^{Q -1} multiplier 520 in FIG. 8 And an information vector restoration device characterized by comprising:

According to the present invention, the decoder inputs k + e codeword symbols and outputs a k-dimensional vector. The inverse matrix Q ⁻¹ multiplier multiplies the k-dimensional vector by the inverse matrix Q ⁻¹ of the regular matrix Q and decodes it into an l-dimensional information vector and a kl-dimensional random number vector. Therefore, strong secret protection characteristics can be obtained by post-processing of the existing C decoder without converting the code itself.

(5) The present invention is an information vector encoding method that receives an l-dimensional information vector and a kl-dimensional random number vector and outputs n codeword symbols, and includes an arbitrary [l + n, k] linear code C and the generator matrix G of a first step of configuring the partial matrix [a B] and MDS-rank e of dual codes C ^⊥ of C as an input (e.g., corresponding to step S101 of FIG. 3), a partial matrix a second step of converting [a B] to echelon matrix _[a RRE QB] (e.g., corresponding to step S102 of FIG. 3) and performs column swapping of echelon matrix _[a RRE QB], converted a third step of outputting the systematic generator matrix _{G t} of the code _{C t} (e.g., corresponding to step S103 of FIG. 3) and, by the conversion code _{C t} systematic generator matrix _{G t} and l-dimensional information vector k- l-dimensional random number vector And Goka, fourth step of outputting the n number of code word symbol (e.g., corresponding to step S104 of FIG. 3) proposes a, the information vector coding method characterized by comprising the.

According to the present invention, configured with any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input submatrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB]. Further, performs column swapping of echelon matrix _[A RRE QB], transform coding _{C t} of outputs systematic generator matrix _{G t,} transform coding _{C t} l dimensional information vector and k by systematic generator matrix _{G t} of Encode l-dimensional random number vector and output n codeword symbols. Therefore, it has a strong secret protection characteristic that any element of the information vector cannot be deterministically decoded from any d1 ( ^C⊥ ) -2 codeword symbols. Here, d1 (C ^⊥ ) represents the first generalized Hamming weight of the code C ^⊥ .

(6) The present invention inputs k + e codeword symbols out of n codeword symbols generated by the information vector encoding method of (5), and inputs them into an l-dimensional information vector and k−1 dimensions. an information vector reconstruction method for decoding a random vectors, any [l + n, k] submatrix and generator matrix G of linear code C, and MDS-rank e of dual codes C ^⊥ of C as an input [a B] And a second step (for example, the step of FIG. 5) for converting the partial matrix [A B] into the _reduced staircase matrix [A _RRE QB] (for example, the step of FIG. 5). And a third step (for example, corresponding to step S203 in FIG. 5) of performing column exchange of the reduced staircase matrix [A _RRE QB] and outputting the systematic generation matrix G _t of the conversion code C _t And the conversion code C _t And a fourth step (for example, corresponding to step S204 in FIG. 5) for decoding into an l-dimensional information vector and a k-l-dimensional random number vector using the systematic generator matrix G _t Proposed method.

According to the present invention, configured with any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input submatrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB]. Further, it performs column swapping of echelon matrix _[A RRE QB], the conversion code _{C t} of outputs systematic generator matrix _{G t,} transform coding _{C t} systematic generator matrix _{G t,} and l-dimensional information vector Decode into k-1 dimensional random number vector. Therefore, the transformed matrix G _t is obtained by performing basic transformation of the C generation matrix G with the parameter e. By using the code C ′ generated by G _t , an information vector can be decoded from any k + e codeword symbols.

  (7) In the present invention, a first step (for example, corresponding to step S301 in FIG. 7) of performing linear transformation using a regular matrix Q using an l-dimensional information vector and a k−1-dimensional random number vector as inputs, and the linear Proposing an information vector encoding method characterized by comprising a second step (for example, corresponding to step S302 in FIG. 7) of inputting a converted k-dimensional vector and outputting n codeword symbols. ing.

  According to the present invention, an l-dimensional information vector and a kl-dimensional random number vector are input, linear transformation is performed using a regular matrix Q, the linearly transformed k-dimensional vector is input, and n codeword symbols are output. . Therefore, strong secret protection characteristics can be obtained by preprocessing of the existing C encoder without converting the code itself.

(8) The present invention inputs k + e codeword symbols out of the n codeword symbols generated by the information vector encoding method of (7), and outputs them as an l-dimensional information vector and k−1 dimensions. An information vector restoration method for decoding into a random vector, the first step of inputting the k + e codeword symbols and outputting a k-dimensional vector (for example, corresponding to step S401 in FIG. 9), the k dimension A second step (for example, corresponding to step S402 in FIG. 9) of multiplying the vector by the inverse matrix Q ⁻¹ of the regular matrix Q and decoding into an l-dimensional information vector and a k−1-dimensional random number vector. A feature information vector restoration method is proposed.

According to the present invention, k + e codeword symbols are input, a k-dimensional vector is output, and the k-dimensional vector is multiplied by the inverse matrix Q ⁻¹ of the regular matrix Q to obtain an l-dimensional information vector and a k−1-dimensional random number. Decode into a vector. Therefore, strong secret protection characteristics can be obtained by post-processing of the existing C decoder without converting the code itself.

(9) The present invention is a program for causing a computer to execute an information vector encoding method that receives an l-dimensional information vector and a kl-dimensional random vector and outputs n codeword symbols. step [l + n, k] and the generator matrix G of the linear code C, the first step (e.g., FIG. 3 constituting the partial matrix [a B] and MDS-rank e of dual codes C ^⊥ of C as an input and corresponding to S101), the step of converting the partial matrix [a B] in echelon matrix _[a RRE QB] (e.g., corresponding to step S102 of FIG. 3) and, echelon matrix _[a RRE QB performs column swapping of, transform coding C _t third step of outputting the systematic generator matrix G _t (e.g., corresponding to step S103 of FIG. 3) and the transform coding C _t systematic generator matrix G _t By the first order A program for causing a computer to execute a fourth step (for example, corresponding to step S104 in FIG. 3) of encoding an information vector and a k−1-dimensional random number vector and outputting n codeword symbols. is suggesting.

According to the present invention, configured with any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input submatrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB]. Further, performs column swapping of echelon matrix _[A RRE QB], transform coding _{C t} of outputs systematic generator matrix _{G t,} transform coding _{C t} l dimensional information vector and k by systematic generator matrix _{G t} of Encode l-dimensional random number vector and output n codeword symbols. Therefore, it has a strong secret protection characteristic that any element of the information vector cannot be deterministically decoded from any d1 ( ^C⊥ ) -2 codeword symbols. Here, d1 (C ^⊥ ) represents the first generalized Hamming weight of the code C ^⊥ .

(10) The present invention inputs k + e codeword symbols out of n codeword symbols generated by the program of (9), and decodes them into an l-dimensional information vector and a k−1-dimensional random number vector. A computer program for causing a computer to execute the information vector restoration method, comprising: a generation matrix G of an arbitrary [l + n, k] linear code C and an MDS-rank e of a dual code C ^⊥ of C as a submatrix A first step (for example, corresponding to step S201 in FIG. 5) constituting [A B], and a second step (for example, converting the partial matrix [A B] into the _reduced staircase matrix [A _RRE QB] (for example, and equivalent) in step S202 in FIG. 5 performs column swapping of echelon matrix _[a RRE QB], a third step of outputting the systematic generator matrix _{G t} of the transformation code _{C t} (e.g., the steps of FIG. 5 And corresponding to 203), by systematic generator matrix _{G t} of the transformation code _{C t,} and a fourth step of decoding the l-dimensional information vector and k-l-dimensional random vector (e.g., corresponding to step S204 of FIG. 5), Has proposed a program to make a computer execute.

According to the present invention, configured with any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input submatrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB]. Further, it performs column swapping of echelon matrix _[A RRE QB], the conversion code _{C t} of outputs systematic generator matrix _{G t,} transform coding _{C t} systematic generator matrix _{G t,} and l-dimensional information vector Decode into k-1 dimensional random number vector. Therefore, the transformed matrix G _t is obtained by performing basic transformation of the C generation matrix G with the parameter e. By using the code C ′ generated by G _t , an information vector can be decoded from any k + e codeword symbols.

  (11) In the present invention, a first step (for example, corresponding to step S301 in FIG. 7) for performing linear transformation using a regular matrix Q using an l-dimensional information vector and a kl-dimensional random number vector as inputs, and the linear A program for causing a computer to execute a second step (for example, corresponding to step S302 in FIG. 7) of inputting a converted k-dimensional vector and outputting n codeword symbols is proposed.

  According to the present invention, an l-dimensional information vector and a kl-dimensional random number vector are input, linear transformation is performed using a regular matrix Q, the linearly transformed k-dimensional vector is input, and n codeword symbols are output. . Therefore, strong secret protection characteristics can be obtained by preprocessing of the existing C encoder without converting the code itself.

(12) The present invention inputs k + e codeword symbols out of n codeword symbols generated by the program of (11), and decodes them into an l-dimensional information vector and a k−1-dimensional random number vector. A first step of inputting the k + e codeword symbols and outputting a k-dimensional vector (for example, corresponding to step S401 of FIG. 9); A second step (for example, corresponding to step S402 in FIG. 9) of multiplying the k-dimensional vector by the inverse matrix Q ⁻¹ of the regular matrix Q and decoding into an l-dimensional information vector and a kl-dimensional random number vector; Has proposed a program to make a computer execute.

According to the present invention, k + e codeword symbols are input, a k-dimensional vector is output, and the k-dimensional vector is multiplied by the inverse matrix Q ⁻¹ of the regular matrix Q to obtain an l-dimensional information vector and a k−1-dimensional random number. Decode into a vector. Therefore, strong secret protection characteristics can be obtained by post-processing of the existing C decoder without converting the code itself.

  According to the present invention, a secret sharing code having strong secret protection characteristics can be configured using various linear codes. Therefore, depending on the code to be used, high-speed encoding and decoding are possible, or by using a code with a long code length, much more distributed information can be generated than the size of a finite field. It is possible to detect falsification of distributed information using the error correction capability of the code.

It is a figure which shows the structure of the information vector encoding apparatus which concerns on the 1st Embodiment of this invention. It is a figure which shows the structure of the code converter which concerns on the 1st Embodiment of this invention. It is a figure which shows the process of the information vector encoding apparatus which concerns on the 1st Embodiment of this invention. It is a figure which shows the structure of the information vector decompression | restoration apparatus which concerns on the 1st Embodiment of this invention. It is a figure which shows the process of the information vector decompression | restoration apparatus which concerns on the 1st Embodiment of this invention. It is a figure which shows the structure of the information vector encoding apparatus which concerns on the 2nd Embodiment of this invention. It is a figure which shows the process of the information vector encoding apparatus which concerns on the 2nd Embodiment of this invention. It is a figure which shows the structure of the information vector decompression | restoration apparatus which concerns on the 2nd Embodiment of this invention. It is a figure which shows the process of the information vector decompression | restoration apparatus which concerns on the 2nd Embodiment of this invention.

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.
Note that the constituent elements in the present embodiment can be appropriately replaced with existing constituent elements and the like, and various variations including combinations with other existing constituent elements are possible. Therefore, the description of the present embodiment does not limit the contents of the invention described in the claims.

<First Embodiment>
A first embodiment of the present invention will be described with reference to FIGS.

<Configuration of Information Vector Encoding Device>
As shown in FIG. 1, the information vector encoding device according to the present embodiment includes an encoder 100, and an l-dimensional information vector and k by a systematic generation matrix G _t of a conversion code C _t obtained from the code conversion device. Encode l-dimensional random number vector and output n codeword symbols.

<Configuration of Code Conversion Device>
As shown in FIG. 2, the code conversion apparatus 200 according to the present embodiment includes a partial matrix A selector 201, a row basic deformer 202, and a column exchanger 203.

The submatrix A selector 201 forms a submatrix [A B] with an arbitrary [l + n, k] linear code C generator matrix G and an MDS-rank e of C dual code C ^{入 力} as inputs. The row basic deformer 202 converts the submatrix [A B] into the _reduced staircase matrix [A _RRE QB]. The column exchanger 203 performs column exchange of the reduced staircase matrix [A _RRE QB] and outputs a systematic generation matrix G _t of the conversion code C _t .

<Processing of Information Vector Encoding Device>
The process of the information vector encoding device will be described with reference to FIG.

First, configurations and any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input (step S101), the partial matrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB] (step S102).

Then, a column exchange echelon matrix _[A RRE QB], and outputs the systematic generator matrix _{G t} of the transformation code _{C t} (step S103), l dimension by systematic generator matrix _{G t} of the transformation code _{C t} The information vector and the kl-dimensional random number vector are encoded, and n codeword symbols are output (step S104).

<Configuration of information vector restoration device>
As shown in FIG. 4, the information vector restoration apparatus according to the present embodiment includes a decoder 300 and is generated by the information vector encoding apparatus using the systematic generation matrix G _t of the conversion code C _t obtained from the code conversion apparatus. Among the n codeword symbols that have been generated, k + e codeword symbols are input and decoded into an l-dimensional information vector and a k−1-dimensional random number vector.

<Processing of information vector restoration device>
The processing of the information vector restoration device will be described with reference to FIG.

First, configurations and any [l + n, k] linear code C generator matrix G of the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as input (step S201), the partial matrix [ A B] is converted into a _reduced staircase matrix [A _RRE QB] (step S202).

Performs column swapping of echelon matrix _[A RRE QB], the conversion code _{C t} of outputs systematic generator matrix _{G t} (step S203), transform coding _{C t} systematic generator matrix _{G t,} l dimension information The vector and the k-1 dimensional random number vector are decoded (step S204).

As described above, according to the present embodiment, strong secret protection that any element of the information vector cannot be definitely decoded from any d1 ( ^Cｄ ) −2 codeword symbols. Has characteristics. Here, d1 (C ^⊥ ) represents the first generalized Hamming weight of the code C ^⊥ . Also, the transformed matrix G _t is obtained by performing basic transformation of the C generation matrix G with the parameter e. By using the code C ′ generated by G _t , an information vector can be decoded from any k + e codeword symbols.

<Second Embodiment>
A second embodiment of the present invention will be described with reference to FIGS.

<Configuration of Information Vector Encoding Device>
As shown in FIG. 6, the information vector encoding apparatus according to the present embodiment includes a regular matrix Q multiplier 410 and an encoder 420.

  The regular matrix Q multiplier 410 receives the l-dimensional information vector and the kl-dimensional random number vector as input, and performs linear transformation using the regular matrix Q. The encoder 420 receives the k-dimensional vector linearly converted in the regular matrix Q multiplier and outputs n codeword symbols.

<Processing of Information Vector Encoding Device>
The process of the information vector encoding device will be described with reference to FIG.

  First, an l-dimensional information vector and a kl-dimensional random number vector are input, and linear transformation is performed using a regular matrix Q (step S301). Then, the linearly converted k-dimensional vector is input and n codeword symbols are output (step S302).

<Configuration of information vector restoration device>
As shown in FIG. 8, the information vector restoration apparatus according to this embodiment includes a decoder 510 and an inverse matrix Q ⁻¹ multiplier 520.

Decoder 510 receives k + e codeword symbols and outputs a k-dimensional vector. The inverse matrix Q ⁻¹ multiplier 520 multiplies the k-dimensional vector by the inverse matrix Q ⁻¹ of the regular matrix Q and decodes it to the l-dimensional information vector and the k−1-dimensional random number vector.

<Processing of information vector restoration device>
The process of the information vector restoration device will be described with reference to FIG.

First, k + e codeword symbols are input, and a k-dimensional vector is output (step S401). Then, the k-dimensional vector is multiplied by the inverse matrix Q ⁻¹ of the regular matrix Q and decoded into an l-dimensional information vector and a kl-dimensional random number vector (step S402).

  As described above, according to the present embodiment, a strong secret protection characteristic can be obtained by preprocessing of an existing C encoder without converting the code itself. Further, strong secret protection characteristics can be obtained by post-processing of the existing C decoder without converting the code itself.

<Example>
In order to describe the present invention specifically, one embodiment of the present invention will be shown below.

<Definition of symbols>
First, _let F _{q be} a finite field with order q. At this time, q is a power of a prime number. In addition, a set support of indices of non-zero elements of the vector v = [v ₁ ,..., V _n ] εF ⁿ _q is defined as in Expression 1.

  Further, the Hamming weight of the vector v is defined as Equation 2.

[N, k] linear symbol C is defined by a k-dimensional linear subspace of F ⁿ _q . Each row of the generator matrix GεFk × nq that generates the [n, k] linear symbol C forms the basis of the [n, k] linear symbol C. [N, k] The dual code C ^の of the linear symbol C is defined as ^Equation 3.

  In addition, the r-th generalized Hamming weight (GHW) of C is defined by Equation 4.

Here, sup (D) is defined by sup (D) = {i: ∃ [x ₁ ,..., X _n ] ∈D, x _i ≠ 0}. {D _r : r = 1, ..., k} is called Hamming Weight Hierarchy (HWH) of [n, k] linear code C, and the smallest e satisfying d _{e + 1} (C) = n−k + e + 1 This is called MDS-rank. For i = 1,..., k,
μ _i (C) = n−k + i−d _i (C) is called C defect.

<Secret sharing code>
The number 5 and the information vector concealing (1 ≦ l ≦ k) C and [l + n, k] linear code on F _q. Further, the generator matrix of C is expressed as Equation 6. At this time, Equation 7 is the i-th column (1 ≦ i ≦ l + n) of G, and is assumed to be Equation 8.

When encoding the vector s using G, first, a number 10 satisfying the number 9 is randomly selected for ∀i = 1,. In the case where G is a systematic generator matrix represented by G = [I _k X], it may be configured as shown in Equation 12 using the k−1 dimensional random number vector of Equation 11. Here, I _k is a k × k unit matrix.

Next, a code word shown in Expression 13 is generated from the vector u. Then, the codeword symbols c _{l + 1} ,..., C _{l + n} are transmitted to the receiver through the transmission path, or are stored as distributed information by n managers. Hereinafter, this configuration is used as a basis.

<Anti-access set>
First, an anti-access set (Anti-Access Structure) and strong secret protection characteristics (Strongly-Secure) are defined below for a set of shared information of secret sharing.

<Definition 1; Anti-access set>
Let S ₁ ,..., S _{1 be} random variables that are statistically independent and follow a uniform distribution on F _q . At this _time, representing S 1, · · ·, the realization value taken by the _{S l s} 1, ···, and _{s l. Further,} c l + _{1, ···,} a _{c l + n} is a random variable on _{F q,} representing each of realizations _c l + _1, ···, and _{c l + n.} A set of random variables {C _i : iεX} for a set of indexes X is expressed as C _x .

Next, J⊂As. t. , | J | = β. [l + n, k] In the secret sharing scheme composed of the linear code C, when J satisfies the “strong secret protection characteristic” defined below, J is called an “anti-access set”.
Strong-Secure property; ∀t = 0,..., L-1, ∀D = {r ₁ ,..., R _β-t } ⊆J, ∀ε = {i ₁ ,. .., I _{t + 1} } ⊂ {1,..., L}, I (Sε; C _D ) = 0 holds. Here, I () indicates the mutual information amount.

<Definition 2; a-Method with strong secret protection characteristics>
[l + n, k] In the secret sharing scheme composed of the linear code C, ∀J⊂As. t. , | J | = a is an anti-access set, the method is called a method having a-strong secret protection characteristic. In the method having the a-strong secret protection characteristic, a characteristic that no element of the vector s is definitely decoded from a or less codeword symbols is guaranteed. a—Threshold-like access structure with strong secret protection characteristics is defined below.

<Definition 3; a-Threshold-like access structure with strong secret protection characteristics>
a-Threshold-like access structure having strong secret protection characteristics satisfies the following properties.
1. Availability: ∀I⊆As. t. , | I | ≧ T, the vector s can be uniquely decoded from the BI.
2. Confidentiality: ∀I⊆As. t. , | I | ≧ a is the anti-access set of definition 1.

<Configuration>
An arbitrary [l + n, k] linear code C is converted to construct a secret sharing code that satisfies definition 3. Here, the generator matrix of C is expressed as Equation 14. Further, the HWH of the dual code C ^⊥ of C is represented as {di (C⊥) = k + i−μi (C⊥): i = 1,..., K}, and MDS-rank is represented as e.

In the following manner, G is converted to G _t , that is, C is converted to obtain C _t .
1. Arbitrary k + e columns of G are selected to construct a submatrix A. The rows of G are exchanged so that G ′ = [A B]. (First k + e column may be sufficient)
2. _Let [A _RRE Q] be a matrix obtained by row-basic transformation of [A I _k ]. At this time, I _k is a k × k unit matrix, and A _RRE is an _irreducible step matrix obtained by subjecting A to a row basic transformation. Further, there is a relationship shown in Formula 15.

3. G ″ = GG ′ = [A _RRE QB] is obtained.
4). Column exchange is performed inside the block matrix A _RRE of G ″ to obtain A ′ _RRE = [I _k X]. At this time, there is a relationship of Formula 16.

5. Let G _t = [A ′ _RRE QB] = [I _k X QB]. Secret sharing code consists of code _{C t} by the _{G t} is the parameter T = k + e, in ^{_{a = d l (C ⊥)}} -2, satisfies the definition 3. That is, the vector s can be decoded by decoding processing from arbitrary k + e codeword symbols (distributed information). Further, no element of the vector s is definitely decoded from any code word symbol (distributed information) less than d ₁ (C ^⊥ ) -2.

  The processing of the information vector encoding device or the information vector restoration device is recorded on a computer-readable recording medium, and the program recorded on the recording medium is read by the information vector coding device or the information vector restoration device and executed. As a result, the information vector encoding apparatus or information vector restoration apparatus of the present invention can be realized. The computer system here includes an OS and hardware such as peripheral devices.

  Further, the “computer system” includes a homepage providing environment (or display environment) if a WWW (World Wide Web) system is used. The program may be transmitted from a computer system storing the program in a storage device or the like to another computer system via a transmission medium or by a transmission wave in the transmission medium. Here, the “transmission medium” for transmitting the program refers to a medium having a function of transmitting information, such as a network (communication network) such as the Internet or a communication line (communication line) such as a telephone line.

  The program may be for realizing a part of the functions described above. Furthermore, what can implement | achieve the function mentioned above in combination with the program already recorded on the computer system, and what is called a difference file (difference program) may be sufficient.

  The embodiments of the present invention have been described in detail with reference to the drawings. However, the specific configuration is not limited to the embodiments, and includes designs and the like that do not depart from the gist of the present invention. For example, although no particular mention was made regarding the decoding algorithm, for example, syndrome decoding or maximum likelihood decoding may be used, but the present invention is not limited to this.

100; Encoder 200; Code converter 210; Submatrix A selector 220; Row basic transformer 230; Column exchanger 300; Decoder 410; Regular matrix Q multiplier 420; Encoder 510; Decoder 520; Regular matrix Q ^-1 multiplier

Claims (12)

An information vector encoding device which receives an l-dimensional information vector and a k-l-dimensional random number vector and outputs n codeword symbols by an encoder,
The encoder is
A submatrix selector that constructs a submatrix [A B] by using a generator matrix G of an arbitrary [l + n, k] linear code C and an MDS-rank e of a dual code C ^⊥ of C, and a submatrix [A performs a row basic deformation for converting B] to echelon matrix _[a RRE QB], the column replacement of the echelon matrix _[a RRE QB], and outputs the systematic generator matrix _{G t} of the transformation code _{C t} The l-dimensional information vector and the kl-dimensional random number vector are encoded by the systematic generation matrix G _t of the conversion code C _t obtained from the code conversion apparatus including the column exchanger, and n codeword symbols are output. Information vector encoding device. Out of n codeword symbols generated by the information vector encoding device according to claim 1, k + e codeword symbols are input, and a decoder obtains them as an l-dimensional information vector and a k-l-dimensional random vector. An information vector restoration device for decoding into
The decoder comprises:
A submatrix selector that constructs a submatrix [A B] by using a generator matrix G of an arbitrary [l + n, k] linear code C and an MDS-rank e of a dual code C ^⊥ of C, and a submatrix [A performs a row basic deformation for converting B] to echelon matrix _[a RRE QB], the column replacement of the echelon matrix _[a RRE QB], and outputs the systematic generator matrix _{G t} of the transformation code _{C t} An information vector restoration device that decodes an l-dimensional information vector and a kl-dimensional random number vector using a systematic generation matrix G _t of a conversion code C _t obtained from a code conversion device including a column exchanger. a regular matrix Q multiplier that receives an l-dimensional information vector and a kl-dimensional random number vector as input and performs a linear transformation using the regular matrix Q;
An arbitrary [l + n, k] linear code C encoder that inputs a k-dimensional vector linearly transformed in the regular matrix Q multiplier and outputs n codeword symbols;
An information vector encoding device comprising: Information for inputting k + e codeword symbols out of n codeword symbols generated by the information vector encoding device according to claim 3 and decoding them into an l-dimensional information vector and a k-l-dimensional random vector. A vector restoration device,
A decoder that inputs the k + e codeword symbols and outputs a k-dimensional vector;
Is multiplied by the inverse matrix Q ^-1 of the regular matrix Q in the k-dimensional vectors, the inverse matrix Q ^-1 multiplier for decoding the l-dimensional information vector and k-l-dimensional random vector,
An information vector restoration device comprising: An information vector encoding method which receives an l-dimensional information vector and a kl-dimensional random number vector and outputs n codeword symbols,
And any [l + n, k] linear code C generator matrix G of a first step of constituting the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as an input,
A second step of converting the submatrix [A B] to an _irreducible staircase matrix [A _RRE QB];
A third step of performing column exchange of the reduced staircase matrix [A _RRE QB] and outputting a systematic generator matrix G _t of the conversion code C _t ;
A fourth step of encoding an l-dimensional information vector and a kl-dimensional random number vector with the systematic generator matrix G _t of the transform code C _t and outputting n codeword symbols;
An information vector encoding method comprising: Information for inputting k + e codeword symbols out of n codeword symbols generated by the information vector encoding method of claim 5 and decoding them into an l-dimensional information vector and a kl-dimensional random vector. A vector restoration method,
And any [l + n, k] linear code C generator matrix G of a first step of constituting the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as an input,
A second step of converting the submatrix [A B] to an _irreducible staircase matrix [A _RRE QB];
A third step of performing column exchange of the reduced staircase matrix [A _RRE QB] and outputting a systematic generator matrix G _t of the conversion code C _t ;
A fourth step of decoding into an l-dimensional information vector and a kl-dimensional random number vector by the systematic generator matrix G _t of the transform code C _t ;
An information vector restoration method comprising: a first step of taking a linear transformation by a regular matrix Q using an l-dimensional information vector and a k-l dimensional random vector as inputs;
A second step of inputting the linearly transformed k-dimensional vector and outputting n codeword symbols;
An information vector encoding method comprising: Information for inputting k + e codeword symbols out of n codeword symbols generated by the information vector encoding method of claim 7 and decoding them into an l-dimensional information vector and a kl-dimensional random vector. A vector restoration method,
A first step of inputting the k + e codeword symbols and outputting a k-dimensional vector;
A second step of multiplying the k-dimensional vector by an inverse matrix Q ⁻¹ of the regular matrix Q and decoding it into an l-dimensional information vector and a kl-dimensional random vector;
An information vector restoration method comprising: A program for causing a computer to execute an information vector encoding method that receives an l-dimensional information vector and a k−1-dimensional random number vector and outputs n codeword symbols,
And any [l + n, k] linear code C generator matrix G of a first step of constituting the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as an input,
A second step of converting the submatrix [A B] to an _irreducible staircase matrix [A _RRE QB];
A third step of performing column exchange of the reduced staircase matrix [A _RRE QB] and outputting a systematic generator matrix G _t of the conversion code C _t ;
A fourth step of encoding an l-dimensional information vector and a kl-dimensional random number vector with the systematic generator matrix G _t of the transform code C _t and outputting n codeword symbols;
A program that causes a computer to execute. An information vector restoration method for inputting k + e codeword symbols out of n codeword symbols generated by the program of claim 9 and decoding the codeword symbols into an l-dimensional information vector and a kl-dimensional random number vector. A program for causing a computer to execute,
And any [l + n, k] linear code C generator matrix G of a first step of constituting the partial matrix [A B] and MDS-rank e of dual codes C ^⊥ of C as an input,
A second step of converting the submatrix [A B] to an _irreducible staircase matrix [A _RRE QB];
A third step of performing column exchange of the reduced staircase matrix [A _RRE QB] and outputting a systematic generator matrix G _t of the conversion code C _t ;
A fourth step of decoding into an l-dimensional information vector and a kl-dimensional random number vector by the systematic generator matrix G _t of the transform code C _t ;
A program that causes a computer to execute. a first step of taking a linear transformation by a regular matrix Q using an l-dimensional information vector and a k-l dimensional random vector as inputs;
A second step of inputting the linearly transformed k-dimensional vector and outputting n codeword symbols;
A program that causes a computer to execute. An information vector restoration method for inputting k + e codeword symbols out of n codeword symbols generated by the program of claim 11 and decoding them into an l-dimensional information vector and a k-l-dimensional random vector. A program for causing a computer to execute,
A first step of inputting the k + e codeword symbols and outputting a k-dimensional vector;
A second step of multiplying the k-dimensional vector by an inverse matrix Q ⁻¹ of the regular matrix Q and decoding it into an l-dimensional information vector and a kl-dimensional random vector;
A program that causes a computer to execute.

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