Deterministic Interpolation of Sparse Black-box Multivariate Polynomials using Kronecker Type Substitutions
aa r X i v : . [ c s . S C ] A ug Deterministic Interpolation of Sparse Black-box MultivariatePolynomials using Kronecker Type Substitutions ∗ Qiao-Long Huang , and Xiao-Shan Gao , KLMM, Academy of Mathematics and Systems ScienceChinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China
Abstract
In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariatepolynomial given as a standard black-box by introducing new Kronecker type substitutions. Let f ∈ R [ x , . . . , x n ] be a sparse black-box polynomial with a degree bound D . When R = C ora finite field, our algorithms either have better bit complexity or better bit complexity in D than existing deterministic algorithms. In particular, in the case of deterministic algorithms forstandard black-box models, our second algorithm has the current best complexity in D which isthe dominant factor in the complexity. Keywords . Sparse polynomial interpolation, Kronecker substitution, black-box model.
Sparse polynomial interpolation is a basic computational problem which attracts lots of attentionrecently. There exist two main classes of sparse polynomial interpolation algorithms: the directapproaches [6, 20, 13, 23, 6, 20, 22], which recover the exponents of the multivariate polynomialsdirectly and the reduction approaches [2, 9, 24, 10, 17, 28], which reduce multivariate interpolationsto univariate interpolations.The sparse polynomial can be given as a standard black-box model or more special models suchas the straight-line program (SLP) model. Since the value of a polynomial of degree D at anypoint other than 0 , ± D bits or more, any algorithm whose complexity is polynomial inlog D cannot perform such an evaluation over Q or Z . Even for polynomials over the finite field F q , there exist no interpolation algorithms whose complexity is polynomial in log D and log q forthe standard black-box model. For special models such as the SLP model [3, 4, 5, 9, 10, 17], theprecision accuracy black-box model [1, 8, 26, 13], and the modular black-box model [11, 12], thereexist interpolation algorithms whose complexity is polynomial in log D .In this paper, we propose two new deterministic interpolation algorithms for standard black-boxmultivariate polynomials by introducing new Kronecker type substitutions. ∗ Partially supported by a grant from NSFC (11688101). .1 Main results Let f ∈ R [ x , . . . , x n ] be a sparse polynomial given by a black-box B f , where R is an integraldomain. Suppose that f has a degree bound D and a term bound T . Our algorithms are based onthe following Koronecker type substitutions: f mod ( d,p ) = f ( x, x d , . . . , x d n − ) mod ( x p −
1) (1) f ( d,p ) = f ( x, x mod ( d,p ) , . . . , x mod ( d n − ,p ) ) (2) f ( d,p,k ) = f ( x, . . . , x mod ( d k − ,p )+ p , . . . , x mod ( d n − ,p ) ) (3)where d, p ∈ Z > .In our first algorithm, we first interpolate the univariate polynomials f ( d,p ) for a fixed prime p and d = 1 , . . . , n − T −
1) + 1, and then recover f from these f ( d,p ) . In this case, f is recoveredwith the base changing Kronecker substitutions : x i = x mod ( d i − ,p ) , d = 1 , . . . , n − T −
1) + 1.In our second algorithm, we first interpolate the univariate polynomials f ( D,p k ) , k = 1 , . . . , N and then recover f from these f ( d,p ) , where N is an integer of size O ( nT log D ) and p k are differentprime numbers. In this case, f is recovered with the modulus changing Kronecker substitutions : x i = x mod ( D i − ,p k ) , k = 1 , . . . , N . Probes Arithmetic Height of Bitoperations data complexityBen-or & Tiwarri [6, 20]
T nT D T D nT D Zippel [28] nT D nT D T D nT D Klivans & Spielman [24] nT nT log D T D max( nT , D ) nT D max( nT , D )This paper (Cor. 3.11) nT nT log D T D max( nT, D ) nT D max( nT, D )This paper (Cor. 4.8) nT log D nT log D nT D n T D Table 1: A “soft-Oh” comparison of deterministic interpolation algorithms over C Table 1 is a comparison with other deterministic algorithms when R = C , where “Probes” is thenumber of calls to B f . We assume that the size of the coefficients is O (1) to simplify the results.From the table, our second algorithm has the lowest complexity in D . Probabilistic algorithms andalgorithms for special black-box models, such as the SLP model, are not compared here. Probes Bit complexity Size of F q Grigoriev-Karpinski-Singer [14] n T log ( q ) + q . log q q ≥ O ∼ ( n T )Klivans & Spielman [24] nT nT D max( nT , D ) log q q ≥ O ∼ ( D max( nT , D ))This paper (Cor. 3.12) nT nT D max( nT, D ) log q q ≥ O ∼ ( D max( nT, D ))This paper (Cor. 4.9) nT log D n T D log q q ≥ O ∼ ( nDT ) Table 2: “Soft-Oh” comparison of deterministic interpolation algorithms F q Table 2 is a comparison with other deterministic algorithms over finite fields, where “Size of F q ”means that this algorithm works for F q satisfying the condition. If the elements of the extensionfield of F q can be used, algorithms work for any finite field. From the table, our first algorithm hasbetter complexity than that in [24]. In [14], it is assumed that deg( f, x i ) < q −
1. Under the sameassumption, D ≤ n ( q −
1) and our second algorithm has complexity O ∼ ( n T q ). So, our secondalgorithm is linear in q and the algorithm in [14] is linear in q . . Also, our algorithm has worsecomplexity in n but better complexity in T . There are recent works on the probabilistic algorithmsover finite fields [2, 15, 18, 19, 28], which have better complexities than deterministic algorithms.Note that the size of f is about O ( nT log D ). Therefore, D is the exponential part in the2omplexity and hence the dominant factor. Our second algorithm has the best complexity in D among the deterministic algorithms for standard black-box multivariate polynomials. Our algorithm has three major steps. First, we compute a set of univariate polynomials f mod ( d,p ) and show that if f mod ( d ,p ) has the maximal number of terms, then at least half of the terms of f do not merge or collide with other terms of f in f mod ( d ,p ) . The d in the base changing Kroneckersubstitution and p in the modulus changing Kronecker substitution are called “ok” base and “ok”prime, respectively. Second, we interpolate extra n univariate polynomials f ( d ,p ,k ) to find a set ofterms containing these non-colliding terms. Finally, we give a criterion to test whether f contains agiven term and use this criterion to pickup the terms of f from the terms found in the second step.The main contributions of this paper are: new Kronecker substitutions, criteria for term testing,and methods to find “ok” base and “ok” prime.In the rest of this section, we compare our algorithms with [9, 10, 3, 24, 17]. Our work buildson and is inspired by these works.The idea of base changing Kronecker substitutions given in Section 3 is introduced in this paperfor the first time. The idea of modulus changing Kronecker substitutions used in Section 4 wasgiven in [9, 24, 17], but the method in this paper is different from them and will be explained below.The substitution f mod ( D,p j ) in (1) was introduced by Grag and Schost [9] to interpolate an SLPpolynomial f by recovering f from f mod ( D,p j ) for O ( T log D ) different primes p j . Our second methodis similar to this method, but works for black-box models and has the following differences. First,for a black-box polynomial f , f mod ( D,p ) cannot be computed to keep all degrees in x less than p .In order to work for black-box polynomials, we use the substitution f ( D,p ) to compute f mod ( D,p ) = f ( D,p ) mod ( x p − f mod ( D,p j ) for O ( T log D ) different primes p j . Third, wegive a new criterion to check whether a term belongs to f . Finally, a new Kronecker substitution f ( d,p,k ) is introduced to recover the exponents.The substitution f ( d,p ) in (2) was introduced by Klivans-Spielman [24] to interpolate black-box polynomials. Instead of f ( d,p,k ) , they used the substitution f ( KS ) = f ( q x, q x mod ( d,p ) , . . . ,q n x mod ( d n − ,p ) ) for different primes q i . Our substitution f ( d,p,k ) has the following advantages: (1)For the complex field, the size of coefficients is not changed after our substitution, while the size ofcoefficients in f ( KS ) is increased by a factor of D . (2) Our algorithm works for general rings, whilethe substitution in [24] needs an element ω ∈ R such that ω i = 1 , i = 1 , , . . . , D .The substitution f ( d,p,k ) in (3) was introduced in [17] for interpolating SLP polynomials. Themethod based on modulus changing Kronecker substitutions given in Section 4 of this paper couldbe considered as a generalization of the method given in [17] from SLP model to black-box model.The major difference is that for black-box model, f mod ( D,p ) is computed from f ( D,p ) mod ( x p − pD instead of p . As a consequence,we need to give new methods for the two key ingredients of the algorithm: the criterion for termtesting and the method of finding the “ok” prime.In Arnold, Giesbrecht, and Roche [4], the concept of “ok” prime is introduced. A prime p is“ok” if at most T terms of f are collisions in f mod ( D,p ) . The “ok” prime in [4] is probabilistic. In thispaper, we give similar notions of “ok” base and “ok” prime, and determinist methods to computethem. Also, the randomized Kronecker substitution is used in [3], while our Kronecker reduction is3eterministic.For interpolation over finite fields, Grigoriev-Karpinski-Singer [14] gave the first deterministicalgorithm in finite fields. Prony’s algorithm is dominated by the cost of discrete logarithms in F q .Ben-or & Tiwarri’s algorithm and Zippel’s algorithm meet the Zero Avoidance Problem [15, 28, 19].Klivans-Spielman’s algorithm needs to compute a factorization and thus only works for the integraldomain which contains an element ω such that ω i = 1 , i = 1 , , . . . , D . Our algorithms are moregeneral comparing to existence methods in the sense that they do not need to solve the zero avoidanceproblem, the discrete logarithm problem, and the factorization problem, which is possible mainlydue to the substitutions f ( d,p ) and f ( d,p,k ) . Throughout this paper, let f = c m + · · · + c t m t ∈ R [ X ] be a black-box multivariate polynomialwith terms c i m i , where R is an integral domain, X = { x , . . . , x n } be a set of n indeterminates, and m i , i = 1 , . . . , t are distinct monomials. Denote f = t to be the number of terms of f , deg f = t to be the total degree of f , and M f = { c m , . . . , c t m t } to be the set of terms of f . Let D, T ∈ N such that D > deg( f ) and T ≥ f .Let d ∈ N , p ∈ N > and x a new indeterminate. Consider the univariate polynomials in R [ x ]: f ( d,p ) = f ( x, x mod ( d,p ) , . . . , x mod ( d n − ,p ) ) (4) f ( d,p,k ) = f ( x, . . . , x mod ( d k − ,p )+ p , . . . , x mod ( d n − ,p ) ) (5) f mod ( d,p ) = f ( d,p ) mod ( x p −
1) = f ( d,p,k ) mod ( x p −
1) (6)where k = 1 , , . . . , n . (4) comes from the modified Kronecker substitutions x i = x mod ( d i − ,p ) , i = 1 , , . . . , n and (5) comes from another modified Kronecker substitutions x i = x mod ( d i − ,p ) , i =1 , , . . . , n, i = k, x k = x mod ( d k − ,p )+ p , respectively. We have the following key concept. Definition 2.1
A term cm ∈ M f is called a collision in f mod ( d,p ) ( f ( d,p ) or f ( d,p,k ) ) if there exists an aw ∈ M f \{ cm } such that m mod ( d,p ) = w mod ( d,p ) ( m ( d,p ) = w ( d,p ) or m ( d,p,k ) = w ( d,p,k ) ). The following fact is obvious.
Lemma 2.2
Let cm ∈ M f . If cm is not a collision in f mod ( d,p ) , then cm is not a collision in f ( d,p ) ,and cm is not a collision in any f ( d,p,k ) for k = 1 , , . . . , n . Let f mod ( d,p ) = a x d + a x d + · · · + a r x d r ( d < · · · < d r ) (7)Since f ( d,p ) mod ( x p −
1) = f ( d,p,k ) mod ( x p −
1) = f mod ( d,p ) , for k = 1 , , . . . , n , we can write f ( d,p ) = f + f + · · · + f r + g (8) f ( d,p,k ) = f k, + f k, + · · · + f k,r + g k (9)where f i mod ( x p −
1) = f k,i mod ( x p −
1) = a i x d i , g mod ( x p −
1) = g k mod ( x p −
1) = 0. Wedefine the following key notation.TS fd,p,D = { a i x e i, · · · x e i,n n | a i is from (7) , and4 : In (8) and (9) , f i = a i x γ i , f k,i = a i x β k,i , k = 1 , , . . . , n. (10)T : e i,k = β k,i − γ i p ∈ N , k = 1 , , . . . , n. T : γ i = n X k =1 e i,k mod ( d k − , p ) } . T : e i, + · · · + e i,n < D } . Similar to [17, Lemma 5.3], we can prove the following result about TS fd,p,D . Lemma 2.3
Let f = P ti =1 c i m i ∈ R [ X ] , d, p ∈ Z > , D > deg( f ) . If c i m i is not a collision in f mod ( d,p ) , then c i m i ∈ TS fd,p,D . The following algorithm is used to find the set TS fd,p . Algorithm 2.4 (TSTerms)Input:
Univariate polynomials f mod ( d,p ) , f ( d,p ) , f ( d,p,k ) ( k = 1 , . . . , n ) in R [ x ], a positive integer d , aprime p , D > deg( f ). Output: TS fd,p,D . Step 1:
Write f mod ( d,p ) , f ( d,p ) , f ( d,p,k ) in the following form f mod ( d,p ) = a x d + · · · + a r x d r f ( d,p ) = a x γ + · · · + a α x γ α + gf ( d,p,k ) = a x β k, + · · · + a α x β k,α + g k such that for i = 1 , . . . , α, k = 1 , . . . , n , mod ( γ i , p ) = mod ( β k,i , p ) = d i and g mod ( d,p ) =( g k ) mod ( d,p ) = 0. Step 2:
Let S = {} . For i = 1 , , . . . , α , do a: for k = 1 , , . . . , n , dolet e i,k = β k,i − γ i p . If e i,k is not in N , then break; b: If P nk =1 e i,k mod ( d k − , p ) = γ i , then break; c: If P nk =1 e i,k ≥ D , then break; d: Let S = S S { a i x e i, · · · x e i,n n } . Step 3:
Return S .Similar to [17, Lemma 5.5], we can prove the following lemma. Lemma 2.5
Algorithm 2.4 needs O ( nT ) arithmetic operations in R and O ∼ ( nT log( pD )) bit oper-ations. An interpolation algorithm based on base changing Kroneckersubstitution
In this section, we give the first interpolation algorithm. The basic idea of the algorithm is torecover f from f ( d,p ) for a fixed prime p and sufficiently many values for d , that is, we will changethe degree base d in the Kronecker substitution x d i . In this subsection, we give a criterion to check whether a given term belongs to a polynomial. Let F p be the finite field with p elements. Lemma 3.1
Let L i = a i, + · · · + a i,n x n − ∈ F p [ x ] \ { } , i = 1 , . . . , l , p a prime such that p ≥ max { n, ( n − l } . If δ is an integer satisfying ( n − l ≤ δ ≤ p , then there exist at least δ − ( n − l integers k in [1 , δ ] such that L i ( k ) = 0 , for all i = 1 , , . . . , l .Proof. For each L i , since deg( L i ) = n −
1, there exist at most ( n −
1) integers k in [1 , δ ] such that L i ( k ) = 0. Since we have l nonzero functions L i , there exist at most ( n − l integers k in [1 , δ ] suchthat for some L i ( k ) = 0. So the rest integers in [1 , p ] does not vanish L i , i = 1 , , . . . , l . Lemma 3.2
Let f = P ti =1 c i m i ∈ R [ X ] , T ≥ f, D > deg( f ) , δ = ( n − T − , and p a primesuch that p ≥ max { n, δ , D } . Then for any integer δ satisfying δ ≤ δ ≤ p and any integer i ∈ [1 , t ] ,there exist at least δ − δ integers d in [1 , δ ] , such that c i m i is not a collision in f mod ( d,p ) .Proof. If t = 1, the proof is obvious. Now we consider the case t ≥
2. It suffices to consider c m . Assume m i = x e i, x e i, · · · x e i,n n , i = 1 , , . . . , t . Let L s ( x ) = ( e , − e s, ) + · · · + ( e ,n − e s,n ) x n − mod p , where s = 2 , , . . . , t . Since m , m s are different monomials, at least one of e ,k − e s,k = 0 , k ∈ { , , . . . , n } . Since p ≥ D and | e ,i − e s,i | < D , we have L s ( x ) = 0. We claim thatif d ∈ [1 , δ ] such that L s ( d ) = 0, then m mod d,p ) = m mod s ( d,p ) . Since deg( m mod d,p ) ) = P nk =1 e ,k d k − mod p ,deg( m mod s ( d,p ) ) = P nk =1 e s,k d k − mod p , we have m mod d,p ) = m mod s ( d,p ) . We proved the claim. By Lemma3.1, there are ≥ δ − ( n − T − ≥ δ − δ integers in [1 , δ ] such that all L s ( x ) , s = 2 , , . . . , t arenon-zero.Now we give a criterion for testing whether a term cm is in M f . Theorem 3.3
Let f = P ti =1 c i m i ∈ R [ X ] , T ≥ f, D > deg( f ) , δ = ( n − T − , δ = ( n − T ,and p a prime with p ≥ max { n, δ + δ + 1 , D } . For a term cm satisfying deg( m ) < D , cm ∈ M f ifand only if there exist at least δ + 1 integers d ∈ [1 , δ + δ + 1] such that f − cm ) mod ( d,p ) < f mod ( d,p ) .Proof. Let cm ∈ M f . If d is an integer such that cm is not a collision in f mod ( d,p ) , then f − cm ) mod ( d,p ) = f mod ( d,p ) −
1, and hence f − cm ) mod ( d,p ) < f mod ( d,p ) . By Lemma 3.2, there exist at most δ + δ + 1 − δ = δ + 1 integers d ∈ [1 , δ + δ + 1] such that cm is not a collision in f mod ( d,p ) . Sothere exist at least δ + 1 integers d such that f − cm ) mod ( d,p ) < f mod ( d,p ) .For the other direction, assume cm / ∈ M f . We show there exist at most δ integers d ∈ [1 , δ + δ + 1] such that f − cm ) mod ( d,p ) < f mod ( d,p ) . Consider two cases: Case 1: m is not a monomial in f .Then f − cm has at most T + 1 terms. By Lemma 3.2, there exist at least δ + δ + 1 − δ = δ + 1integers d ∈ [1 , δ + δ + 1] such that cm is not collision in ( f − cm ) mod ( d,p ) . So there are at least δ + 16ntegers d such that f − cm ) mod ( d,p ) > f mod ( d,p ) . So there are at most δ + δ +1 − ( δ +1) = δ integers d such that f − cm ) mod ( d,p ) < f mod ( d,p ) . Case 2: m is a monomial in f , but cm is not a term in f .Then f − cm has the same number of terms as f . Let the term of f with monomial m be c m . ByLemma 3.2, there exist at least δ + δ + 1 − δ = δ + 1 integers j such that ( c − c ) m is not collisionin ( f − cm ) mod ( d,p ) . So there are at least δ + 1 integers j such that f − cm ) mod ( d,p ) = f mod ( d,p ) . Since δ + δ + 1 − ( δ + 1) = δ < δ , there are at most δ integers d such that f − cm ) mod ( d,p ) < f mod ( d,p ) . In this subsection, we show how to find an “ok” degree d such that half of terms in f do not collidein f mod ( d,p ) for certain prime p . Lemma 3.4 [18] Let B j , j = 1 , , . . . , s be nonempty sets of integers and a i , i = 1 , , . . . , t all thedistinct elements in ∪ sj =1 B j . Let c be the number of a i satisfying a i ∈ B j and B j ≥ for some j .Then t − c ≤ s and for s ∈ [ t − c, s ] ∩ N , we have ( t − s ) ≤ c ≤ t − s ) . Denote C f ( d,p ) to be the number of collision terms of f in f mod ( d,p ) . Then, we have Lemma 3.5
Let f ∈ R [ X ] , f mod ( d ,p ) = s , and f mod ( d ,p ) = s . If s ≥ s , then C f ( d ,p ) ≤ C f ( d ,p ) .Proof. Assume f mod ( d ,p ) = a x e + · · · + a s x e s , e i = e j , when i = j . Let f = f + · · · + f s + g , where f mod i ( d ,p ) = a i x e i , i = 1 , . . . , s and g mod ( d ,p ) = 0. Let B i , i = 1 , . . . , s be the sets of terms in f i and B be the set of terms in g . So by Lemma 3.4, we have ( t − s ) < C f ( d ,p ) ≤ t − s ). By the samereason, we have ( t − s ) ≤ C f ( d ,p ) ≤ t − s ). Now C f ( d ,p ) ≤ t − s ) ≤ t − s ) ≤ C f ( d ,p ) . The lemmais proved. Lemma 3.6
Let f = P ti =1 c i m i ∈ R [ X ] , m i = x e i, · · · x e i,n n , D > deg( f ) , p a prime such that p ≥ max { n, D } , A u,v = P ns =1 ( e u,s − e v,s ) x s − mod p , where u, v ∈ { , . . . , t } , and u < v . If C f ( d ,p ) = s and d is an integer in [1 , p ] , then at least ⌈ s ⌉ of A u,v satisfy A u,v ( d ) = 0 .Proof. Let n i be the number of collision blocks with i terms in f mod ( d ,p ) . Assume c j m j + · · · + c j i m j i is a collision block with i terms. For any u, v ∈ { j , . . . , j i } , u < v , we have m mod u ( d ,p ) = m mod v ( d ,p ) . So( e u, + · · · + e u,n d n − ) mod p = ( e v, + · · · + e v,n d n − ) mod p , which implies that d is a root of A u,v .There exist C i = i ( i − such pairs ( u, v ) and n i such collision blocks. Let K = P ti =1 12 ( i − i ) n i .So there are K different A u,v with root d . Now we give a lower bound of K . First we see that t = P ti =1 in i , s = P ti =2 in i . K = P ti =1 12 ( i − i ) n i = P ti =1 i n i − P ti =1 in i = P ti =1 i n i − t = n + P ti =2 i n i − t ≥ n + t − n − t = t − n = s . Since K is an integer, K ≥ ⌈ s ⌉ . Theorem 3.7
Let f = P ti =1 c i m i ∈ R [ X ] , T ≥ f, D > deg f , δ = ( n − T − , and p be a primesuch that p ≥ max { n, δ + 1 , D } . Let d be an integer in [1 , δ + 1] such that f mod ( d ,p ) ≥ f mod ( d,p ) for all d = 1 , . . . , δ + 1 . Then at least ⌈ t ⌉ terms of f do not a collide in f mod ( d ,p ) .Proof. If t = 1, the proof is obvious. So now we assume t ≥
2. We first claim that there existsat least one integer d in [1 , δ + 1] such that C f ( d,p ) < t . We prove it by contradiction. As-sume for d = 1 , . . . , δ + 1, C f ( d,p ) ≥ t . Then by Lemma 3.6, there exist ⌈ C f ( d,p ) ⌉ polynomials7 u,v ( x ) , u, v ∈ { , . . . , t } , u < v such that A u,v ( d ) = 0. Since d = 1 , . . . , δ + 1 are different ele-ments in F p , the polynomials A u,v ( x ) , u, v ∈ { , . . . , t } , u < v have at least P δ +1 d =1 ⌈ C f ( d,p ) ⌉ roots.Now P δ +1 d =1 ⌈ C f ( d,p ) ⌉ > δ ⌈ · t ⌉ ≥ δ t = ( n − T − t , which contradicts to the fact that thesum of the degrees of { A u,v , u, v ∈ { , , . . . , t } , u < v } is at most t ( t − n − C f ( d ,p ) ≤ C f ( d,p ) , d = 1 , . . . , δ + 1. So C f ( d ,p ) < · t = t . So thenumber of no collision terms of f in f mod ( d ,p ) is at least > t − t = t , which is ≥ ⌈ t ⌉ . Now, we give a reduction algorithm which reduces multivariate interpolation to univariate interpo-lation, where we assume that a univariate interpolation algorithm exists.
Algorithm 3.8 (MIPolyBase)Input:
A black-box procedure B f that computes f ∈ R [ X ], T ≥ f , D > deg( f ). Output:
The exact form of f . Step 1:
Let δ = ( n − T − , δ = ( n − T, N = max { δ + 1 , δ + δ + 1 } , and p a prime suchthat p ≥ max { n, N, D } . Step 2:
For d = 1 , , . . . , N , find f ( d,p ) via a univariate interpolation algorithm. Let f d = f ( d,p ) ,f mod d = f mod ( d,p ) = f ( d,p ) mod ( x p − Step 3:
Let α = max { f mod d | d = 1 , , . . . , N } and d satisfying f mod d = α . Let h = 0. Step 4:
While α = 0, do a: For k = 1 , , . . . , n , find f ( d ,p,k ) via a univariate interpolation algorithm and let g k = f ( d ,p,k ) − h ( d ,p,k ) . b: Let TS f − hd ,p,D = TSTerms ( f mod d , f d , g , . . . , g n , d , p, D ). c: Let s = 0. For each u ∈ TS fd ,p,D , set s := s + u if { d | f mod d − u mod ( d,p ) ) < f mod d , d = 1 , . . . , δ + δ + 1 } ≥ δ + 1 . d: Let h = h + s , T = T − s , δ = ( n − T − , δ = ( n − T, N = max { δ +1 , δ + δ +1 } . e: For d = 1 , . . . , N , let f d = f d − s ( d,p ) , f mod d = f mod d − s mod ( d,p ) . f: Let α = max { f mod d | d = 1 , . . . , N } and d satisfying f mod d = α . Step 5:
Return h Theorem 3.9
Algorithm 3.8 is correct and needs interpolating O ( nT ) univariate polynomials withdegrees less than O ( D max { nT, D } ) and terms less than T . Besides this, it still needs O ∼ ( nT ) ringoperations in R and O ∼ ( n T log D ) bit operations. roof. By Theorem 3.7 and Lemma 2.3, at least half of terms of f are in TS fd ,p obtained in b ofStep 4. In c of Step 4, Theorem 3.3 is used to select the elements of M f from TS fd ,p . Then in eachloop of Step 4, at least half of the terms of f are obtained and f will be obtained by running atmost log T loops of Step 4. The correctness of the algorithm is proved.We now analyse the complexity. We call N univariate interpolations in Step 2 and at most n log t univariate interpolations in a of Step 4. Note that deg( f ( d,p ) ) ≤ D ( p −
1) and deg f ( d,p,k ) ≤ D ( p − p is max { O ( nT ) , O ( D ) } and f ≤ T , the first part of the theorem is proved.In Step 2, obtaining one f mod d needs O ( t ) ring operations and O ∼ ( t log( pD )) bit operations. Sinceit totally has N polynomials f d , the complexity is O ( nT ) ring operations and O ∼ ( nT log D ) bitoperations. In Step 3, since N is O ( nT ) and f mod ( d ,p ) has no more than T terms, it needs O ∼ ( nT )bit operations. For Step 4, first we analyse the complexity of one cycle. In b , by Theorem 2.5,the complexity is O ( nT ) ring operations and O ∼ ( nT log D ) bit operations. In c , since every checkneeds to compare δ + δ + 1 polynomials and to use δ + δ + 1 substitutions, the complexity is O ( fd ,p n ( δ + δ ) log( pD )+ fd ,p ( δ + δ ) log T log( pD )) bit operations and O ( fd ,p ( δ + δ ))ring operations. Since fd ,p ≤ T , the complexity is O ( nT ) ring operations and O ∼ ( n T log D )bit operations. In e , it needs n ( s ) operations to obtain s ( d,p ) , s mod ( d,p ) . Since we need to update N polynomials, the complexity is O ∼ ( n ( s ) N log( pD )) bit operations and O ( sN ) ring operations.Since we obtain at least half of the terms in f in every recursion, at most log t recursions areneed. Since the sum of s is t , the total complexity of Step 4 is O ∼ ( nT ) ring operations and O ∼ ( n T log D ) bit operations. Algorithm 3.8 reduces multivariate interpolations to univariate interpolations. In this section, wecombine Algorithm 3.8 with three univariate interpolation algorithms to give multivariate interpo-lation algorithms, which are written as corollaries of Theorem 3.9.For a univariate polynomial with degree D , the Lagrange algorithm works over any integraldomain with more than D + 1 elements and has arithmetic complexity O ∼ ( D ) [27]. CombiningTheorem 3.9 with the Lagrange algorithm, we have the following result. Corollary 3.10
Let R be an integral domain with more than O ∼ ( D max( nT, D )) elements. ThenAlgorithm 3.8 needs O ∼ ( nT D max( nT, D )) queries of B f and O ∼ ( nT D max( nT, D )) arithmeticoperations over R , plus a similar number of bit operations if using the Lagrange interpolation algo-rithm. If R is the field of complex numbers, we may use the Ben-or and Tiwari algorithm, whosecomplexity is dominated by the root finding step [6, 20]. In [16, Lemma 2.3 ], we show that forunivariate polynomials, the root can be found by factoring the coefficients of an auxiliary polynomial.As a consequence, the univariate Ben-or and Tiwarri’s algorithm costs O ∼ ( T log D ) R operations,which has better complexities than the Lagrange algorithm since T ≤ D for univariate polynomials.This result, combining with Theorem 3.9, gives the following result. Corollary 3.11
Let R = C . Then Algorithm 3.8 needs O ( nT ) queries of B f and costs O ∼ ( nT D max( nT, D )) bit operations if Ben-or and Tiwarri’s algorithm is used for univariate interpolation.Proof. Since f ( d,p ) , f ( d,p,k ) ≤ T and deg( f ( d,p ) ) , deg( f ( d,p,k ) ) are O ∼ ( D max( nT, D )), the arith-metic complexity of interpolating univariate polynomials is O ∼ ( nT log D ). Since we evaluate at9he points 2 , . . . , T − , the height of the data is O ∼ ( T D max( nT, D ). So the bit complexity ofAlgorithm 3.8 is O ∼ ( nT D max( nT, D )).If R = F q , the Ben-or and Tiwari algorithm can be changed into a deterministic algorithm whichneeds 2 T queries and O ∼ ( D log q ) bit operations [18]. Note that the Lagrange algorithm has thesame complexity O ∼ ( D log q ) but needs O ( D ) queries. Based on this modified Ben-or and Tiwarialgorithm, we have Corollary 3.12
Let R = F q . Then Algorithm 3.8 needs O ( nT ) queries of B f and O ∼ ( nT D max( nT, D ) log q ) extra bit operations if the modified Ben-or and Tiwari’s algorithm is used forunivariate interpolations.Proof. Since the degree of f ( d,p ) and f ( d,p,k ) are O ∼ ( D max( nT, D )), f ( d,p ) , f ( d,p,k ) ≤ T , by [18]and Theorem 3.9, the bit complexity is O ∼ ( n T D log q ).Among the three algorithms, the one in Corollary 3.10 works for more general rings, but has highcomplexities. Both the algorithms in Corollaries 3.11 and 3.12 use Ben-or and Tiwari’s algorithmfor univariate interpolations, but work for different coefficient rings. In this section, we give a method to recover f from f ( D,p ) , where D is a degree bound for f and p will be a sufficiently many primes, that is, we change the modulus p in the Kronecker substitution.The algorithm is quite similar to that given in Section 3, but we need to give a different criterionfor term testing: instead of finding a “ok” degree, we need to find a “ok” prime. We first give a new criterion for term testing.
Lemma 4.1
Let f = P ti =1 c i m i , T ≥ f, D > deg( f ) , N be the smallest number such that p p · · · p N ≥ D n ( T − , where p i is the i -th prime. Then for each cm ∈ M f , there exist at most N − primes p such that cm is a collision in f mod ( D,p ) .Proof. If t = 1, then N ≥
1. The proof is obvious. Now we assume t ≥
2. It suffices to showthat for any N different primes q , q , . . . , q N , there exists at least one q j , such that cm is not acollision in f mod ( D,q j ) . Assume m i = x e i, · · · x e i,n n , i = 1 , . . . , t . It suffices to consider the case of c m .We prove it by contradiction. Assume that for every q j , j = 1 , . . . , N , c m is a collision in f mod ( D,q j ) .Let B = Q ts =2 ( P ni =1 ( e ,i − e s,i ) D i − ) . First, we show that if c m is a collision in f mod ( D,q j ) , then mod ( B, q j ) = 0. Since c m is a collision in f mod ( D,q j ) , without loss of generality, assume m mod D,q j ) = m mod D,q j ) . Then 0 = deg( m mod D,q j ) ) − deg( m mod D,q j ) ) = mod ( P ni =1 e ,i D i − , q j ) − mod ( P ni =1 e ,i D i − , q j ). So mod ( P ni =1 ( e ,i − e ,i ) D i − , q j ) = 0, which implies that mod ( B, q j ) = 0. Since q , . . . , q N are different primes, Q N j =1 q j divides B . Note that | P ni =1 ( e ,i − e s,i ) D i − | ≤ ( D − P ni =1 D i − ) = D n −
1. So | B | = | Q ts =2 P ni =1 ( e ,i − s s,i ) D i − | ≤ ( D n − t − . Thus Q N j =1 q j ≥ Q N j =1 p j ≥ D n ( T − > ( D n − T − ≥ | B | , which contradicts the fact that Q N j =1 q j divides B . Soat least one of q j , j = 1 , . . . , N does not divide B . Without loss of generality, assume q does not10ivide it, then mod ( P ni =1 e ,i D i − − P ni =1 e s,i D i − , q ) = 0. So mod ( P ni =1 e ,i D i − , q ) = mod ( P ni =1 e s,i D i − , q ), for s = 2 , , . . . , t . So c m is not a collision in f mod ( D,q ) . We proved the lemma.We give a new criterion for cm ∈ C f . Theorem 4.2
Let f = P ti =1 c i m i , T ≥ f, D > deg f , N ( N ) be the smallest number such that p p · · · p N ≥ D n ( T − ( p p · · · p N ≥ D nT ) , and p i be the i -th prime. For a term cm satisfying deg( m ) < D , cm ∈ M f if and only if there exist at least N integers j ∈ [1 , N + N − such that f − cm ) mod ( D,p j ) < f mod ( D,p j ) .Proof. Let cm ∈ M f . If p j is a prime such that cm is not a collision in f mod ( D,p j ) , then f − cm ) mod ( D,p j ) = f mod ( D,p j ) −
1. So f − cm ) mod ( D,p j ) < f mod ( D,p j ) . By Lemma 4.1, there exist at most N − cm is a collision in f . In p , . . . , p N + N − , as N + N − − ( N −
1) = N , there exist atleast N primes such that f − cm ) mod ( D,p j ) < f mod ( D,p j ) . For the other direction, assume cm / ∈ M f .Consider two cases: Case 1: m is not a monomial in f . In this case, f − cm has at most T + 1terms. By Lemma 4.1, there exist at most N − j ∈ [1 , N + N −
1] such that cm is acollision in ( f − cm ) mod ( D,p j ) . In p , . . . , p N + N − , since N + N − − ( N −
1) = N , there are at least N primes such that f − cm ) mod ( D,p j ) > f mod ( D,p j ) . So there are at most N + N − − N = N − f − cm ) mod ( D,p j ) < f mod ( D,p j ) . Case 2: m is a monomial in f , but cm is not a termin f . In this case, f − cm has the same number of terms as f . Assume the term of f with monomial m is c m . By Lemma 4.1, there exist at most N − c − c ) m is a collision in( f − cm ) mod ( D,p j ) . In p , . . . , p N + N − , since N + N − − ( N −
1) = N , there are at least N primessuch that f − cm ) mod ( D,p j ) = f mod ( D,p j ) . So there are at most N + N − − N = N − ≤ N − f − cm ) mod ( D,p j ) < f mod ( D,p j ) .A prime p is called a “ok” prime, if at least half of terms of f do not collide in f mod ( D,p ) . We have Lemma 4.3 [17] Let f = P ti =1 c i m i ∈ R [ X ] , m i = x e i, · · · x e i,n n , D > deg( f ) , A = Q i Theorem 4.4 Let f = P ti =1 c i m i ∈ R [ X ] , T ≥ f, D > deg f , N be the smallest number suchthat p · · · p N ≥ D n ( T − , where p i is the i -th prime. Let j be an integer in [1 , N ] such that f mod ( D,p j ) ≥ f mod ( D,p j ) for all j = 1 , . . . , N . Then p j is an “ok” prime.Proof. We first claim that there exists at least one p j in p , . . . , p N such that C f ( D,p j ) < t . We proveit by contradiction. Assume for j = 1 , . . . , N , C f ( D,p j ) ≥ t . Then by Lemma 4.3, p ⌈ C ( D,pj ) ⌉ j divides A . Since p j , j = 1 , . . . , N are different primes, then Q N j =1 p ⌈ C ( D,pj ) ⌉ j divides A .Now Q N j =1 p ⌈ C ( D,pj ) ⌉ j ≥ D nt ( T − , which contradicts to the fact that A ≤ ( D n − t ( t − . Weproved the claim. By Lemma 3.5, we have C f ( D,p j ) ≤ C f ( D,p j ) , j = 1 , . . . , N . So C f ( D,p j ) < t . Sothe number of no collision terms of f in f mod ( D,p j ) is > t − t = t , which is at least ⌈ t ⌉ . The theoremis proved. 11 .2 Reduction from multivariate interpolation to univariate interpolation We now give the a new reduction algorithm based on the results given in the preceding section. Algorithm 4.5 (MIPoly2)Input: A black-box procedure B f for f ∈ R [ X ], T ≥ f , D > deg f . Output: The exact form of f . Step 1: Let K = max { , ⌈ n ( T − 1) log D ⌉ + ⌈ nT log D ⌉ , ⌈ nT log D ⌉} . Let p , . . . , p K be thefirst K primes. Step 2: Find the N , N , N be the integers defined in Lemma 4.1, Theorem 4.2, Theorem 4.4. Let N = max { N + N − , N } . Step 3: For j = 1 , . . . , N , compute f ( D,p j ) via a univariate interpolation algorithm. Let f j = f ( D,p j ) , f mod j = f mod ( D,p j ) = f ( D,p j ) mod ( x p − Step 4: Let α = max { f mod j | j = 1 , , . . . , N } and j satisfies f mod j = α . Let h = 0. Step 5: While α = 0 do a: For k = 1 , , . . . , n , compute f ( D,p j ,k ) via a univariate interpolation algorithm. Let g k = f ( D,p j ,k ) − h ( D,p j ,k ) . b: Let TS f − hD,p j ,D = TSTerms ( f mod j , f j , g , . . . , g n , D, p j , D ). c: Let s = 0. For each u ∈ TS fD,p j ,D , set s = s + u if { j | f mod j − u mod ( D,p j ) ) < f mod j ) , j = 1 , . . . , N + N − } ≥ N . d: Let h = h + s , T = T − s , update N , N , N for the new T following Step 2. Let N = max { N , N + N − } . e: For j = 1 , , . . . , N , let f j = f j − s ( D,p j ) , f mod j = f mod j − s mod ( D,p j ) . f: Let α = max { f mod j | j = 1 , , . . . , N } and j satisfies f mod j = α . Step 6: Return h . Theorem 4.6 Algorithm 4.5 is correct and needs interpolating O ( nT log D ) univariate polynomialswith degrees less than O ∼ ( nDT ) and terms less than T . Besides this, we still need O ∼ ( nT log D ) ring operations in R and O ∼ ( n T log D ) bit operations.Proof. Since N is the smallest number such that p · · · p N ≥ D n ( T − , we have 2 N − ≤ p · · · p N − < D n ( T − . So N ≤ max { , ⌈ n ( T − 1) log D ⌉} . Similarly, we have N ≤ max { , ⌈ nT log D ⌉} , N ≤ { max(1 , ⌈ nT log D ⌉ ) } . So in Step 2, we always have K ≥ N . By Theorem 4.4 and Lemma2.3, at least half of terms of f are in TS fD,p j . In c of Step 5, Theorem 4.2 is used to select theelements of M f from TS fD,p j . The correctness of the algorithm is proved.We call N univariate interpolations in Step 3 and at most n log t univariate interpolations in a of Step 5. Note that deg( f ( D,p i ) ) ≤ D ( p i − 1) and deg( f ( D,p i ,k ) ) ≤ D ( p i − i -th primeis O ( i log i ), N , N , N are O ( nT log D ), the first part of the theorem is proved.12n Step 1, the bit complexity of finding the first K primes is O ∼ ( K ) by [27, p.500,Them.18.10].Since K is O ∼ ( nT log D ), the bit complexity of Step 1 is O ∼ ( nT log D ). In Step 2, we need tocompute D n ( T − , D nT , D n ( T − . Since the height of data is O ( nT log D ) and the arithmetic oper-ation to compute them is O (log ( nT )), the bit complexity is O ∼ ( nT log D ). Since N , N , N are O ( nT log D ), the bit complexity of finding N , N , N is O ( n T log D ). In Step 3, computing one f mod j needs O ∼ ( T log( pD )) bit operations and O ( T ) ring operations. Since we need to compute N polynomials f mod j , the complexity is O ∼ ( nT log D ) bit operations and O ( nT log D ) ring op-erations. In Step 4, since N is O ( nT log D ) and the terms of f mod j is no more than T , we need O ∼ ( nT log D ) bit operations. Now we consider Step 5. We first analyse the complexity of one cy-cle. In b , by Lemma 2.5, the complexity is O ∼ ( nT log D ) bit operations and O ( nT ) ring operationsin R . In c , since every check needs to compare N + N − N + N − O ( f − hD,p j n ( N + N ) log( pD )+ f − hD,p j ( N + N ) log T log( pD ))bit operations and O ( f − hD,p j ( N + N )) ring operations. Since f − hD,p j ≤ T , the complexity is O ∼ ( n T log D ) bit operations and O ( nT log D ) ring operations. In d , for the same reason as inStep 2, the complexity is O ( n T log D ) bit operations. In e , we need n s operations to obtain s ( D,p j ) and s mod ( D,p j ) . Since we need to update N polynomials, the complexity is O ∼ ( n ( s ) N log( pD ))bit operations and O ( sN ) ring operations. Since in every recursion, we obtain at least half of theterms in f , at most log t recursions are needed. Since the sum of s is t , the complexity of Step 5is O ∼ ( n T log D ) bit operations and O ∼ ( nT log D ) ring operations. Similar to section 3.4, we gave three multivariate interpolation algorithms by combining Algorithm4.5 with three different univariate interpolation algorithms.Similar to Corollary 3.10, we have the following result. Corollary 4.7 Let R be an integral domain with more than O ∼ ( nT D ) elements and f ∈ R [ X ] .Then Algorithm 4.5 needs O ∼ ( n T D ) queries of B f and O ∼ ( n T D ) arithmetic operations over R plus a similar number of bit operations if using the Lagrange interpolation algorithm. Similar to Corollary 3.11, we have the following result. Corollary 4.8 Let R = C . Then Algorithm 4.5 needs O ( nT log D ) queries of B f and O ∼ ( n T D ) bit operations if Ben-or and Tiwarri’s algorithm is used for univariate interpolations. Similar to Corollary 3.12, we have the following result. Corollary 4.9 Let R = F q . Then Algorithm 4.5 needs O ( nT log D ) queries of B f and O ∼ ( n T D log q ) bit operations if Ben-or and Tiwari’s algorithm is used for univariate interpolations. In this paper, we revisit the approach of reducing the black-box multivariate polynomial interpo-lation to that of the univariate polynomial by introducing a new modified Kronecker Substitution.Over the field C , the bit complexity of the algorithm is linear in D , while all existing determin-istic algorithms are quadratic in D . Over finite fields, the new algorithm has better complexitiescomparing to existing deterministic algorithms in T and D and has the same complexity in n .13inally, we compare the complexities of Algorithm 3.8 and Algorithm 4.5 in the case of finitefields. In other cases, the results are the same. The complexities of Algorithm 3.8 and Algorithm 4.5are O ∼ ( nT D max( nT, D ) log q ) and O ∼ ( n T D log q ), respectively. If nT > D , the two algorithmshave the same complexities. If nT < D , the ratio of the complexities of Algorithm 3.8 and Algorithm4.5 is DnT , that is, Algorithm 4.5 performs better for large D . References [1] N. Alon, Y. Mansour, Epsilon-discrepancy sets and their application for interpolation of sparsepolynomials, Inform. Process. Lett. 54(6) (1995) 337-342.[2] A. Arnold and D.S. Roche “Multivariate sparse interpolation using randomized Kroneckersubstitutions,” ISSAC’14, ACM Press, 35-42, 2014.[3] A. Arnold, M. Giesbrecht, D.S. Roche, “Faster sparse multivariate polynomial interpolation ofstraight-line programs,” J. of Sym. Comp., 75, 4-24, 2016.[4] A. Arnold, M. Giesbrecht, D.S. Roche, “Faster sparse interpolation of straight-line programs,”In CASC13, LNCS 8136, 61-74. 2013[5] M. Avenda˜no, T. Krick, A. Pacetti, “Newton-Hensel interpolation lifting,” Foundations ofComputational Mathematics, 6(1), 82-120, 2006.[6] M. Ben-Or and P. Tiwari, “A deterministic algorithm for sparse multivariate polynomialinterpolation,” Proc. STOC’88, ACM Press, 301-309, 1988.[7] M. Clausen, A. Dress, J. Grabmeier, M. Karpinski, “On zero-testing and interpolation of kk