Efficiently inverting bijections given by straight line programs

Sturtivant, Carl; Zhang, Z.-L.

doi:10.1109/fscs.1990.89551

Cited by 9 publications

(5 citation statements)

References 11 publications

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“…Acknowledgments: Thanks go to Erich Kaltofen for communicating to me his paper [22] and to an anonymous referee for pointing out the reference [38]. I am grateful to Alan Selman for answering my questions about the complexity of one-way functions.…”

Section: Resultsmentioning

confidence: 99%

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The Complexity of Factors of Multivariate Polynomials

Bürgisser

2004

Found Comput Math

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The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen [22]. The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.

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Section: Resultsmentioning

confidence: 99%

“…Coming back to the discussion of one-way functions, we remark that Sturtivant and Zhang [38] obtained the following related result, which excludes the existence of certain one-way functions in the algebraic framework of computation. Let ψ : k n → k n be bijective such that ψ as well as ψ −1 are polynomial mappings.…”

Section: Remark 14mentioning

confidence: 95%

The Complexity of Factors of Multivariate Polynomials

Bürgisser

2004

Found Comput Math

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“…Straight-line programs, or equivalently algebraic circuits, are important both as a computational model and as a data structure for polynomial computation. Their rich history includes both algorithmic advances and practical implementations (Kaltofen, 1989;Sturtivant and Zhang, 1990;Bruno et al, 2002).…”

Section: Previous Workmentioning

confidence: 99%

Computer Algebra in Scientific Computing

England

Boulier

Sadykov

et al. 2013

Lecture Notes in Computer Science

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We give a new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program. Our algorithm constructs an approximation f * of f , such that f − f * probably has at most half the number of terms of f , then recurses on the difference f −f * . Our approach builds on previous work by Garg andSchost (2009), andRoche (2011), and is asymptotically more efficient in terms of the total cost of the probes required than previous methods, in many cases.

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“…One of the most fundamental questions in cryptanalysis is to characterize the class of permutations (or functions) whose inverse can be computed in polynomial time or by a polynomial-size circuit [1], [16], [18], [25]. Much research in theoretical cryptography has centered around finding the weakest possible cryptographic assumptions that enable the implementation of major cryptographic primitives [8], [7], [ 17]; however, little progress has been made on the characterization of permutations with small inversion circuits [18].…”

Section: Introductionmentioning

confidence: 99%

Functional inversion and communication complexity

Teng

1994

J. Cryptology

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We study the relationship between the multiparty communication complexity of functions over certain communication topologies and the complexity of inverting those functions. We show that if a function of n variables has a ringprotocol or a tree-protocol of communication complexity bounded by q~, then there is a circuit of size 2°~n that computes an inverse of the function. Consequently, we prove that although inverting NC ° Boolean circuits is NP-hard, planar NC 1 Boolean circuits can be inverted in NC, and hence in polynomial time. From the ring-protocol theorem, we derive an ~(n 10g n) lower bound on the VLSI area required to lay out any one-way function. Our results on inverting boolean circuits can be extended to algebraic circuits over finite rings. We prove that on certain topologies no one-way function can be computed with low communication complexity.

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Efficiently inverting bijections given by straight line programs

Cited by 9 publications

References 11 publications

The Complexity of Factors of Multivariate Polynomials

The Complexity of Factors of Multivariate Polynomials

Computer Algebra in Scientific Computing

Functional inversion and communication complexity

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