Fast Encoding of AG Codes Over C_abCurves

Abstract

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called C ab curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some C ab curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity Õpn 3{2 q resp. Õpqnq for AG codes over any C ab curve satisfying very mild assumptions, where n is the code length and q the base field size, and Õ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity Õpnq for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities Õpn 5{4 q and Õpn 3{2 q respectively.

Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation

“…The following lemma bounds the cost of computing a reduced Gröbner basis of Γ(P), which we use only to bound the complexity of our precomputation; it rests on [2] which is currently in review. The interpolation algorithm of that paper is a small generalisation of Pan's interpolation algorithm on grids [17].…”

“…We prove that if P ⊆ K 2 is random of fixed cardinality n, and if |K| ≫ n 2 log(n) then P is balanced with high probability 2 . Similarly, if G is square-free and R is uniformly random, then ⟨G, y − R⟩ is balanced with high probability.…”

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

“…In this direction, one-point AG codes on some family of curves, including Kummer type curves, are especially appealing. For instance, they have recently been shown to have subquadratic encoding [BRS20].…”

Fast Decoding of AG Codes

“…Then F " F q px, yq and x b `αy a `gpx, yq " 0, where α P F q zt0u and gpX, Y q P F q rX, Y s has pa, bq-weighted degree strictly less then ab. The curve defined by the equation X b `αY a `gpX, Y q " 0 is sometimes called a C ab -curve or a Miura-Kayima curve [28]; codes defined over such curves are of particular interest for practical applications, as they can be encoded efficiently [6]. When it comes to decoding, the additional assumptions that G " mP 8 and that D ´nP 8 is a principal divisor were used in [4] to decode the code C L pD, Gq in complexity Õpℓ 5 a 3 pn `gqq.…”