A Selection of Lower Bounds for Arithmetic Circuits

Kayal, Neeraj; Saptharishi, Ramprasad

doi:10.1007/978-3-319-05446-9_5

Cited by 7 publications

(7 citation statements)

References 28 publications

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“…A similar result also holds for homogeneous depth four circuits. The known Ω(2 n 1/2 log n ) lower bounds for the restricted versions of these circuits [53] do not imply any nontrivial result for NNL for explicit varieties. Thus there is a sharp phase transition in the difficulty of the lower bound problem in this model at the exponent 1/2.…”

Section: Remarkmentioning

confidence: 91%

Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry.In particular, we show that:(1) The categorical quotient for any finite dimensional representation V of SL m , with constant m, is explicit in characteristic zero.(2) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in the dimension of V .(3) The categorical quotient of the space of r-tuples of m×m matrices by the simultaneous conjugation action of SL m is explicit in any characteristic.(4) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in m and r in any characteristic p ∈ [2, ⌊m/2⌋].(5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory.The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

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Section: Remarkmentioning

confidence: 91%

Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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“…Since we use polynomials to construct k-independent (almost) universal hash functions, we must prove their concrete optimality in cWRAM evaluations. However, all concrete optimality results for polynomial evaluation are known only over infinite (e.g., rational) fields [9], and the gap between these bounds and the lower bounds over finite fields (e.g., Z p ) is very large [35]. Furthermore, optimality is obtained using only two operations (i.e., +, ×) and cannot hold in computation models with large instruction sets like the cWRAM and real processors.…”

Section: B Proving Optimality Of Universal Hash Functions In Cwrammentioning

confidence: 99%

“…If the only operations allowed for polynomial evaluation are the addition and multiplication, Horner rule's bound of 2d operations for degree-d polynomials was shown to be uniquely optimal in one-time evaluations [9], [61]. However, this bound does not hold in finite fields, where the minimum number of modular additions and multiplications is Ω( (d + 1)) [35]. Furthermore, these bounds do not hold in any WRAM models or any real computer where many more operations are implemented by the ISA.…”

Section: B Polynomial Evaluationmentioning

confidence: 99%

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A rest stop on the unending road to provable security

2021

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Root-of-Trust (RoT) establishment ensures either that the state of an untrusted system contains all and only content chosen by a trusted local verifier and the system code begins execution in that state, or that the verifier discovers the existence of unaccounted for content. This ensures program booting into system states that are free of persistent malware. An adversary can no longer retain undetected control of one's local system. We establish RoT unconditionally; i.e., without secrets, trusted hardware modules and instructions, or bounds on the adversary's computational power. The specification of a system's chipset and device controllers, and an external source of true random numbers, such as a commercially available quantum RNG, is all that is needed. Our system specifications are those of a concrete Word Random Access Machine (cWRAM) model -the closest computation model to a real system with a large instruction set.We define the requirements for RoT establishment and explain their differences from past attestation protocols. Then we introduce a RoT establishment protocol based on a new computation primitive with concrete (non-asymptotic) optimal space-time bounds in adversarial evaluation on the cWRAM. The new primitive is a randomized polynomial, which has kindependent uniform coefficients in a prime order field. Its collision properties are stronger than those of a k-independent (almost) universal hash function in cWRAM evaluations, and are sufficient to prove existence of malware-free states before RoT is established. Preliminary measurements show that randomizedpolynomial performance is practical on commodity hardware even for very large k.To prove the concrete optimality of randomized polynomials, we present a result of independent complexity interest: a Hornerrule program is uniquely optimal whenever the cWRAM execution space and time are simultaneously minimized.

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“…In particular, it is known that there is no sub-exponential family of monotone circuits for the permanent. This was first shown for the field of real numbers [17] and a proof for general fields, with a suitably adapted notion of monotonicity is given in [18]. An exponential lower bound for the permanent is also known for depth-3 arithmetic circuits [15] for all finite fields.…”

Section: Introductionmentioning

confidence: 97%

Symmetric Arithmetic Circuits

Dawar,

Wilsenach

2020

Preprint

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We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characterisitic zero.

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A Selection of Lower Bounds for Arithmetic Circuits

Cited by 7 publications

References 28 publications

Geometric complexity theory V: Efficient algorithms for Noether normalization

Geometric complexity theory V: Efficient algorithms for Noether normalization

A rest stop on the unending road to provable security

Symmetric Arithmetic Circuits

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