Polynomial Linear System Solving with Random Errors: new bounds and early termination technique
aa r X i v : . [ c s . S C ] F e b Polynomial Linear System Solving with Random Errors: newbounds and early termination technique
Eleonora Guerrini [email protected], U. Montpellier, CNRSMontpellier, France
Romain Lebreton [email protected], U. Montpellier, CNRSMontpellier, France
Ilaria Zappatore [email protected], LIXPalaiseau, France
ABSTRACT
This paper deals with the polynomial linear system solving witherrors (PLSwE) problem. Specifically, we focus on the evaluation-interpolation technique for solving polynomial linear systems andwe assume that errors can occur in the evaluation step. In thisframework, the number of evaluations needed to recover the so-lution of the linear system is crucial since it affects the number ofcomputations. It depends on the parameters of the linear system(degrees, size) and on a bound on the number of errors.Our work is part of a series of papers about PLSwE aiming toreduce this number of evaluations. We proved in [Guerrini et al.,Proc. ISIT’19] that if errors are randomly distributed, the bound ofthe number of evaluations can be lowered for large error rate.In this paper, following the approach of [Kaltofen et al., Proc. IS-SAC’17], we improve the results of [Guerrini et al., Proc. ISIT’19]in two directions. First, we propose a new bound of the number ofevaluations, lowering the dependency on the parameters of the lin-ear system, based on work of [Cabay, Proc. SYMSAC’71]. Second,we introduce an early termination strategy in order to handle theunnecessary increase of the number of evaluations due to overes-timation of the parameters of the system and on the bound on thenumber of errors.
Solving polynomial linear systems (PLS) of the form 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) where A is a nonsingular square matrix and 𝒃 is a vectorof polynomials over a finite field F 𝑞 is a classical computer alge-bra problem. The solution 𝒚 ( 𝑥 ) is a vector of rational functions.This problem can be efficiently solved by parallelizing the classi-cal evaluation-interpolation technique considering a network of 𝐿 nodes that independently compute the evaluations 𝐴 ( 𝛼 𝑗 ) and 𝒃 ( 𝛼 𝑗 ) at a given evaluation point 𝛼 𝑗 ∈ F 𝑞 and the solution ofthe evaluated system 𝒚 𝑗 = 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) . The nodes then sendthe so-obtained 𝒚 , . . . , 𝒚 𝐿 to the master node which finally per-forms a Cauchy interpolation to recover the solution 𝒚 ( 𝑥 ) . As in[BK14, KPSW17], this paper focuses on a scenario in which thenodes could make errors, possibly computing 𝒚 𝑗 ≠ 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) .After receiving all these evaluations, the master node performs a Cauchy interpolation with errors in order to recover the solution 𝒚 ( 𝑥 ) . The problem that the master node has to face, i.e. recoveringthe solution 𝒚 ( 𝑥 ) of the PLS given its evaluations, some of whicherroneous, is what we call Polynomial Linear System Solving witherrors (PLSwE).In order to solve PLSwE, we can exploit decoding techniquesof Reed-Solomon (RS) codes, as shown in [BK14, KPSW17]. Basi-cally they set out a system of linear equations ( key equation ) (as theWelch-Berlekamp decoding method, [BW86]) and bound 𝐿 ( i.e. the number of nodes, which coincides with the number of evaluationpoints 𝛼 𝑗 in some way to guarantee the uniqueness of the solution.The goal is to minimize the number 𝐿 of evaluation points neededto recover the solution or equivalently to maximize the bound onthe number of errors ( decoding radius ) that we could correct. In[BK14, KPSW17], as for classical RS codes they can correct up tothe unique decoding radius .In [KPSW17] it is shown that there are mainly two ways tobound 𝐿 : first, considering the problem as a generalized rationalfunction reconstruction (RFR) and analyze it in terms of some esti-mations of the solution degree. Second, exploit the linear algebrastructure of the problem taking also into account the degrees ofthe input matrix 𝐴 and of the vector 𝑏 . More recently, [GLZ19]presented an algorithm that corrects errors beyond the unique de-coding radius (equivalently with less evaluation points than [BK14,KPSW17]) recovering the solution for almost all errors. The ideais to remark that the PLSwE problem can be viewed as a gener-alization of the decoding of the Interleaved Reed Solomon codes(IRS).IRS can be seen as the simultaneous evaluation of a vector ofpolynomials.Results from decoding IRS codes show that, if errors are uni-formly distributed, the larger the dimension of the vector is, themore errors we can correct, exceeding the standard unique decod-ing radius (see [BKY03, BMS04, SSB07, SRM09, SSB10]), asymptot-ically reaching the optimal error capability of the Shannon bound([Sha48]).A first contribution of this work consists in the combination ofthe advantages of IRS decoding techniques from [GLZ19] with thecounting of [KPSW17] which exploits the linear algebra setting asin [Cab71] (see Section 3).Recall that our goal is to lower the number of evaluations in or-der to reduce the nodes computations, at the expense of potentiallyincreasing the complexity of interpolation by the master node.All the bounds of the number of evaluations introduced for PLSwEsolving depend on some upper bounds on the degree of the solu-tion 𝒚 ( 𝑥 ) that we want to recover and on the number of errors.These upper bounds could overestimate the actual degrees of 𝒚 ( 𝑥 ) and number of errors. The discrepancy between these quantitiesmay significantly overestimate the number of evaluations neededfor the computations compared to the actual number needed to re-cover the solution. We propose an early termination technique (asin [KPSW17]), an adaptive strategy which, starting from a mini-mal value of evaluation points, iteratively increments this numberuntil a nontrivial result is found. leonora Guerrini, Romain Lebreton, and Ilaria Zappatore In Section 4.1 we present an early termination technique for afixed bound 𝜏 on the number of errors, which is the classical er-ror correcting codes framework. However, the number of errorscould grow with the number of evaluations 𝐿 , which is graduallyincremented in the early termination technique. For this purpose,we present in Section 4.2 a scenario in which the error bound lin-early depends on 𝐿 . Compared to the early termination techniquesof [KPSW17], we decrease the number of evaluation points. In re-turn, our algorithm may fail for a small fraction of errors; we givean estimation of the success probability of our algorithm in pres-ence of random errors. To sum up, our second contribution is topropose an early termination strategy which benefits from the IRSdecoding approach, is sensitive to the real number of errors, andadapts to linear error bound. To the best of our knowledge, the de-pendency on the real number of errors is original in the literature.The paper is organized as follows: in Section 2 we recall the sce-nario of PLSwE with results revisited from literature, in Section 3we present a new bound of the number of evaluation points neededfor PLSwE solving in presence of random errors and finally in Sec-tion 4 we introduce an early termination algorithm that succeedsfor almost all errors. Let F 𝑞 a finite field of order 𝑞 . Consider a polynomial linear system(PLS), 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) (1)where 𝐴 ∈ F 𝑞 [ 𝑥 ] 𝑛 × 𝑛 is nonsingular and 𝒃 ∈ F 𝑞 [ 𝑥 ] 𝑛 × . This sys-tem admits only one solution 𝒚 = 𝒗 𝑑 ∈ F 𝑞 ( 𝑥 ) 𝑛 × , i.e. a vectorof rational functions with the same denominator. We assume thatgcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 is monic.The evaluation-interpolation [McC77] is a classic technique forsolving PLS. It consists in • (evaluation) the evaluation of 𝐴 and 𝒃 at 𝐿 ≤ 𝑞 distinct eval-uation points { 𝛼 , . . . , 𝛼 𝐿 } . In this work, for simplicity weomit the rank drop case study, i.e. we suppose that for any 𝛼 𝑗 the corresponding evaluated matrix 𝐴 ( 𝛼 𝑗 ) is still full rank.All the results of this work can be extended to the generalcase of rank drops (as in [KPSW17] for more details). • (Pointwise interpolation of the evaluated systems) Compute 𝒚 𝑗 = 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) , for any 𝑗 . • (Interpolation) Reconstruct ( 𝒗 , 𝑑 ) ∈ F 𝑞 [ 𝑥 ] ( 𝑛 + )× , given theevaluated solutions 𝒚 𝑗 for any 𝑗 and the degree bounds 𝑁 > deg ( 𝒗 ) : = max ≤ 𝑖 ≤ 𝑛 { deg ( 𝑣 𝑖 )} and 𝐷 > deg ( 𝑑 ) .In order to minimize the number of evaluation points needed to uniquely recover the solution one can consider L( 𝑁 , 𝐷 ) : = min { 𝑁 + 𝐷 − | {z } L 𝑅𝐹𝑅 , max { deg ( 𝐴 ) + 𝑁 , deg ( 𝒃 ) + 𝐷 } | {z } L 𝑃𝐿𝑆 } , (2)where deg ( 𝐴 ) : = max ≤ 𝑖,𝑗 ≤ 𝑛 deg ( 𝑎 𝑖,𝑗 ( 𝑥 )) . We use this notation tostress out the dependency of the degree bounds 𝑁 and 𝐷 . Recallthat L 𝑅𝐹𝑅 is the minimum number of evaluation points needed touniquely interpolate a rational function ( i.e. the Cauchy interpola-tion problem) [GG13, Section 5.7]. On the other hand, L 𝑃𝐿𝑆 is the minimum number of evaluation points needed to uniquely recovera rational function which is a solution of a PLS ([Cab71]). Indeed, inthis case we also assume to know the degrees of 𝐴, 𝒃 or their upperbound.We remark that if the bounds are tight, i.e. deg ( 𝒗 ) + = 𝑁 anddeg ( 𝑑 ) + = 𝐷 , then 𝐿 𝑃𝐿𝑆 < 𝐿 𝑅𝐹𝑅 if and only if deg ( 𝑑 ) > deg ( 𝐴 ) (see [KPSW17, Theorem 3.1]). In the following section we formal-ize and describe our error scenario. Fix 𝐿 pairwise distinct evaluation points { 𝛼 , . . . , 𝛼 𝐿 } . Assume thatany node, computes 𝒚 𝑗 = 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) ∈ F 𝑛 × 𝑞 for any 1 ≤ 𝑗 ≤ 𝐿 .These nodes could make some errors and compute 𝒚 𝑗 ≠ 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) . Notice that, in this error model, the number of errors coin-cides with the number of nodes which compute an incorrect result.We assume that after the nodes computations the master node re-ceives the following matrix 𝑌 = (cid:18) 𝒗 ( 𝛼 ) 𝑑 ( 𝛼 ) , . . . , 𝒗 ( 𝛼 𝐿 ) 𝑑 ( 𝛼 𝐿 ) (cid:19) + Ξ (3)where Ξ ∈ F 𝑛 × 𝐿𝑞 is the error matrix. We denote by Ξ ∗ ,𝑗 the 𝑗 thcolumn of Ξ . The error support of the error matrix is 𝐸 : = { 𝑗 | Ξ ∗ ,𝑗 ≠ } . After receiving 𝑌 , the master node has to recover thesolution ( 𝒗 , 𝑑 ) of the PLS (1). In this work, we focus on this step,which we call polynomial linear system solving with errors (PLSwE).Formally PLSwE refers to the problem of recovering ( 𝒗 , 𝑑 ) thesolution of (1) given, • ≤ 𝐿 ≤ 𝑞 distinct evaluation points in F 𝑞 , { 𝛼 , . . . , 𝛼 𝐿 } , • the degree bounds 1 ≤ 𝑁 , 𝐷 ≤ 𝐿 , such that 𝑁 > deg ( 𝒗 ) , 𝐷 > deg ( 𝑑 ) and deg ( 𝐴 ) , deg ( 𝒃 ) , • an upper bound on the number of errors occurred at theparallelization step, i.e. 𝜏 ≥ | 𝐸 | , where 𝐸 = { 𝑗 | Ξ ∗ ,𝑗 ≠ } , • the matrix 𝑌 as in (3). As in [BK14, KPSW17, GLZ19], in order to solve PLSwE, we searchfor solutions ( 𝝋 ,𝜓 ) = ( 𝜑 , . . . , 𝜑 𝑛 ,𝜓 ) ∈ F 𝑞 [ 𝑥 ] ( 𝑛 + )× of the follow-ing key equations 𝜑 𝑖 ( 𝛼 𝑗 ) = 𝑦 𝑖,𝑗 𝜓 ( 𝛼 𝑗 ) , deg ( 𝜑 𝑖 ) < 𝑁 + 𝜏, deg ( 𝜓 ) < 𝐷 + 𝜏. (4)for any 1 ≤ 𝑖 ≤ 𝑛 and 1 ≤ 𝑗 ≤ 𝐿 .Note that the key equations (4) are the vector generalization ofthe classic computer algebra problem of the Cauchy interpolation ([GG13, Section 5.7]).This approach is the generalization of the Welch-Berlekamp de-coding method [BW86] for Reed-Solomon codes. In this frame-work, it is crucial to determine the smallest number of evaluationpoints 𝐿 needed to guarantee the uniqueness of a solution of thesekey equations, in the sense that we explain in what follows.We denote S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 : = {( 𝝋 ,𝜓 ) ∈ F 𝑞 [ 𝑥 ] ( 𝑛 + )× satisfying (4) } . We now consider the F 𝑞 [ 𝑥 ] -module M spanned by the solu-tions of the key equations (4). More specifically, any element of M is a linear combination with polynomial coefficients of rank (M) olynomial Linear System Solving with Random Errors: new bounds and early termination technique solutions ( 𝝋 ,𝜓 𝑖 ) ∈ S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 ( i.e. a basis of M ). The case inwhich M is uniquely generated, i.e. rank (M) =
1, correspondsto what we refer to uniqueness of the solution of the key equa-tions. Indeed, in this case M is generated by only one element ( 𝝋 ,𝜓 ) ∈ S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 , i.e. for any ( 𝝋 ′ ,𝜓 ′ ) ∈ M , there exists a poly-nomial 𝑅 ∈ F 𝑞 [ 𝑥 ] such that ( 𝝋 ′ ,𝜓 ′ ) = ( 𝑅 𝝋 , 𝑅𝜓 ) . In other terms,if the polynomials 𝜓,𝜓 ′ are nonzero, the two vectors of rationalfunctions 𝝋 / 𝜓 and 𝝋 ′ / 𝜓 ′ are equal. Remark . Let Λ : = Î 𝑗 ∈ 𝐸 ( 𝑥 − 𝛼 𝑗 ) be the error locator polynomial , i.e. the monic polynomial of degree deg ( Λ ) = | 𝐸 | whose roots arethe erroneous evaluations .We have that ( Λ 𝒗 , Λ 𝑑 ) ∈ S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 . Indeed, for any 1 ≤ 𝑖 ≤ 𝑛 and 1 ≤ 𝑗 ≤ 𝐿 , Λ ( 𝛼 𝑗 ) 𝑣 𝑖 ( 𝛼 𝑗 ) = 𝑦 𝑖,𝑗 Λ ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) and we also havethat deg ( Λ 𝒗 ) < 𝑁 + 𝜏 and deg ( Λ 𝑑 ) < 𝐷 + 𝜏 .In [BK14, KPSW17] is provided the minimum number of pointswhich guarantees the uniqueness of the solutions of the key equa-tions (4). The following proposition is a restatement of this resultusing definitions and notations of this paper. We will later provethis result in a more general context (see Proposition 4.1. Proposition 2.2. If 𝐿 ≥ 𝐿 𝐾𝑃𝑆𝑊 : = L( 𝑁 + 𝜏, 𝐷 + 𝜏 ) + 𝜏 (see (2) ), then the rank of M , the F 𝑞 [ 𝑥 ] -module generated by solutions,is . By Remark 2.1, if ( 𝝋 ,𝜓 ) is a generator of M with 𝜓 monic, then ( 𝝋 ,𝜓 ) = ( Λ 𝒗 , Λ 𝑑 ) . This proposition tells us that if 𝐿 ≥ L( 𝑁 + 𝜏, 𝐷 + 𝜏 )+ 𝜏 = min { 𝑁 + 𝐷 − , max { deg ( 𝐴 ) + 𝑁 , deg ( 𝒃 ) + 𝐷 }} + 𝜏 , then rank (M) = ( 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 ) , i.e. S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 where 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 : = min { 𝑁 + 𝜏 − ( deg ( 𝒗 ) + | 𝐸 |) , 𝐷 + 𝜏 − ( deg ( 𝑑 ) + | 𝐸 |)} (5)By convention, if 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 ≤ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 = {( , )} .Indeed, notice thatdeg ( 𝑥 𝑖 Λ 𝒗 ) = 𝑖 + | 𝐸 | + deg ( 𝒗 ) ≤ 𝑁 − + 𝜏 deg ( 𝑥 𝑖 Λ 𝑑 ) = 𝑖 + | 𝐸 | + deg ( 𝑑 ) ≤ 𝐷 − + 𝜏 and so 𝑖 ≤ min { 𝑁 − + 𝜏 − ( deg ( 𝒗 )+| 𝐸 |) , 𝐷 − + 𝜏 − ( deg ( 𝑑 )+| 𝐸 |)} = 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 − . Remark . Let V 𝑚,𝑝 = ( 𝛼 𝑗 − 𝑖 ) ≤ 𝑖 ≤ 𝑚 ≤ 𝑗 ≤ 𝑝 . Consider the homogeneouslinear system related to (4). We observe that the set of solutions S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 is the kernel of the matrix 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = © « V 𝐿,𝑁 + 𝜏 − 𝐷 V 𝐿,𝐷 + 𝜏 . . . ... V 𝐿,𝑁 + 𝜏 − 𝐷 𝑛 V 𝐿,𝐷 + 𝜏 ª®®¬ (6)where for any 1 ≤ 𝑖 ≤ 𝑛 , 𝐷 𝑖 is the diagonal matrix whose elementson the diagonal are 𝑦 𝑖, , . . . , 𝑦 𝑖,𝐿 .In [BK14, KPSW17] was proposed an algorithm which computesa column echelon form of 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 in order to find the minimaldegree solution of S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 , i.e. ( Λ 𝒗 , Λ 𝑑 ) . Another approach tofind this solution could be to compute a basis (as for instance in[OS07, RS16]) of the F 𝑞 [ 𝑥 ] -module M . Indeed, we have seen inProposition 2.2 that if 𝐿 ≥ 𝐿 𝐾𝑃𝑆𝑊 , the module is generated by ( Λ 𝒗 , Λ 𝑑 ) .Note that we can recover ( 𝒗 , 𝑑 ) from ( Λ 𝒗 , Λ 𝑑 ) by dividing by Λ = gcd ( Λ 𝒗 , Λ 𝑑 ) . We denote by FindSolution ( 𝑌, 𝑁 + 𝜏, 𝐷 + 𝜏 ) the algorithm that computes ( 𝒗 , 𝑑 ) from S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 using one of theabove methods, followed by the division by the 𝑔𝑐𝑑 .We now observe that if 𝑑 ∈ F 𝑞 ( i.e. 𝐷 = vector of 𝑛 polynomials , given its evaluations,some of which erroneous. This problem can be viewed as the prob-lem of decoding 𝑛 codewords of a Reed-Solomon code of length 𝐿 and dimension 𝑁 . In this specific case, Proposition 2.2 tells us thatwith 𝐿 ≥ 𝑁 + 𝜏 − | 𝐸 | ≤ 𝜏 ≤ 𝐿 − 𝑁 = : 𝜏 wecan uniquely decode the 𝑛 − vector of RS codewords. From a codingtheory point of view, 𝜏 the unique decoding radius of an RS codeof length 𝐿 and dimension 𝑁 . We now recall that an 𝑛 -InterleavedRS code of length 𝐿 and dimension 𝑁 is the direct sum of 𝑛 -RScodes with the same length and dimension. An 𝑛 -IRS codewordis the evaluation of a vector of polynomials at 𝐿 distinct evalua-tion points. Therefore, PLSwE with constant 𝑑 can also be seenas the decoding of an 𝑛 -IRS codeword. The advantage of consid-ering the problem under this interleaving point of view consistsin the fact that we can extend the results of the decoding of IRS([BKY03, BMS04, SSB07, SRM09, SSB10] and correct beyond theunique decoding radius (or equivalently reduce the number of eval-uations needed to recover the solution of our PLS).Indeed, in [GLZ19] we proved that in the general case of PLSwE,we can reconstruct the solution ( 𝒗 , 𝑑 ) of the PLS with 𝐿 𝐺𝐿𝑍 : = 𝑁 + deg ( 𝑑 ) − + | 𝐸 | + l | 𝐸 | 𝑛 m evaluation points for almost all errors.In this result we assume to know exactly the actual degree of thedenominator 𝑑 and the actual number of errors | 𝐸 | = |{ 𝑗 | Ξ 𝑗, ∗ ≠ }| . Notice that this assumption is quite strong. For this reasonin this work we introduce a new bound on 𝐿 which generalizes 𝐿 𝐺𝐿𝑍 by assuming to know some upper bounds on the degreeof 𝑑 and of | 𝐸 | . Our new bound also takes into account the linearalgebra setting of the problem (see Equation (2)). There are two new contributions in this section. First, we relax theconstraint of [GLZ19] introducing a number of evaluations whichonly depends on some upper bounds on the degree of the denom-inator and on the number of errors. We also introduce another in-dependent counting on the number of evaluations that takes intoaccount deg ( 𝐴 ) , deg ( 𝒃 ) of the PLS (1) as in [Cab71, KPSW17]. Weprove that with 𝐿 ≥ 𝐿 𝐺𝐿𝑍 : = L( 𝑁 + 𝜏, 𝐷 + 𝜏 ) + l 𝜏𝑛 m (7)evaluation points, where L( 𝑁 + 𝜏, 𝐷 + 𝜏 ) is defined as in (2), wecan uniquely reconstruct the solution for almost all errors (Theo-rem 3.1). Theorem 3.1.
Let 𝜏 ≥ and 𝑛, 𝑁 , 𝐷 ≥ . Let 𝐿 ≥ 𝐿 𝐺𝐿𝑍 , considerthe set of evaluation points { 𝛼 , . . . , 𝛼 𝐿 } and 𝐸 ⊆ { , . . . , 𝐿 } , with | 𝐸 | ≤ 𝜏 . Moreover, fix 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) and denote 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) with gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = and 𝑑 monic. Let deg ( 𝒗 ) < 𝑁 and deg ( 𝑑 ) < 𝐷 .Consider the random matrix 𝑌 , where we denote by 𝒚 𝑗 : = 𝑌 ∗ ,𝑗 forany ≤ 𝑗 ≤ 𝐿 , constructed as follows: • if 𝑗 ∈ 𝐸 , 𝒚 𝑗 is a uniformly distributed element of F 𝑛 × 𝑞 , • if 𝑗 ∉ 𝐸 , 𝒚 𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) ,then leonora Guerrini, Romain Lebreton, and Ilaria Zappatore S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 , where 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 definedas in (5) , with probability at least − 𝐷 + 𝜏𝑞 . Proof.
First notice that ( Λ 𝒗 , Λ 𝑑 ) ∈ S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 and so h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 ⊆ ker ( 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 ) = S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 wherethe last equality was already observed in the Remark 2.3. The proofis based on the following two steps:(1) we prove that there exists a draw of columns of 𝑌 , 𝒚 𝑗 for 𝑗 ∈ 𝐸 , for which the corresponding solution space S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 . Notice that we only need to provethis inclusion ⊂ since the other one is always true.(2) In the second part, we derive the bound on fraction of errorsfor which the solution space is not of the form h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i .1) First assume that 𝐿 𝐺𝐿𝑍 = 𝑁 + 𝐷 − + 𝜏 + (cid:6) 𝜏𝑛 (cid:7) . Consider apartition of the set of error positions 𝐸 , i.e. 𝐸 = ∪ 𝑛𝑖 = 𝐼 𝑖 , such thatfor any 1 ≤ 𝑖 ≤ 𝑛 , | 𝐼 𝑖 | ≤ ⌈| 𝐸 |/ 𝑛 ⌉ . Note that such a partition existssince 𝑛 ⌈| 𝐸 |/ 𝑛 ⌉ ≥ | 𝐸 | . For any 𝑗 ∈ 𝐸 , denote by 𝑖 𝑗 the unique indexsuch that 𝑗 ∈ 𝐼 𝑖 𝑗 .Construct a matrix 𝑉 , such that 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) if 𝑗 ∉ 𝐸 , and for 𝑗 ∈ 𝐸 consider 𝑉 ∗ ,𝑗 ∈ F 𝑛 × 𝑞 chosen so that 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝜺 𝑖 𝑗 (where 𝜺 𝑖 is the 𝑖 th element of the canonical basis of F 𝑛 × 𝑞 ).Consider ( 𝝋 ,𝜓 ) ∈ S 𝑉,𝑁 + 𝜏,𝐷 + 𝜏 and multiply 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝜺 𝑖 𝑗 by 𝜓 ( 𝛼 𝑗 ) , for 𝑗 ∈ 𝐸 . Since ( 𝝋 ,𝜓 ) ∈ S 𝑉,𝑁 + 𝜏,𝐷 + 𝜏 , by the keyequations (4), we get 𝜓 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 = 𝜓 ( 𝛼 𝑗 ) 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜓 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = ( 𝜓 𝒗 − 𝝋 )( 𝛼 𝑗 ) . Fix 1 ≤ 𝑖 ≤ 𝑛 , we claim that for any 𝑗 ∉ 𝐼 𝑖 then 𝜓 ( 𝛼 𝑗 ) 𝑣 𝑖 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜑 𝑖 ( 𝛼 𝑗 ) =
0. Indeed, if 𝑗 ∉ 𝐸 , then 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) and so byreplacing 𝑉 ∗ ,𝑗 in the key equations (4), we have 𝜓 ( 𝛼 𝑗 ) 𝑣 𝑖 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜑 𝑖 ( 𝛼 𝑗 ) =
0. Now if 𝑗 ∈ 𝐸 \ 𝐼 𝑖 , by the choice of 𝑉 ∗ ,𝑗 , then 𝜓 ( 𝛼 𝑗 ) 𝑣 𝑖 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜑 𝑖 ( 𝛼 𝑗 ) = ≤ 𝑖 ≤ 𝑛 , deg ( 𝜓𝑣 𝑖 − 𝑑𝜑 𝑖 ) < 𝑁 + 𝐷 + 𝜏 −
1. On theother hand the number of roots of these polynomials are 𝐿 − | 𝐼 𝑖 | ≥ 𝐿 − ⌈| 𝐸 |/ 𝑛 ⌉ ≥ 𝐿 𝐺𝐿𝑍 − 𝜏 / 𝑛 and since 𝐿 𝐺𝐿𝑍 ≥ 𝑁 + 𝐷 − + 𝜏 + 𝜏 / 𝑛 it isthen 𝐿 − | 𝐼 𝑖 | ≥ 𝑁 + 𝐷 + 𝜏 −
1. Therefore, since all these polynomialshave more roots than their degree they are the zero polynomials.Hence, 𝜓 ( 𝑥 ) 𝒗 ( 𝑥 ) − 𝑑 ( 𝑥 ) 𝝋 ( 𝑥 ) = .On the other hand, assume that 𝐿 𝐺𝐿𝑍 = max { deg ( 𝐴 )+ 𝑁 , deg ( 𝒃 )+ 𝐷 } + (cid:6) 𝜏𝑛 (cid:7) + 𝜏 . As before, we can consider a partition of 𝐸 , 𝐸 = ∪ 𝑛𝑖 = 𝐼 𝑖 ,such that for any 1 ≤ 𝑖 ≤ 𝑛 , | 𝐼 𝑖 | ≤ ⌈| 𝐸 |/ 𝑛 ⌉ .Construct a matrix 𝑉 , such that 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) if 𝑗 ∉ 𝐸 , and so that 𝑉 ∗ ,𝑗 satisfies 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = − 𝐴 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 when 𝑗 ∈ 𝐸 .For 𝑗 ∈ 𝐸 , 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = − 𝐴 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 or equiv-alently 𝐴 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 − 𝐴 ( 𝛼 𝑗 ) 𝒗 ( 𝛼 𝑗 ) = 𝑑 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 . Hence, 𝜺 𝑖 𝑗 = 𝐴 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 − 𝐴 ( 𝛼 𝑗 ) 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) = 𝐴 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 − 𝒃 ( 𝛼 𝑗 ) . Notice that by as-sumption 𝑑 ( 𝛼 𝑗 ) ≠ 𝜓 ( 𝛼 𝑗 ) we get 𝐴 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 𝜓 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) 𝜓 ( 𝛼 𝑗 ) = 𝜓 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 and since ( 𝝋 ,𝜓 ) ∈ S 𝑉,𝑁,𝐷,𝜏 , it satisfies 𝝋 ( 𝛼 𝑗 ) = 𝑉 ∗ ,𝑗 𝜓 ( 𝛼 𝑗 ) and so we have ( 𝐴 𝝋 − 𝒃 𝜓 )( 𝛼 𝑗 ) = 𝜓 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 . We now denote 𝒑 : = 𝐴 ( 𝑥 ) 𝝋 ( 𝑥 ) − 𝜓 ( 𝑥 ) 𝒃 ( 𝑥 ) ∈ F 𝑞 [ 𝑥 ] 𝑛 × . Fix1 ≤ 𝑖 ≤ 𝑛 , we claim that for any 𝑗 ∉ 𝐼 𝑖 then 𝑝 𝑖 ( 𝛼 𝑗 ) =
0, where 𝑝 𝑖 is the 𝑖 th component of 𝒑 . Indeed, if 𝑗 ∉ 𝐸 , then 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) = 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) and so since 𝝋 ( 𝛼 𝑗 ) = 𝑉 ∗ ,𝑗 𝜓 ( 𝛼 𝑗 ) , we get 𝒑 ( 𝛼 𝑗 ) = 𝐴 ( 𝛼 𝑗 ) 𝝋 ( 𝛼 𝑗 ) − 𝜓 ( 𝛼 𝑗 ) 𝒃 ( 𝛼 𝑗 ) =
0. On the other hand, if 𝑗 ∈ 𝐸 \ 𝐼 𝑖 thenby the choice of 𝑉 ∗ ,𝑗 , then 𝑝 𝑖 ( 𝛼 𝑗 ) = ≤ 𝑖 ≤ 𝑛 and 𝑗 ∉ 𝐼 𝑖 , then 𝑝 𝑖 ( 𝛼 𝑗 ) =
0. Notethat deg ( 𝑝 𝑖 ( 𝑥 )) < max { deg ( 𝐴 ) + 𝑁 , deg ( 𝒃 ) + 𝐷 } . On the otherhand the roots of this polynomial are 𝐿 − | 𝐼 𝑖 | ≥ 𝐿 𝐺𝐿𝑍 − 𝜏 / 𝑛 = max { deg ( 𝐴 ) + 𝑁 , deg ( 𝒃 ) + 𝐷 } + 𝜏 . So we can conclude that 𝒑 ( 𝑥 ) = 𝐴 ( 𝑥 ) 𝝋 ( 𝑥 ) − 𝜓 ( 𝑥 ) 𝒃 ( 𝑥 ) = .Now, since 𝐴 ( 𝑥 ) 𝒗 ( 𝑥 ) = 𝑑 ( 𝑥 ) 𝒃 ( 𝑥 ) if we multiply this equationby 𝜓 ( 𝑥 ) and also 𝒑 ( 𝑥 ) by 𝑑 ( 𝑥 ) and we subtract both the equationswe finally get 𝝋 ( 𝑥 ) 𝑑 ( 𝑥 ) − 𝜓 ( 𝑥 ) 𝒗 ( 𝑥 ) = .Therefore, in both cases we have 𝝋 ( 𝑥 ) 𝑑 ( 𝑥 ) − 𝜓 ( 𝑥 ) 𝒗 ( 𝑥 ) = .Now, since 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) and gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 is monic,there exists 𝑅 ∈ F 𝑞 [ 𝑥 ] such that 𝝋 = 𝑅 𝒗 and 𝜓 = 𝑅𝑑 . Noticethat for any 1 ≤ 𝑗 ≤ 𝐿 by the key equations (4) we get, 0 = 𝝋 ( 𝛼 𝑗 ) − 𝜓 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝑅 ( 𝛼 𝑗 )[ 𝒗 ( 𝛼 𝑗 ) − 𝑉 ∗ ,𝑗 𝑑 ( 𝛼 𝑗 )] . By construction,if 𝑗 ∈ 𝐸 , then 𝒗 ( 𝛼 𝑗 ) − 𝑉 ∗ ,𝑗 𝑑 ( 𝛼 𝑗 ) ≠ and so 𝑅 ( 𝛼 𝑗 ) =
0. There-fore, the error locator polynomial Λ = Î 𝑗 ∈ 𝐸 ( 𝑥 − 𝛼 𝑗 ) divides 𝑅 and so ( 𝝋 ,𝜓 ) ∈ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 . Hence, S 𝑉 ,𝑁 + 𝜏,𝐷 + 𝜏 ⊆h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 and so the equality holds.Hence, we finally get S 𝑉 ,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 for a draw 𝑉 of 𝑌 .2) We now conclude the proof by estimating the fraction of er-rors for which the solution space is exactly of the form S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 . Now for a generic instance of 𝑌 recallthat h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 ⊆ S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = ker ( 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 ) , then dim ( ker ( 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 )) ≥ 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 . By the Rank-Nullity The-orem we have that rank ( 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 ) ≤ 𝑛 ( 𝑁 + 𝜏 )+ 𝐷 + 𝜏 − 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 = : 𝜌. On the other hand, as proved above, there exists a draw 𝑉 ∗ ,𝑗 of 𝑌 ∗ ,𝑗 , for 𝑗 ∈ 𝐸 , such that rank ( 𝑀 𝑉 ,𝑁 + 𝜏,𝐷 + 𝜏 ) = 𝜌 . This meansthat there exists a nonzero 𝜌 -minor in 𝑀 𝑉 ,𝑁 + 𝜏,𝐷 + 𝜏 . We considerthe same nonzero 𝜌 -minor in 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 as a multivariate poly-nomial 𝐶 whose indeterminates are ( 𝑦 𝑖,𝑗 ) ≤ 𝑖 ≤ 𝑛𝑗 ∈ 𝐸 . We remark thatwe showed the existence of a draw 𝑉 ∗ ,𝑗 of 𝑌 ∗ ,𝑗 , for 𝑗 ∈ 𝐸 , such that 𝐶 ( 𝑉 ∗ ,𝑗 ) is non zero. Hence, the polynomial 𝐶 is nonzero. For anymatrix 𝑌 such that ( 𝑌 ∗ ,𝑗 ) 𝑗 ∈ 𝐸 is not a root of 𝐶 , then S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 . Note that the total degree of the polyno-mial 𝐶 is at most 𝐷 + 𝜏 , since only the last 𝐷 + 𝜏 columns of the matrix 𝑀 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 contains the variables ( 𝑦 𝑖,𝑗 ) ≤ 𝑖 ≤ 𝑛𝑗 ∈ 𝐸 (see Remark 2.3).Finally, by the Schwartz-Zippel Lemma, the polynomial 𝐶 can-not be zero in more than ( 𝐷 + 𝜏 )/ 𝑞 fractions of its domain. There-fore, we can conclude that the probability that S 𝑌,𝑁 + 𝜏,𝐷 + 𝜏 ≠ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝑁 + 𝜏,𝐷 + 𝜏 is at most ( 𝐷 + 𝜏 )/ 𝑞 . (cid:3) All the bounds on the number of evaluations 𝐿 introduced so far, i.e. 𝐿 𝐾𝑃𝑆𝑊 and 𝐿 𝐺𝐿𝑍 (see Proposition 2.2 and Theorem 3.1), de-pend on the bounds
𝑁 , 𝐷 and 𝜏 . Therefore, if 𝑁 , 𝐷, 𝜏 overestimatethe degrees 𝒗 and 𝑑 and the actual number of errors, the corre-sponding 𝐿 𝐾𝑃𝑆𝑊 , 𝐿
𝐺𝐿𝑍 would be too big compared to the numberwe really need. An approach to overcome this problem consists inthe introduction of an early termination strategy whose goal is todecrease the number of evaluations needed to recover a solutionwithout knowing the actual degrees of the solution and the num-ber of errors. This strategy was first proposed in [KPSW17] and it olynomial Linear System Solving with Random Errors: new bounds and early termination technique was based on the introduction of some parameters that try to esti-mate the degrees of the solution 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 )/ 𝑑 ( 𝑥 ) of the PLS (1). Inthis work, we revisit this method (Proposition 4.1), by introducingsome parameters 𝜈, 𝜗 which represent attempts to find the actualdegrees of the key equation solution ( Λ 𝒗 , Λ 𝑑 ) , and a criterion whichallows us to check if these parameters 𝜈, 𝜗 upper bound these de-grees. This makes our strategy sensitive to the actual number oferrors (instead of the upper bound 𝜏 which can be imprecise) andto the actual degrees of the solution 𝒚 ( 𝑥 ) (see Remark 4.3).Another significant difference from [KPSW17] consists in the re-duction of the number of evaluations which guarantees to uniquelyrecover the solution in presence of random errors.We divide this section into two parts. First we assume to knowa fixed upper bound on the number of errors that the nodes couldmake. We then notice that in the early termination algorithms (Al-gorithms 1, 2, 3, 4) the number of evaluations is iteratively incre-mented. The number of errors depends on this number of evalua-tions and so it can be hard to find a valid upper bound for it. For thisreason (as in [KPSW17]) in the second part, we introduce a linearerror bound which depends on the variable number of evaluationsand on an error rate 𝜌 𝐸 . In both cases, we propose two countingfor 𝐿 ; one that can correct any error and one which derives fromour Theorem 3.1 for the scenario where errors are random. We start by recalling and introducing some useful notations thatwe will use throughout this section. Let 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) be a PLSand 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) with gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 monic. Con-sider 1 ≤ 𝑁 , 𝐷 ≤ 𝐿 such that 𝑁 > deg ( 𝒗 ) and 𝐷 > deg ( 𝑑 ) anddeg ( 𝐴 ) , deg ( 𝒃 ) .In this first part of the section we also assume to know 𝜏 ≥ | 𝐸 | (see Section 2). Let 𝜈, 𝜗 ≥ L( 𝜈, 𝜗 ) : = min { max { 𝑁 − + 𝜗, 𝐷 − + 𝜈 } , max { deg ( 𝐴 )+ 𝜈, deg ( 𝒃 )+ 𝜗 }} . (8) 𝐿 for any error. From [KPSW17] we can derive thefollowing proposition, adapted to our choice of the parameters 𝜈, 𝜗 that gives a criterion for an early termination algorithm that cancorrect any error.
Proposition 4.1.
Let 𝜗, 𝜈, ≥ and consider 𝐿 ≥ L( 𝜈, 𝜃 ) + 𝜏 eval-uation points { 𝛼 , . . . , 𝛼 𝐿 } , where 𝜏 ≥ | 𝐸 | . Fix 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) anddenote 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) with gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = and 𝑑 monic.Then the solution space of the key equations (4) with input 𝑌, 𝜈, 𝜗 is S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 where 𝛿 𝜈,𝜗 = min { 𝜈 − ( deg ( 𝒗 ) +| 𝐸 |) , 𝜗 − ( deg ( 𝑑 ) + | 𝐸 |)} . Proof.
We now prove that S 𝑌,𝜈,𝜗 ⊂ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 , theother inclusion being straightforward. Let ( 𝝋 ,𝜓 ) ∈ S 𝑌,𝜈,𝜗 . For any1 ≤ 𝑖 ≤ 𝑛 and 1 ≤ 𝑗 ≤ 𝐿 , we have 𝜑 𝑖 ( 𝛼 𝑗 ) = 𝑦 𝑖,𝑗 𝜓 ( 𝛼 𝑗 ) , ( Λ 𝑣 𝑖 )( 𝛼 𝑗 ) = 𝑦 𝑖,𝑗 ( Λ 𝑑 )( 𝛼 𝑗 ) (9)Assume that 𝐿 ≥ max { 𝐷 − + 𝜈, 𝑁 − + 𝜗 } + 𝜏 . If we multiply thefirst equation in (9) by ( Λ 𝑑 )( 𝛼 𝑗 ) and the second by 𝜓 ( 𝛼 𝑗 ) and wesubtract them, we get ( Λ ( 𝑣 𝑖 𝜓 − 𝑑𝜑 𝑖 ))( 𝛼 𝑗 ) = ≤ 𝑗 ≤ 𝐿 .Now, since the polynomial Λ ( 𝑣 𝑖 𝜓 − 𝑑𝜑 𝑖 ) has 𝐿 roots and degree Algorithm 1:
Early Termination for PLSwE for fixed errorbound 𝜏 . Input : a stream of vectors 𝑌 = ( 𝒚 𝑗 ) for 𝑗 = , , . . . whichis extensible on demand, where 𝒚 𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) + 𝒆 𝑗 , 𝑁 > deg ( 𝒗 ) , 𝐷 > deg ( 𝑑 ) , 𝜏 ≥ | 𝐸 ( 𝐿 )| = |{ 𝑗 | 𝒆 𝑗 ≠ }| , deg ( 𝐴 ) , deg ( 𝒃 ) Output : ( 𝒗 , 𝑑 ) the solution of (1) 𝐿 ← L( , ) + 𝜏 ; while true do foreach 𝜈, 𝜗 with L( 𝜈, 𝜗 ) + 𝜏 == 𝐿 do if Check ( 𝐿, 𝜈, 𝜗 ) then return FindSolution (( 𝒚 𝑗 ) , 𝐿, 𝜈, 𝜗 ) ; 𝐿 ← 𝐿 +
1; Require a new 𝒚 𝑗 ; < | 𝐸 | + max ( deg ( 𝑣 ) + 𝜗, deg ( 𝑑 ) + 𝜈 ) which is smaller than 𝐿 byassumption, so it is the zero polynomial.On the other hand, assume 𝐿 ≥ max { deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 } + 𝜏 .Since for any 𝑗 ∉ 𝐸 , 𝒚 𝑗 = 𝐴 ( 𝛼 𝑗 ) − 𝒃 ( 𝛼 𝑗 ) . Combining this equationwith 𝝋 ( 𝛼 𝑗 ) = 𝒚 𝑗 𝜓 ( 𝛼 𝑗 ) we get ( 𝐴 𝝋 − 𝜓 𝒃 )( 𝛼 𝑗 ) =
0. Now, notice thatthe vector of polynomials 𝐴 ( 𝑥 ) 𝝋 ( 𝑥 ) − 𝜓 ( 𝑥 ) 𝒃 ( 𝑥 ) has 𝐿 − | 𝐸 | rootsand degree < max { deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 } which is smaller than 𝐿 − | 𝐸 | by assumption. So 𝐴 ( 𝑥 ) 𝝋 ( 𝑥 ) = 𝜓 ( 𝑥 ) 𝒃 ( 𝑥 ) . Combined with 𝐴 ( 𝑥 ) 𝒗 ( 𝑥 ) = 𝑑 ( 𝑥 ) 𝒃 ( 𝑥 ) , we get 𝐴 ( 𝑥 )[ 𝝋 ( 𝑥 ) 𝑑 ( 𝑥 ) − 𝒗 ( 𝑥 ) 𝜓 ( 𝑥 )] = . Since 𝐴 ( 𝑥 ) is full rank, we obtain 𝝋 ( 𝑥 ) 𝑑 ( 𝑥 ) − 𝒗 ( 𝑥 ) 𝜓 ( 𝑥 ) = .Since gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 monic, there exists 𝑃 ∈ F 𝑞 [ 𝑥 ] such that ( 𝝋 ,𝜓 ) = ( 𝑃 𝒗 , 𝑃𝑑 ) .Now 𝝋 ( 𝛼 𝑗 ) = 𝒚 𝑗 𝜓 ( 𝛼 𝑗 ) yields ( 𝑃 ( 𝒗 − 𝒚 𝑑 ))( 𝛼 𝑗 ) = 𝑃 ( 𝛼 𝑗 ) = 𝑗 ∈ 𝐸 . This means that ∃ 𝑃 ′ ∈ F 𝑞 [ 𝑥 ] , 𝑃 = Λ 𝑃 ′ . Finally, ( 𝝋 ,𝜓 ) = 𝑃 ′ ( Λ 𝒗 , Λ 𝑑 ) and the degree constraints on ( 𝝋 ,𝜓 ) implydeg 𝑃 ′ < 𝛿 𝜈,𝜗 which concludes our proof. (cid:3) Proposition 4.1 gives a criterion to check if ( 𝜈, 𝜗 ) are upper boundson the degree of the solution ( Λ 𝒗 , Λ 𝑑 ) . Indeed, 𝜈 > deg ( 𝒗 ) + | 𝐸 | and 𝜗 > deg ( 𝑑 ) + | 𝐸 | ) iff 𝛿 𝜈,𝜗 > iff S 𝑌,𝜈,𝜗 ≠ {( , )} .Let Check ( 𝐿, 𝜈, 𝜗 ) be the function that returns the Boolean S 𝑌,𝜈,𝜗 != {( , )} . If Check returns true , then we can call
FindSolution ( 𝑌, 𝜈, 𝜗 ) to recover ( 𝒗 , 𝑑 ) from S 𝑌,𝜈,𝜗 (see Remark 2.3).We are now ready to introduce Algorithm 1, whose correctness fol-lows from Proposition 4.1.Notice that
Check and
FindSolution perform the same compu-tation: they compute a basis of the module generated by solutionsof the key equation. Note also that in Algorithm 1 the number ofevaluations varies, which could affect the number of errors. There-fore, we denote | 𝐸 ( 𝐿 )| : = |{ ≤ 𝑗 ≤ 𝐿 | 𝒆 𝑗 ≠ }| instead of | 𝐸 | tostress out the dependency in 𝐿 .We now analyze the termination of Algorithm 1. Proposition 4.2.
Algorithm 1 terminates when 𝐿 equals 𝐿 𝑠 : = L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + + 𝜏 . Proof.
We start by proving by contraposition that if L( 𝜈, 𝜗 ) + 𝜏 < L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + + 𝜏 then 𝛿 𝜈,𝜗 ≤ i.e. the algorithm do not stop (since in this case Check returns false). In-deed, if 𝛿 𝜈,𝜗 >
0, then deg ( 𝒗 )+| 𝐸 ( 𝐿 𝑠 )| < 𝜈 and deg ( 𝑑 )+| 𝐸 ( 𝐿 𝑠 )| < 𝜗 and so, L( 𝜈, 𝜗 ) ≥ L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + leonora Guerrini, Romain Lebreton, and Ilaria Zappatore Algorithm 2:
Early Termination for PLSwE for fixed errorbound 𝜏 for random errors.Same as Algorithm 1 exceptLine 1: 𝐿 ← L( , ) + (cid:6) 𝜏𝑛 (cid:7) Line 3: 𝜈, 𝜗 with L( 𝜈, 𝜗 ) + (cid:6) 𝜏𝑛 (cid:7) == 𝐿 Then, note that for 𝜈 = deg ( 𝒗 ) + | 𝐸 ( 𝐿 𝑠 )| + 𝜗 = deg ( 𝑑 ) +| 𝐸 ( 𝐿 𝑠 )| +
1, the number of evaluations L( 𝜈, 𝜗 ) + 𝜏 equals to 𝐿 𝑠 andthe algorithm stops ( 𝛿 𝜈,𝜗 > (cid:3) Remark . Compared to the number of evaluations L( deg ( 𝒗 ) , deg ( 𝑑 )) + + 𝜏 of [KPSW17, Equations 5 and 9] (tak-ing 𝑅 ∗ =
0, omitting the rank drops), we can lower the evaluationbound of Proposition 1 due to a dependency on the real numberof errors | 𝐸 ( 𝐿 𝑠 )| . To the best of our knowledge, this dependency isoriginal in the literature. 𝐿 for random errors. We now present an early ter-mination strategy applied to the PLSwE problem (Algorithm 2)which allows us to further reduce the number of evaluation pointscompared to [KPSW17]. Notice that we slightly modify the struc-ture of Algorithm 1 considering 𝐿 : = L( 𝜈, 𝜗 ) + (cid:6) 𝜏𝑛 (cid:7) evaluationpoints (step 1).Algorithm 2 is based on the following new result. Theorem 4.4.
Let 𝜗, 𝜈 ≥ , consider 𝐿 ≥ L( 𝜈, 𝜃 ) + (cid:6) 𝜏𝑛 (cid:7) distinctevaluation points { 𝛼 , . . . , 𝛼 𝐿 } . Let 𝐸 ⊆ { , . . . , 𝐿 } . Moreover, fix 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) and denote 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) with gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = and 𝑑 monic.Consider the random matrix 𝑌 where we denote the columns as 𝒚 𝑗 : = 𝑌 ∗ ,𝑗 , such that 𝒚 𝑗 is a uniformly distributed element of F 𝑛 × 𝑞 if 𝑗 ∈ 𝐸 , and 𝒚 𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) if 𝑗 ∉ 𝐸 . Then the solution space satisfies S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 where 𝛿 𝜈,𝜗 = min { 𝜈 − ( deg ( 𝒗 ) +| 𝐸 |) , 𝜗 − ( deg ( 𝑑 ) + | 𝐸 |)} , with probability at least − 𝜗𝑞 . Proof.
The structure of the proof is the same as the proof ofTheorem 3.1. In the first part we prove that there exists a draw ofcolumns 𝒚 𝑗 for 𝑗 ∈ 𝐸 for which the corresponding solution space S 𝑌,𝜈,𝜗 is generated by elements of the form h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i . Morespecifically recall that h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ⊆ S 𝑌,𝜈,𝜗 is always true and sowe need to prove the other inclusion in order to get the equality.In the second part, by using the Schwartz-Zippel Lemma we deter-mine the bound on fraction of errors for which the solution spaceis not of the form h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i .Since here we are considering some general parameters 𝜈, 𝜗 in-stead of the bounds 𝑁 + 𝜏, 𝐷 + 𝜏 , the only difference between thisproof and the previous one consists in the first part. We still con-sider 𝐸 = ∪ 𝑛𝑖 = 𝐼 𝑖 , such that for any 1 ≤ 𝑖 ≤ 𝑛 , | 𝐼 𝑖 | ≤ ⌈| 𝐸 |/ 𝑛 ⌉ . Recallthat for any 𝑗 ∈ 𝐸 , we denote by 𝑖 𝑗 the unique index such that 𝑗 ∈ 𝐼 𝑖 𝑗 .We first assume that L( 𝜈, 𝜗 ) = max { 𝑁 − + 𝜗, 𝐷 − + 𝜈 } . Con-struct a matrix 𝑉 (as in the proof of Theorem 3.1) such that 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) if 𝑗 ∉ 𝐸 , and 𝑉 ∗ ,𝑗 ∈ F 𝑛 × 𝑞 is chosen so that 𝒗 ( 𝛼 𝑗 )− 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝜺 𝑖 𝑗 if 𝑗 ∈ 𝐸 ( 𝜺 𝑖 is the 𝑖 th canonical vector). Let ( 𝝋 ,𝜓 ) ∈ S 𝑉,𝜈,𝜗 . We now denote 𝒑 : = 𝜓 𝒗 − 𝑑 𝝋 , and 𝑝 𝑖 its 𝑖 thcomponent. We now show that 𝒑 ( 𝑥 ) = .We already have 𝒑 ( 𝛼 𝑗 ) = ( 𝜓 𝒗 − 𝑑 𝝋 )( 𝛼 𝑗 ) = 𝑗 ∉ 𝐸 . For 𝑗 ∈ 𝐸 ,we can combine 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝜺 𝑖 𝑗 and 𝝋 ( 𝛼 𝑗 ) = 𝑉 ∗ ,𝑗 𝜓 ( 𝛼 𝑗 ) toget 𝒑 ( 𝛼 𝑗 ) = 𝜓 ( 𝛼 𝑗 ) 𝒗 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜓 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = 𝜓 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 . Fix 1 ≤ 𝑖 ≤ 𝑛 , then for any 𝑗 ∉ 𝐼 𝑖 then 𝑝 𝑖 ( 𝛼 𝑗 ) =
0. Now, no-tice that 𝑝 𝑖 has degree ≤ max { 𝜗 + 𝑁 − , 𝐷 − + 𝜈 } − 𝐿 − | 𝐼 𝑖 | ≥ 𝐿 − ⌈| 𝐸 |/ 𝑛 ⌉ ≥ L( 𝜈, 𝜗 ) − ⌈| 𝜏 |/ 𝑛 ⌉ = max { 𝜈 + 𝐷 − , 𝜗 + 𝑁 − } and so it is the zero polynomial. Therefore, 𝒑 ( 𝑥 ) = 𝜓 ( 𝑥 ) 𝒗 ( 𝑥 ) − 𝑑 ( 𝑥 ) 𝝋 ( 𝑥 ) = . The rest follows by observingthat 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) is such that gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 is monic.So, we can conclude that S 𝑉 ,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 .We now assume that L( 𝜈, 𝜗 ) = max { deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 } .Construct a matrix 𝑉 such that 𝑉 ∗ ,𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) if 𝑗 ∉ 𝐸 , and 𝑉 ∗ ,𝑗 ∈ F 𝑛 × 𝑞 is chosen so that 𝒗 ( 𝛼 𝑗 )− 𝑑 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 = − 𝐴 ( 𝛼 𝑗 ) − 𝑑 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 . As inthe proof of Theorem 3.1, for 𝑗 ∈ 𝐸 then 𝜓 ( 𝛼 𝑗 ) 𝜺 𝑖 𝑗 = 𝐴 ( 𝛼 𝑗 ) 𝜓 ( 𝛼 𝑗 ) 𝑉 ∗ ,𝑗 − 𝒃 ( 𝛼 𝑗 ) 𝜓 ( 𝛼 𝑗 ) = ( 𝐴 𝝋 − 𝒃 𝜓 )( 𝛼 𝑗 ) We now denote 𝒑 : = 𝐴 𝝋 − 𝒃 𝜓 and by 𝑝 𝑖 its 𝑖 th component. Fix1 ≤ 𝑖 ≤ 𝑛 , then we observe that for any 𝑗 ∉ 𝐼 𝑖 then 𝑝 𝑖 ( 𝛼 𝑗 ) = ( 𝑝 𝑖 ) ≤ max { deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 } − 𝐿 − | 𝐼 𝑖 | ≥ max { deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 } and so 𝑝 𝑖 =
0. There-fore, 𝒑 ( 𝑥 ) = 𝐴 ( 𝑥 ) 𝝋 ( 𝑥 ) − 𝒃 ( 𝑥 ) 𝜓 ( 𝑥 ) = . The proof then follows byobserving that 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) is the only solution of the linear system 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) and is such that gcd ( gcd ( 𝑣 𝑖 ) , 𝑑 ) = 𝑑 monic(with the same argument than the proof of Theorem 3.1). (cid:3) Therefore, under the assumptions of Theorem 4.4, with 𝐿 ≥L( 𝜈, 𝜗 ) + (cid:6) 𝜏𝑛 (cid:7) evaluation points, then S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿𝜈,𝜗 | {z } with probability at least 1 − 𝜗𝑞 (cid:26) = {( , ) } ⇐⇒ 𝛿 𝜈,𝜗 ≤ ≠ {( , ) } ⇐⇒ 𝛿 𝜈,𝜗 > (10) Therefore, the
Check ( 𝐿, 𝜈, 𝜗 ) function remains unchanged butit could return an incorrect answer with probability ≤ 𝜗𝑞 . On theother hand, the FindSolution is slightly different from the onethat we have previously defined. Indeed, it could happen that the F 𝑞 [ 𝑥 ] -module M generated by solutions in S 𝑌,𝜈𝜗 has rank (M) >
1, in which case the function outputs a failure message. Note thateven if rank (M) = FindSolution could return an incorrectoutput; for instance if S 𝑌,𝜈,𝜗 ≠ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 = {( , )} .Remark that both these problems can only happen if S 𝑌,𝜈,𝜗 ≠ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 , so they have probability ≤ 𝜗𝑞 .We now better analyze how Algorithm 2 works: • if S 𝑌,𝜈,𝜗 = {( , )} , by (10), the parameters 𝜈, 𝜗 must be toosmall compared to deg ( 𝑣 ) + | 𝐸 | , deg ( 𝑑 ) + | 𝐸 | . Hence, in thiscase Algorithm 2 increments the number of evaluations. • Otherwise, if S 𝑌,𝜈,𝜗 ≠ {( , )} then – by Theorem 4.4, with probability at least 1 − 𝜗𝑞 we have S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 which is nontrivial. By (10), 𝛿 𝜈,𝜗 > F 𝑞 [ 𝑥 ] -module generated by solutionsof PLSwE is uniquely generated. Hence, the FindSolution function returns ( 𝒗 , 𝑑 ) . olynomial Linear System Solving with Random Errors: new bounds and early termination technique – On the other hand, with probability at most ≤ 𝜗𝑞 we have S 𝑌,𝜈,𝜗 ≠ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 , and FindSolution eitherreturns a failure message or another solution ( 𝒗 ′ , 𝑑 ′ ) ≠ ( 𝒗 , 𝑑 ) . Remark . We can optimize the steps 2 and 3 of both Algorithms 1and 2 respectively by only testing two specific ( 𝜈, 𝜗 ) instead ofall those giving a fixed 𝐿 . The goal is to make early terminationalgorithms have a smaller failure probability, and incidentally tomake them faster.So we should try to find which ( 𝜈, 𝜗 ) maximize 𝛿 𝜈,𝜗 amongthose that give the same number of evaluations 𝐿 . Indeed, our goalis to have 𝛿 𝜈,𝜗 >
0. The two candidates are ( 𝜈 , 𝜗 ) = (L − ( 𝐷 − ) , L − ( 𝑁 − )) and ( 𝜈 , 𝜗 ) = (L − deg ( 𝐴 ) , L − deg ( 𝒃 )) where L : = 𝐿 − (cid:6) 𝜏𝑛 (cid:7) .We now show that any ( 𝜈, 𝜗 ) such that L = L( 𝜈, 𝜗 ) , we have ( 𝜈, 𝜗 ) ≤ ( 𝜈 , 𝜗 ) or ( 𝜈, 𝜗 ) ≤ ( 𝜈 , 𝜗 ) (for the partial component-wise ordering). Indeed, either L = L( 𝜈, 𝜗 ) = 𝑚𝑎𝑥 ( 𝐷 − + 𝜈, 𝑁 − + 𝜗 ) and then ( 𝜈, 𝜗 ) ≤ ( 𝜈 , 𝜗 ) , or L = L( 𝜈, 𝜗 ) = 𝑚𝑎𝑥 ( deg ( 𝐴 ) + 𝜈, deg ( 𝒃 ) + 𝜗 ) and then ( 𝜈, 𝜗 ) ≤ ( 𝜈 , 𝜗 ) .Remark that if 𝐷 − ≤ deg ( 𝐴 ) , 𝑁 − ≤ deg ( 𝒃 ) then we shouldonly try ( 𝜈 , 𝜗 ) because ( 𝜈 , 𝜗 ) ≥ ( 𝜈 , 𝜗 ) and L( 𝜈 , 𝜗 ) = L .Similarly if 𝐷 − ≥ deg ( 𝐴 ) , 𝑁 − ≥ deg ( 𝒃 ) then we should onlytry ( 𝜈 , 𝜗 ) . However, if 𝐷 − ≤ deg ( 𝐴 ) , 𝑁 − ≥ deg ( 𝒃 ) , weshould try both candidates because they are not comparable andthey both lead to L( 𝜈 , 𝜗 ) = L( 𝜈 , 𝜗 ) = L , and as a consequenceof the same number of evaluations. Proposition 4.6.
Algorithm 2 terminates with at most 𝐿 𝑠 evalua-tion points, where 𝐿 𝑠 is such that 𝐿 𝑠 = L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + + (cid:6) 𝜏𝑛 (cid:7) . Its output is correctwith probability at least − ( 𝐷 + 𝐸 ( 𝐿 𝑠 )+ ) ( deg ( Λ 𝒗 , Λ 𝑑 )+ ) 𝑞 . Proof.
Assume first that we are always in the favorable cases, i.e. S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 for any attempt ( 𝜈, 𝜗 ) . In otherterms we assume that Algorithm 2 returns the solution ( Λ 𝒗 , Λ 𝑑 ) of the PLS (1). Then the proof that Algorithm 2 stops with exactly 𝐿 𝑠 evaluations is similar to the proof of Proposition 4.2.We now analyze the unfortunate cases in order to bound thecorresponding probability:(1) it could happen that for a certain attempt ( 𝜈, 𝜗 ) , then S 𝑌,𝜈,𝜗 ≠ {( , )} and 𝛿 𝜈,𝜗 ≤
0. In this case the
Check function at step4 would return false . Then Algorithm 2 would stop pre-maturely, i.e. before reaching 𝐿 𝑠 evaluations, with a failuremessage or an incorrect output.(2) it could also happen that Algorithm 2 reaches 𝐿 𝑠 evaluationsand parameters ( 𝜈, 𝜗 ) such that 𝛿 𝑌,𝜈,𝜗 > FindSolution returns an incorrect PLS solution. Note first that
Check mustreturn true because S 𝑌,𝜈,𝜗 ⊇ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 ≠ {( , )} .However, if S 𝑌,𝜈,𝜗 ≠ h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 then FindSolution will return a failure message or an incorrect output.The probability of falling in an unfortunate case is related tothe number of ( 𝜈, 𝜗 ) we attempt before reaching 𝐿 𝑠 = L( deg ( 𝒗 ) +| 𝐸 ( 𝐿 𝑠 )| + , deg ( 𝑑 ) + | 𝐸 ( 𝐿 𝑠 )| + ) + (cid:6) 𝜏𝑛 (cid:7) . The number of evalua-tions starts from L( , ) + (cid:6) 𝜏𝑛 (cid:7) and ends at 𝐿 𝑠 : there are at mostmax ( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + = deg ( Λ 𝒗 , Λ 𝑑 ) + ( 𝜈 , 𝜗 ) and ( 𝜈 , 𝜗 ) (see Remark 4.5). Now any attempt ( 𝜈, 𝜗 ) could fail with probability ≤ 𝜗𝑞 . It remains to upper bound 𝜗 among all attempts.Denote by ( 𝜈 𝑠 , 𝜗 𝑠 ) , ( 𝜈 𝑠 , 𝜗 𝑠 ) the candidate parameters correspond-ing to 𝐿 𝑠 .Therefore, max { 𝜗 𝑠 , 𝜗 𝑠 } = L( deg ( 𝒗 ) , deg ( 𝑑 ))+ 𝐸 ( 𝐿 𝑠 )+ − min { 𝑁 − , deg ( 𝒃 )} . First if 𝑁 − ≤ deg ( 𝒃 ) , then since L( deg ( 𝒗 ) , deg ( 𝑑 ))+ 𝐸 ( 𝐿 𝑠 )+ ≤ max { 𝐷 − + deg ( 𝒗 ) , 𝑁 − + deg ( 𝑑 )}+ 𝐸 ( 𝐿 𝑠 ) + ≤ 𝑁 + 𝐷 + 𝐸 ( 𝐿 𝑠 ) , we have max { 𝜗 𝑠 , 𝜗 𝑠 } ≤ 𝐷 + 𝐸 ( 𝐿 𝑠 ) + 𝑁 − ≥ deg ( 𝒃 ) , and we have L( deg ( 𝒗 ) , deg ( 𝑑 )) + 𝐸 ( 𝐿 𝑠 ) + ≤ max { deg ( 𝐴 ) + deg ( 𝒗 ) , deg ( 𝒃 ) + deg ( 𝑑 )} + 𝐸 ( 𝐿 𝑠 ) + = deg ( 𝒃 ) + deg ( 𝑑 ) + 𝐸 ( 𝐿 𝑠 ) + 𝐴 𝒗 = 𝒃 𝑑 ) then max { 𝜗 𝑠 , 𝜗 𝑠 } ≤ deg ( 𝑑 ) + + 𝐸 ( 𝐿 𝑠 ) ≤ 𝐷 + 𝐸 ( 𝐿 𝑠 ) + . Combining all these results wecan conclude that the probability to reach the unfortunate cases isat most ( 𝐷 + 𝐸 ( 𝐿 𝑠 )+ ) ( deg ( Λ 𝒗 , Λ 𝑑 )+ ) 𝑞 . (cid:3) Up until now, our early termination schemes have assumed thatthe number of errors was bounded by a constant 𝜏 . Since earlytermination requires more and more evaluations, it would be in-teresting to have an error bound that depends on the number ofevaluation. In this section, we consider a linear error bound whichdepends on an error rate . Assumption 4.7.
For any number of evaluation points 𝐿 , the num-ber of errors 𝐸 ( 𝐿 ) is bounded by | 𝐸 ( 𝐿 )| ≤ 𝜌 𝐸 𝐿 where 0 ≤ 𝜌 𝐸 < / | 𝐸 ( 𝐿 )| ≤ ⌈ 𝜌 𝐸 𝐿 ⌉ . For the sake of simplicity,we restrict ourselves to Assumption 4.7. However, our results canbe adapted to the alternative linear error bound. 𝐿 for any error. We start by adapting Proposition 4.1to the special case of linear error bound.
Proposition 4.8.
Let 𝜈, 𝜗 ≥ and consider 𝐿 = j L( 𝜈,𝜗 )+ − 𝜌 𝐸 k eval-uation points and error bound 𝜏 = ⌊ 𝜌 𝐸 𝐿 ⌋ . Fix 𝐴 ( 𝑥 ) 𝒚 ( 𝑥 ) = 𝒃 ( 𝑥 ) anddenote 𝒚 ( 𝑥 ) = 𝒗 ( 𝑥 ) 𝑑 ( 𝑥 ) with gcd ( gcd 𝑖 ( 𝑣 𝑖 ) , 𝑑 ) = and 𝑑 monic.Under Assumption 4.7, we have S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 . Proof.
We use the following notations ¯ 𝐿 ∗ ( 𝜈, 𝜗 ) = L( 𝜈,𝜗 )+ − 𝜌 𝐸 , sothat 𝐿 ∗ ( 𝜈, 𝜗 ) = (cid:4) ¯ 𝐿 ∗ ( 𝜈, 𝜗 ) (cid:5) , and 𝜏 ∗ ( 𝜈, 𝜗 ) = ⌊ 𝜌 𝐸 𝐿 ∗ ( 𝜈, 𝜗 )⌋ .We first show that we are under the hypotheses of Proposition 4.1, i.e. that 𝐿 ∗ ( 𝜈, 𝜗 ) ≥ L( 𝜈, 𝜗 ) + 𝜏 ∗ ( 𝜈, 𝜗 ) and 𝜏 ∗ ( 𝜈, 𝜗 ) ≥ | 𝐸 ( 𝐿 ∗ ( 𝜈, 𝜗 )| .First | 𝐸 ( 𝐿 ∗ ( 𝜈, 𝜗 ))| ≤ 𝜌 𝐸 𝐿 ∗ ( 𝜈, 𝜗 ) using Assumption 4.7. Since | 𝐸 ( 𝐿 ∗ ( 𝜈, 𝜗 ))| ∈ N , we get | 𝐸 ( 𝐿 ∗ ( 𝜈, 𝜗 ))| ≤ ⌊ 𝜌 𝐸 𝐿 ∗ ( 𝜈, 𝜗 )⌋ = 𝜏 ∗ ( 𝜈, 𝜗 ) .Then L( 𝜈, 𝜗 ) + 𝜏 ∗ ( 𝜈, 𝜗 ) ≤ L( 𝜈, 𝜗 ) + 𝜌 𝐸 𝐿 ∗ ( 𝜈, 𝜗 )≤ L( 𝜈, 𝜗 ) + 𝜌 𝐸 ¯ 𝐿 ∗ ( 𝜈, 𝜗 ) = ¯ 𝐿 ∗ ( 𝜈, 𝜗 ) − , and so L( 𝜈, 𝜗 ) + 𝜏 ∗ ( 𝜈, 𝜗 ) ≤ (cid:4) ¯ 𝐿 ∗ ( 𝜈, 𝜗 ) (cid:5) = 𝐿 ∗ ( 𝜈, 𝜗 ) . (cid:3) Therefore, we can use the evaluation counting 𝐿 ( 𝜈, 𝜗 ) = j L( 𝜈,𝜗 )+ − 𝜌 𝐸 k to detect if ( 𝜈, 𝜗 ) are good estimations, and eventually return the leonora Guerrini, Romain Lebreton, and Ilaria Zappatore Algorithm 3:
Early Termination for PLSwE for linear er-ror bound 𝜌 𝐸 . Input : a stream of vectors 𝑌 = ( 𝒚 𝑗 ) for 𝑗 = , , . . . whichis extensible on demand, where 𝒚 𝑗 = 𝒗 ( 𝛼 𝑗 ) 𝑑 ( 𝛼 𝑗 ) + 𝒆 𝑗 𝑁 > deg ( 𝒗 ) , 𝐷 > deg ( 𝑑 ) , deg ( 𝐴 ) , deg ( 𝒃 ) ,0 ≤ 𝜌 𝐸 < / | 𝐸 ( 𝐿 )| ≤ 𝜌 𝐸 𝐿 for all 𝐿 . Output : ( 𝒗 , 𝑑 ) the solution of (1) 𝐿 𝑛𝑢𝑚 ← L( , ) + while true do 𝐿 ← j 𝐿 𝑛𝑢𝑚 − 𝜌 𝐸 k ; 𝜏 ← ⌊ 𝜌 𝐸 𝐿 ⌋ ; Require new 𝒚 𝑗 ; foreach 𝜈, 𝜗 with L( 𝜈, 𝜗 ) + == 𝐿 𝑛𝑢𝑚 do if Check ( 𝐿, 𝜈, 𝜗 ) then return FindSolution (( 𝒚 𝑗 ) , 𝐿, 𝜈, 𝜗 ) ; 𝐿 𝑛𝑢𝑚 ← 𝐿 𝑛𝑢𝑚 + ( 𝒗 , 𝑑 ) of the PLS. This is exactly what Algorithm 3 doesand Proposition 4.8 shows its correction.We now prove that Algorithm 3 stops with a certain number ofevaluation points. Proposition 4.9. (1)
Algorithm 3 terminates. (2)
Algorithm 3 stops when it reaches 𝐿 𝑠 evaluation points where 𝐿 𝑠 = (cid:22) L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 )| + − 𝜌 𝐸 (cid:23) (11)(3) We can bound 𝐿 𝑠 ≤ j L( deg ( 𝒗 ) , deg ( 𝑑 ))+ − 𝜌 𝐸 k . (4) If for some reason fewer errors are made, i.e. | 𝐸 ( 𝐿 )| ≤ 𝜌 ′ 𝐸 𝐿 with 𝜌 ′ 𝐸 < 𝜌 𝐸 , then 𝐿 𝑠 ≤ j L( deg ( 𝒗 ) , deg ( 𝑑 ))+ − 𝜌 ′ 𝐸 − 𝜌 𝐸 k . The inequality given in Item 3 relates the performance of ourearly termination algorithm to the literature. Indeed, the right-handbound can be derived from [KPSW17, Algorithm 2.2] with 𝜌 𝑅 = 𝑞 𝑅 = 𝑞 𝐸 = +∞ (for simplicity).Note that 𝜌 𝐸 and 𝜌 ′ 𝐸 don’t play the same role: 𝜌 𝐸 must be knownin advance (it is an input of the algorithm) and be related to a lin-ear error bound that is always true. If Assumption 4.7 fails then thecorrectness of Algorithm 3 may be lost. On the other hand, 𝜌 ′ 𝐸 isused to demonstrate that our early termination technique is sensi-tive to the real number of errors (in addition to real degrees of 𝒗 , 𝑑 ), i.e. that it can stop earlier if fewer errors than expected are made. Proof.
We keep the notations of the proof of Proposition 4.8.(1) We need to prove that Algorithm 3 stops. Consider 𝑓 ( 𝑥 ) : = 𝑥 − L( deg ( 𝒗 ) , deg ( 𝑑 ))+ 𝜌 𝐸 𝑥 + − 𝜌 𝐸 , which is strictly increasing because 1 > 𝜌 𝐸 /( − 𝜌 𝐸 ) . This last inequality is true since 0 ≤ 𝜌 𝐸 < / 𝐿 ∈ N such that 𝑓 ( 𝐿 ) ≥
0. Set 𝜈 ′ = deg ( 𝒗 ) + | 𝐸 ( 𝐿 )| + , 𝜗 ′ = deg ( 𝑑 ) + | 𝐸 ( 𝐿 )| + 𝐿 ′ = 𝐿 ∗ ( 𝜈 ′ , 𝜗 ′ ) . We have 𝐿 ′ = (cid:22) L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 )| + − 𝜌 𝐸 (cid:23) ≤ L( deg ( 𝒗 ) , deg ( 𝑑 )) + 𝜌 𝐸 𝐿 + − 𝜌 𝐸 ≤ 𝐿 Algorithm 4:
Early Termination for PLSwE for linear er-ror bound 𝜌 𝐸 for random errors.Same as Algorithm 3 exceptLine 3: 𝐿 ← j 𝐿 𝑛𝑢𝑚 − 𝜌 𝐸 / 𝑛 k ; 𝜏 ← ⌊ 𝜌 𝐸 𝐿 ⌋ ; Require new 𝒚 𝑗 ;where the last inequality comes from 𝑓 ( 𝐿 ) ≥
0. Hence, 𝜈 ′ ≥ deg ( 𝒗 )+| 𝐸 ( 𝐿 ′ )| + 𝜗 ′ ≥ deg ( 𝑑 ) + | 𝐸 ( 𝐿 ′ )| + 𝛿 𝜈 ′ ,𝜗 ′ >
0. So Algo-rithm 3 would stop with ≤ 𝐿 ′ evaluations.(2) Let 𝐿 𝑠 be the number of evaluations when the algorithmstops. We now prove Equation (11). There exists 𝜈 𝑠 , 𝜗 𝑠 such that 𝐿 𝑠 = 𝐿 ∗ ( 𝜈 𝑠 , 𝜗 𝑠 ) , and we must have 𝛿 𝜈 𝑠 ,𝜗 𝑠 > i.e. 𝜈 > deg ( 𝒗 ) +| 𝐸 ( 𝐿 𝑠 )| and 𝜗 > deg ( 𝑑 ) + | 𝐸 ( 𝐿 𝑠 )| . Define 𝜈 ′ = deg ( 𝒗 ) + | 𝐸 ( 𝐿 𝑠 )| + , 𝜗 ′ = deg ( 𝑑 ) + | 𝐸 ( 𝐿 𝑠 )| +
1. We now prove that 𝐿 𝑠 = 𝐿 ∗ ( 𝜈 ′ , 𝜗 ′ ) bycontradiction, which implies Equation (11). So assume that 𝐿 𝑠 > 𝐿 ′ : = 𝐿 ∗ ( 𝜈 ′ , 𝜗 ′ ) (note that the inequality ≥ is always true since ( 𝜈, 𝜗 ) ≥ ( 𝜈 ′ , 𝜗 ′ ) and 𝐿 ∗ ( 𝜈, 𝜗 ) is increasing). But 𝜈 ′ = deg ( 𝒗 ) +| 𝐸 ( 𝐿 𝑠 )| + > deg ( 𝒗 ) + | 𝐸 ( 𝐿 ′ )| and 𝜗 ′ = deg ( 𝑑 ) + | 𝐸 ( 𝐿 𝑠 )| + > deg ( 𝑑 ) + | 𝐸 ( 𝐿 ′ )| so that 𝛿 𝜈 ′ ,𝜗 ′ > 𝐿 ′ < 𝐿 𝑠 evaluations which is a contradiction.(3) Now let ¯ 𝐿 ′ = L( deg ( 𝒗 ) , deg ( 𝑑 ))+ − 𝜌 𝐸 and 𝐿 ′ = (cid:4) ¯ 𝐿 ′ (cid:5) . ¯ 𝐿 ′ is definedso that 0 = 𝑓 ( ¯ 𝐿 ′ ) . We now prove that 𝑓 ( 𝐿 𝑠 ) ≤
0, which implies 𝐿 𝑠 ≤ ¯ 𝐿 ′ since 𝑓 is strictly increasing, thus 𝐿 𝑠 ≤ (cid:4) ¯ 𝐿 ′ (cid:5) = 𝐿 ′ since 𝐿 𝑠 ∈ N . The claim comes from 𝐿 𝑠 ≤ L( deg ( 𝒗 ) , deg ( 𝑑 ))+| 𝐸 ( 𝐿 𝑠 ) |+ − 𝜌 𝐸 ≤ L( deg ( 𝒗 ) , deg ( 𝑑 ))+ 𝜌 𝐸 𝐿 𝑠 + − 𝜌 𝐸 .(4) If one execution of PLSwE satisfies | 𝐸 ( 𝐿 )| ≤ 𝜌 ′ 𝐸 𝐿 for 𝜌 ′ 𝐸 < 𝜌 𝐸 , we can prove that 𝐿 𝑠 ≤ j L( deg ( 𝒗 ) , deg ( 𝑑 ))+ − 𝜌 ′ 𝐸 − 𝜌 𝐸 k by adapting theprevious proof with 𝑓 ( 𝑥 ) : = 𝑥 − L( deg ( 𝒗 ) , deg ( 𝑑 ))+ 𝜌 ′ 𝐸 𝑥 + − 𝜌 𝐸 . (cid:3) 𝐿 for random errors. As before, we can lower thenumber of evaluation points considering randomly distributed er-rors. Theorem 4.4 can be adapted to the context of a linear errorbound.
Proposition 4.10.
Under Assumption 4.7 and assumptions of The-orem 4.4, and using 𝐿 = j L( 𝜈,𝜗 )+ − 𝜌 𝐸 / 𝑛 k evaluation points and errorbound 𝜏 = ⌊ 𝜌 𝐸 𝐿 ⌋ for any 𝜈, 𝜗 ≥ , we have S 𝑌,𝜈,𝜗 = h 𝑥 𝑖 Λ 𝒗 , 𝑥 𝑖 Λ 𝑑 i ≤ 𝑖 < 𝛿 𝜈,𝜗 with probability at least − 𝜗𝑞 . The proof is similar to the one of Proposition 4.8, except that itis based on Theorem 4.4.With the help of the latter proposition, we can introduce Algo-rithm 4, which returns a correct solution of the PLSwE problemwith high probability.
Proposition 4.11.
Algorithm 4 stops with at most 𝐿 𝑠 evaluations,where 𝐿 𝑠 = (cid:22) L( deg ( 𝒗 ) , deg ( 𝑑 )) + | 𝐸 ( 𝐿 𝑠 ) | + − 𝜌 𝐸 / 𝑛 (cid:23) ≤ (cid:22) L( deg ( 𝒗 ) , deg ( 𝑑 )) + − ( + / 𝑛 ) 𝜌 𝐸 (cid:23) (12) Its output is correct with probability ≥ − ( 𝐷 + 𝐸 ( 𝐿 𝑠 )+ ) ( deg ( Λ 𝒗 , Λ 𝑑 )+ ) 𝑞 .If for some reason fewer errors are made, i.e. | 𝐸 ( 𝐿 )| ≤ 𝜌 ′ 𝐸 𝐿 with 𝜌 ′ 𝐸 < 𝜌 𝐸 , then 𝐿 𝑠 ≤ j L( deg ( 𝒗 ) , deg ( 𝑑 ))+ − 𝜌 ′ 𝐸 − 𝜌 𝐸 / 𝑛 k . olynomial Linear System Solving with Random Errors: new bounds and early termination technique Proof.
The proof concerning all the numbers of evaluation pointsis similar to the one of Proposition 4.9, but it is based on the num-ber of evaluations of Theorem 4.4. The statement of correctnesscan be proved similarly to Proposition 4.6. (cid:3)
REFERENCES [BK14] B. Boyer and E. Kaltofen. Numerical Linear System Solving with ParametricEntries by Error Correction. In
Proceedings of SNC’14 . ACM, 2014.[BKY03] D. Bleichenbacher, A. Kiayias, and M. Yung. Decoding of interleaved Reed-Solomon codes over noisy data. In
Proceedings of ICALP’03 , 2003.[BMS04] A. Brown, L. Minder, and A. Shokrollahi. Probabilistic decoding of Inter-leaved RS-Codes on the Q-ary symmetric channel. In
Proceedings of ISIT’04 .IEEE, 2004.[BW86] E. R. Berlekamp and L. R. Welch. Error Correction of Algebraic Block Codes,U.S. Patent 4 633 470, Dec. 1986.[Cab71] S. Cabay. Exact Solution of Linear Equations. In
Proceedings of SYMSAC’71 ,New York, NY, USA, 1971. Association for Computing Machinery.[GG13] J. von zur Gathen and J. Gerhard.
Modern Computer Algebra . CambridgeUniversity Press, 3rd edition, 2013.[GLZ19] E. Guerrini, R. Lebreton, and I. Zappatore. Polynomial Linear System Solv-ing with Errors by Simultaneous Polynomial Reconstruction of InterleavedReed-Solomon Codes. In
Proceedings of ISIT’19 . IEEE, 2019. [KPSW17] E. Kaltofen, C. Pernet, A. Storjohann, and C. Waddell. Early Terminationin Parametric Linear System Solving and Rational Function Vector Recov-ery with Error Correction. In
Proceedings of ISSAC’17 . ACM, 2017.[KY13] E. L. Kaltofen and Z. Yang. Sparse multivariate function recovery fromvalues with noise and outlier errors. In
Proceedings of ISSAC’13 , 2013.[KY14] E. L. Kaltofen and Z. Yang. Sparse multivariate function recovery with ahigh error rate in the evaluations. In
Proceedings of ISSAC’14 , 2014.[McC77] M. T. McClellan. The exact solution of linear equations with rational func-tion coefficients.
ACM Trans. Math. Softw. , 3(1), 1977.[OS07] Z. Olesh and A. Storjohann. The Vector Rational Function Reconstructionproblem. In
Proceedings of the Waterloo Workshop . World Scientific, 2007.[RS16] J. Rosenkilde and A. Storjohann. Algorithms for simultaneous padé approx-imations. In
Proceedings of ISSAC’2016 , 2016.[Sha48] C.E. Shannon. A mathematical theory of communication.
Bell Syst. Tech. J. ,27(3), 1948.[SRM09] G. Schmidt, V. R.Sidorenko, and M.Bossert. Collaborative Decoding of Inter-leaved Reed–Solomon Codes and Concatenated Code Designs.
IEEE Trans-actions on Information Theory , 55(7), 2009.[SSB07] G. Schmidt, V. Sidorenko, and M. Bossert. Enhancing the Correcting Radiusof Interleaved Reed-Solomon Decoding using Syndrome Extension Tech-niques. In
Proceedings of ISIT’07 . IEEE, 2007.[SSB10] G. Schmidt, V. R. Sidorenko, and M. Bossert. Syndrome Decoding ofReed–Solomon Codes Beyond Half the Minimum Distance Based on Shift-Register Synthesis.