On the minimum distance of elliptic curve codes

Li, Jiyou; Wan, Daqing; Zhang, Jun

doi:10.1109/isit.2015.7282884

Cited by 12 publications

(12 citation statements)

References 17 publications

(22 reference statements)

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“…The following result is a direct consequence of Corollary 3.4 and Lemma 4.1. [22,8,14], and the dual code GC(D, H, a) in Theorem 4.2 of GC(D, G, a) is a LCD code with parameters [22,14,8], which is verified by MAGMA. Table I lists the numbers of rational points of some elliptic curves over F 2 m [17].…”

Section: General Constructions Of Lcd Codes From Elliptic Curvesmentioning

confidence: 72%

Complementary Dual Algebraic Geometry Codes

Mesnager

Tang

2018

IEEE Trans. Inform. Theory

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Linear complementary dual (LCD) codes is a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry (AG) LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from elliptic curves. Then, some explicit LCD codes from elliptic curve are presented. MDS codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we introduce a construction mechanism for obtaining LCD codes from any algebraic curve and derive some explicit LCD codes from hyperelliptic curves and Hermitian curves. Index TermsLinear complementary dual codes, algebraic geometry codes, algebraic curves, elliptic curves, non-special divisors I. INTRODUCTIONLinear complementary dual (LCD) cyclic codes over finite fields were first introduced and studied by Massey [15] in 1964. In the literature LCD cyclic codes were referred to as reversible cyclic codes. It is well-known that LCD codes are asymptotically good. Furthermore, using the full dimension spectra of linear codes, Sendrier showed that LCD codes meet the asymptotic Gilbert-Varshamov bound [21]. Afterwards, LCD codes have been extensively studied in literature. In particular many properties and constructions of LCD codes have been obtained. Yang and Massey have provided in [24] a necessary and sufficient condition under which a cyclic code have a complementary dual. Dougherty et al. have developed in [6] a linear programming bound on the largest size of a LCD code of given length and minimum distance. Esmaeili and Yari analyzed LCD codes that are quasi-cyclic [7]. Muttoo and Lal constructed a reversible code over F q [18]. Tzeng and Hartmann proved that the minimum distance of a class of reversible cyclic codes is greater than the BCH bound [19]. In [13] Li et al. studied a class of reversible BCH codes proposed in [12] and extended the results on their parameters. As a byproduct, the parameters of some primitive BCH codes have been analyzed. Some of the obtained codes are optimal or have the best known parameters. In [3] Carlet and Guilley investigated an application of LCD codes against side-channel attacks, and presented several constructions of LCD codes. In [5], Ding et al. constructed several families of LCD cyclic codes over finite fields and analyzed their paramet...

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Section: General Constructions Of Lcd Codes From Elliptic Curvesmentioning

confidence: 72%

Complementary Dual Algebraic Geometry Codes

Mesnager

Tang

2018

IEEE Trans. Inform. Theory

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“…This problem has been extensively studied in [23]- [25]. By Lemma (3), the asymptotic formula N (t, B, P) has a main term n t /N and an error term M t .…”

Section: A Main Resultsmentioning

confidence: 99%

“…Roughly speaking, this formula significantly improves the classical inclusion-exclusion sieve for distinct coordinate counting problems. We cite it here without proof, and there are many interesting applications of this new sieve method [23]- [25].…”

Section: Technical Toolsmentioning

confidence: 99%

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Algebraic Geometric Secret Sharing Schemes over Large Fields Are Asymptotically Threshold

Fan

Chen²,

Zhao³

2021

Preprint

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In Chen-Cramer Crypto 2006 paper [7] algebraic geometric secret sharing schemes were proposed such that the "Fundamental Theorem in Information-Theoretically Secure Multiparty Computation" by Goldwasser and Wigderson [3] and Chaum, Crépeau and Damgård [6] can be established over constant-size base finite fields. These algebraic geometric secret sharing schemes defined by a curve of genus g over a constant size finite field F q is quasi-threshold in the following sense, any subset of u ≤ T − 1 players (non qualified) has no information of the secret and any subset of u ≥ T + 2g players (qualified) can reconstruct the secret. It is natural to ask that how far from the threshold these quasi-threshold secret sharing schemes are? How many subsets of u ∈ [T, T + 2g − 1] players can recover the secret or have no information of the secret?In this paper it is proved that almost all subsets of u ∈ [T, T + g − 1] players have no information of the secret and almost all subsets of u ∈ [T + g, T + 2g − 1] players can reconstruct the secret when the size q goes to the infinity and the genus satisfies lim g √ q = 0. Then algebraic geometric secret sharing

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“…The sieving argument in [21] has been used to obtain a good asymptotic formula for N (D, k, b) in the more general case that D is close to G, for instance, when |D| ≥ 2 3 |G|. In the case that G is the group of rational points on an elliptical curve over a finite field, please refer to [22] for a concrete example.…”

mentioning

confidence: 99%