A set of multivariate polynomials is algebraically independent if they exhibit no non-trivial algebraic relations, and this notion is fundamental in algebra. When these polynomials are given as algebraic circuits, deciding algebraic independence has several applications in algebraic complexity theory. Over fields of zero (or exponentially large) characteristic, this problem is known to have an efficient randomized algorithm. Over finite fields of small characteristic, a sequence of works has culminated in showing that algebraic independence admits Arthur-Merlin proofs, in particular giving the complexity bound of AM∩coAM ([Guo et al., 2019]). We improve the complexity of deciding algebraic independence over finite fields by showing that it admits zero-knowledge proofs, in particular giving the upper bound of NISZK ⊆ AM∩coAM, the class of problems admitting non-interactive statistical zero-knowledge proofs. This is achieved by arguing that algebraically independent polynomials yield maps whose output distribution has high-entropy, while algebraically dependent polynomials yield maps with low-entropy. We can then reduce to the question of approximating entropy, which is a known NISZK-complete problem. We also more generally show that transcendence degree, which quantifies the independence of a set of possibly dependent polynomials, can be computed in NISZK.

We show that any nonzero polynomial in the ideal generated by the r × r minors of an n × n matrix X can be used to efficiently approximate the determinant. Specifically, for any nonzero polynomial f in this ideal, we construct a small depth-three f-oracle circuit that approximates the Θ(r1/3) × Θ(r1/3) determinant in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by r × r minors is at least as hard to approximately compute as the Θ(r1/3) × Θ(r1/3) determinant. We also prove an analogous result for the Pfaffian of a 2n × 2n skew-symmetric matrix and the ideal generated by Pfaffians of 2r × 2r principal submatrices.

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP , as long as one is willing to make a suitable physical assumption : if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting. In this work, we study the following question: Does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * . Our proof consists of constructing a zero knowledge interactive probabilistically checkable proof with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP* model captures this setting. In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP*. Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP* model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.

A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL = L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan [Nis92], pseudorandom generators with seedlength O(log2n) were constructed (see also [INW94], [GR14]). Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas. A recent line of inquiry (e.g., [BV10], [GMR+12], [IMZ12], [RSV13], [SVW14], [HLV17], [LV17], [CHRT17]) has suggested focusing on a particular limitation of the existing PRGs ([Nis92], [INW94], [GR14]), which is that they only fool roBPs when the variables are read in a particular known order, such as x1n. In comparison, existentially one can obtain logarithmic seed-length for fooling the set of polynomial-size roBPs that read the variables under any fixed unknown permutation xπ(1)xπ(n). While recent works have established novel PRGs in this setting for subclasses of roBPs, there were no known no(1)seed-length explicit PRGs for general polynomial-size roBPs in this setting. In this work, we follow the "bounded independence plus noise" paradigm of Haramaty, Lee and Viola [HLV17], [LV17], and give an improved analysis in the general roBP unknownorder setting. With this analysis we obtain an explicit PRG with seed-length O(log3n) for polynomial-size roBPs reading their bits in an unknown order. Plugging in a recent Fourier tail bound of Chattopadhyay, Hatami, Reingold, and Tal [CHRT17], we can obtain a Õ(log2n) seed-length when the roBP is of constant width.

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC'18) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s. We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS'95) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models.

In this paper we study the complexity of constructing a hitting set for VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of rational n-tuples of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree r that are the limit of size s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known.

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits.

Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP.

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest.

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted boolean circuit classes. Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). We give two general methods of converting certain algebraic lower bounds into proof complexity ones. Our methods require stronger notions of lower bounds, which lower bound a polynomial as well as an entire family of polynomials it defines. Our techniques are reminiscent of existing methods for converting boolean circuit lower bounds into related proof complexity results, such as feasible interpolation. We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof complexity results. We complement our lower bounds by giving short refutations of the previously-studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems.

An emerging theory of "linear algebraic pseudorandomness: aims to understand the linear algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, two-source rank condensers, and rank-metric codes. In particular, with the recent construction of near-optimal subspace designs by Guruswami and Kopparty as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps F^n to F^t for t<

Kayal has recently introduced the method of shifted partial derivatives as a way to give the first exponential lower bound for computing an explicit polynomial as a sum of powers of constant-degree polynomials. This method has garnered further attention because of the work of Gupta, Kamath, Kayal and Saptharishi who used this method to obtain lower bounds that approach the "chasm at depth-4". In this work, we investigate to what extent this method can be used to obtain deterministic polynomial identity testing (PIT) algorithms, which are algorithms that decide whether a given algebraic circuit computes the zero polynomial. In particular, we give a poly(s)(log s)-time deterministic black-box PIT algorithm for a size-s sum of powers of constant-degree polynomials. This is the first sub-exponential deterministic PIT algorithm for this model of computation and the first PIT algorithm based on the method of shifted partial derivatives. We also study the problem of divisibility testing, which when given multivariate polynomials f and g (as algebraic circuits) asks to decide whether f divides g. Using Strassen's technique for the elimination of divisions, we show that one can obtain deterministic divisibility testing algorithms via deterministic PIT algorithms, and this reduction does not dramatically increase the complexity of the underlying algebraic circuits. Using this reduction, we show that deciding divisibility of a constant-degree polynomial f into a sparse polynomial g reduces to PIT of sums of a monomial multiplied by a power of constant-degree polynomials. We then extend the method of shifted partial derivatives to give a poly(s)(log s)-time deterministic black-box PIT algorithm for this model of computation, and thus derive a corresponding deterministic divisibility algorithm. This is the first non-trivial deterministic algorithm for this problem. Previous work on multivariate divisibility testing due to Saha, Saptharishi and Saxena gave algorithms for when f is linear and g is sparse, and essentially worked via PIT algorithms for read-once (oblivious) algebraic branching programs (roABPs). We give explicit sums of powers of quadratic polynomials that require exponentially-large roABPs in a strong sense, showing that techniques known for roABPs have limited applicability in our regime. Finally, by combining our results with the algorithm of Forbes, Shpilka and Saptharishi we obtain poly(s)(log log s)-time deterministic black-box PIT algorithms for various models (translations of sparse polynomials, and sums of monomials multiplied by a power of a linear polynomial) where only poly(s) (Theta(log s))-time such algorithms were previously known.

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in nO(log2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order.

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm. The main result of this work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg2S), this is analogous (in the terminology of boolean pseudorandom ness) to a seed-length of lg2S, which is the seed length of the pseudorandom generators of Nisan and Impagliazzo-Nisan-Wigderson for read-once oblivious boolean branching programs. Thus our work can be seen as an algebraic analogue of these foundational results in boolean pseudorandom ness. Our results are stronger for branching programs of bounded width, where we give a hitting set of size exp(lg2S/lglg S), corresponding to a seed length of lg2S/lglg S. This is in stark contrast to the known results for read-once oblivious boolean branching programs of bounded width, where no pseudorandom generator (or hitting set) with seed length o(lg2S) is known. Thus, while our work is in some sense an algebraic analogue of existing boolean results, the two regimes seem to have non-trivial differences. In follow up work, we strengthened a result of Mulmuley, and showed that derandomizing a particular case of the No ether Normalization Lemma is reducible to black-box PIT of read-once oblivious ABPs. Using the results of the present work, this gives a derandomization of No ether Normalization in that case, which Mulmuley conjectured would difficult due to its relations to problems in algebraic geometry. We also show that several other circuit classes can be black-box reduced to read-once oblivious ABPs, including set-multilinear ABPs (a generalization of depth-3 set-multilinear formulas), non-commutative ABPs (generalizing non-commutative formulas), and (semi-)diagonal depth-4 circuits (as introduced by Saxena). For set-multilinear ABPs and non-commutative ABPs, we give quasi-polynomial-time black-box PIT algorithms, where the latter case involves evaluations over the algebra of (D+1)x(D+1) matrices, where D is the depth of the ABP. For (semi-)diagonal depth-4 circuits, we obtain a black-box PIT algorithm (over any characteristic) whose run-time is quasi-polynomial in the runtime of Saxena's white-box algorithm, matching the concurrent work of Agrawal, Saha, and Saxena. Finally, by combining our results with the reconstruction algorithm of Klivans and Shpilka, we obtain deterministic reconstruction algorithms for the above circuit classes.

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [36]), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for tensors of degree 2 (matrices), and obtain the first quasi-polynomial sized hitting sets for arbitrary tensors. We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank recovery asks (say, over R) to recover a matrix M from few measurements, under the promise that M is rank ≤ r. In this work, we restrict our attention to recovering matrices that are exactly rank ≤ r using deterministic, non-adaptive, linear measurements, that are free from noise. Over R, we provide a set (of size 4nr) of such measurements, from which M can be recovered in O(rn2+r3n) field operations, and the number of measurements is essentially optimal. Further, the measurements can be taken to be all rank-1 matrices, or all sparse matrices. To the best of our knowledge no explicit constructions with those properties were known prior to this work.

The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n]d→ F with rank at least 2n⌊d/2⌋+ n - Θ(d lg n). This improves the lower-order terms in known lower bounds for any odd d ≥ 3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by apply ing known counting lower bounds, that there exist order-3 permutation tensors with super-linear rank as well as order d permutation tensors with high rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor TGd: Gd→ F, by TGd(g1,⋯,gd) = 1 iff g1⋯ gd= 1G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over "large" fields F, showing (among other things) that rankF(TGd) ≤ |G|d/2. In the case that d = 3, we are able to show that rankF(TG3) ≤ O(|G|ω/2) ≤ O(|G|1.19), where ω is the exponent of matrix multiplication. The next upper bound uses interpolation and only works for abelian G, showing that over any field F that rankF(TGd) ≤ O(|G|1+1g dlgd-1|G|). In either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank. We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n]d→ F that have tensor rank at most dn but have monotone tensor rank exactly nd-1. This is a nearly optimal separation.