On Squares of Cyclic Codes

Cascudo, Ignacio

doi:10.1109/tit.2018.2867873

Cited by 20 publications

(59 citation statements)

References 26 publications

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“…We obtain this result by describing the cyclic codes as a subfield subcode of an evaluation code and generalizing Theorem 3.3 in [3]. The proof of this proposition is very similar to the one in [3] and can be found in Appendix A.…”

Section: Constructions From Binary Cyclic Codesmentioning

confidence: 60%

“…Afterwards, we determine the product of two codes when both codes are constructed using the (u, u + v)-construction in Section 3. In Section 4, we restrict ourselves to squares of codes and exemplify what parameters we can achieve using cyclic codes in the (u, u + v)-construction in order to compare the parameters with the codes from [3].…”

Section: Results and Outlinementioning

confidence: 99%

“…Corollary 2.4: If we additionally assume that C 1 ⊇ · · · ⊇ C s , equality occurs in the bound in (3).…”

Section: Proposition 23mentioning

confidence: 99%

“…As a consequence of the BCH-bound, see for example [3], we have that the minimum distance of the code generated by g is greater than or equal to n − Amp(I) + 1.…”

Section: Constructions From Binary Cyclic Codesmentioning

confidence: 99%

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Squares of matrix-product codes

Cascudo

Gundersen²,

Ruano

2020

Finite Fields and Their Applications

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The component-wise or Schur product C * C ′ of two linear error-correcting codes C and C ′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C ′ . When C = C ′ , we call the product the square of C and denote it C * 2 . Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrixproduct codes, a general construction that allows to obtain new longer codes from several "constituent" codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u, u+v)-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).

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Section: Constructions From Binary Cyclic Codesmentioning

confidence: 60%

Section: Results and Outlinementioning

confidence: 99%

“…Corollary 2.4: If we additionally assume that C 1 ⊇ · · · ⊇ C s , equality occurs in the bound in (3).…”

Section: Proposition 23mentioning

confidence: 99%

“…As a consequence of the BCH-bound, see for example [3], we have that the minimum distance of the code generated by g is greater than or equal to n − Amp(I) + 1.…”

Section: Constructions From Binary Cyclic Codesmentioning

confidence: 99%

See 2 more Smart Citations

Squares of matrix-product codes

Cascudo

Gundersen²,

Ruano

2020

Finite Fields and Their Applications

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“…To carry it out, we consider specific BCH codes. Instead of the classical way, our BCH codes are regarded as subfield-subcodes of evaluation codes defined by evaluating univariate polynomials [11]. We consider this construction because it can be extended to evaluation by polynomials in several variables [20,18] which we hope will give better codes in the future.…”

Section: Asymmetric Eaqecc From Bch Codesmentioning

confidence: 99%

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

et al. 2020

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The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.2010 Mathematics Subject Classification. 81P70; 94B65; 94B05.

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Efficient UC Commitment Extension with Homomorphism for Free (and Applications)

Cascudo

Damgård

David

et al. 2019

Lecture Notes in Computer Science

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Homomorphic universally composable (UC) commitments allow for the sender to reveal the result of additions and multiplications of values contained in commitments without revealing the values themselves while assuring the receiver of the correctness of such computation on committed values. In this work, we construct essentially optimal additively homomorphic UC commitments from any (not necessarily UC or homomorphic) extractable commitment. We obtain amortized linear computational complexity in the length of the input messages and rate 1. Next, we show how to extend our scheme to also obtain multiplicative homomorphism at the cost of asymptotic optimality but retaining low concrete complexity for practical parameters. While the previously best constructions use UC oblivious transfer as the main building block, our constructions only require extractable commitments and PRGs, achieving better concrete efficiency and offering new insights into the sufficient conditions for obtaining homomorphic UC commitments. Moreover, our techniques yield public coin protocols, which are compatible with the Fiat-Shamir heuristic. These results come at the cost of realizing a restricted version of the homomorphic commitment functionality where the sender is allowed to perform any number of commitments and operations on committed messages but is only allowed to perform a single batch opening of a number of commitments. Although this functionality seems restrictive, we show that it can be used as a building block for more efficient instantiations of recent protocols for secure multiparty computation and zero knowledge non-interactive arguments of knowledge.

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On Squares of Cyclic Codes

Cited by 20 publications

References 26 publications

Squares of matrix-product codes

Squares of matrix-product codes

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

Efficient UC Commitment Extension with Homomorphism for Free (and Applications)

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