Bases for Riemann–Roch Spaces of One-Point Divisors on an Optimal Tower of Function Fields

Noseda, Francesco; Oliveira, Gilvan; Quoos, Luciane

doi:10.1109/tit.2011.2179519

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…In our opinion, the similarity of settings between foldable codes and (asymptotically good) towers of curves makes the latter one of the most promising candidates for new foldable codes. If one could assert that a tower (for instance of Kummer extensions [NOQ12] or of Artin-Schreier type [GS95,CNT18]) comes with a sequence of compatible divisors, this would provide a sequence of foldable codes.…”

Section: Towards Other Foldable Ag Codes Familiesmentioning

confidence: 99%

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Bordage¹,

Nardi²

2020

Preprint

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In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(C, P, D) is a vector space associated to evaluations on P of functions in the Riemann-Roch space L C (D). The problem of testing proximity to an error-correcting code C consists in distinguishing between the case where an input word, given as an oracle, belongs to C and the one where it is far from every codeword of C. AG codes are good candidates to construct short proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap.We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes, known as the FRI protocol [BBHR18]. We identify suitable requirements for designing efficient IOPP systems for AG codes. In addition to proposing the first proximity test targeting AG codes, our IOPP admits quasilinear prover arithmetic complexity and sublinear verifier arithmetic complexity with constant soundness for meaningful classes of AG codes. We take advantage of the algebraic geometry framework that makes any group action on the curve that fixes the divisor D translate into a decomposition of the code C. Concretely, our approach relies on Kani's result that splits the Riemann-Roch space of any invariant divisor under this action into several explicit Riemann-Roch spaces on the quotient curve [Kan86]. Under some hypotheses, these spaces behave well enough to define an AG code C ′ on the quotient curve so that a proximity test to C can be reduced to one to C ′ . Iterating this process thoroughly, we end up with a membership test to a code with significantly smaller length.

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Section: Towards Other Foldable Ag Codes Familiesmentioning

confidence: 99%

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Bordage¹,

Nardi²

2020

Preprint

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“…We describe the action of the procedure on inputs G = (V, E) and m. We first observe that without loss of generality, we may assume that the graph G is 4-regular, since one can use standard techniques to transform any constraint graph of size n into a 4-regular constraint graph of size 4n in polynomial time 5 . This is similar to the reduction of 3SAT to the special case of 3SAT in which every variable appears at most three times.…”

Section: Proof Of Lemma 52mentioning

confidence: 99%

“…In [3,5,6] polynomial-time algorithms are given for some specific towers. We believe that it is possible to provide such a polynomial time algorithm also for the tower T , and hence to obtain an explicit construction of the codes C i in polynomial time.…”

Section: Remark A19mentioning

confidence: 99%

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Ben–Sasson

Kaplan

Kopparty

et al. 2016

J. ACM

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The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n ε , then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(t log t) for any language in NTIME(t).More specifically, for any ε > 0 we present (non-uniform) probabilistically checkable proofs (PCPs) of length 2 O(1/ε) · n that can be checked using n ε queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this paper for the first time for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth [Trans. Information Theory 2006].

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“…The tower M was introduced by A. Garcia and H. Stichtenoth in [4], where they established its asymptotic optimality. In [8], certain properties of this tower were explored to achieve an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields. Through the implementation of an algorithm in Scilab (www.scilab.org), specific levels of Weierstrass semigroups were explicitly determined.…”

mentioning

confidence: 99%

“…Through the implementation of an algorithm in Scilab (www.scilab.org), specific levels of Weierstrass semigroups were explicitly determined. By examining these semigroups in [8], we derived explicit expressions for the defining functions and minimal generators of Weierstrass semigroups in an algebraic manner, effectively computing essential divisors within the tower M. It is important to recall that the extension M j+1 /M j is a Kummer extension of degree 2, and for each positive integer i, there exists only one place P i (and corresponding places Q i , R i , S i ) of M i over P 1 := (x 1 = ∞) (or Q 1 := (x 1 = 0), R 1 := (x 1 = α), S 1 := (x 1 = −α) with α 2 = −1); for further details, refer to the proof of Proposition 5.3 in [4]. Notably, the places P i , Q i , R i , and S i are F-rational.…”

mentioning

confidence: 99%