5,713 research outputs found
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
Valiant introduced some 25 years ago an algebraic model of computation along
with the complexity classes VP and VNP, which can be viewed as analogues of the
classical classes P and NP. They are defined using non-uniform sequences of
arithmetic circuits and provides a framework to study the complexity for
sequences of polynomials. Prominent examples of difficult (that is,
VNP-complete) problems in this model includes the permanent and hamiltonian
polynomials. While the permanent and hamiltonian polynomials in general are
difficult to evaluate, there have been research on which special cases of these
polynomials admits efficient evaluation. For instance, Barvinok has shown that
if the underlying matrix has bounded rank, both the permanent and the
hamiltonian polynomials can be evaluated in polynomial time, and thus are in
VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded
treewidth several difficult problems (including evaluating the permanent and
hamiltonian polynomials) can be solved efficiently. An earlier result of this
flavour is Kasteleyn's theorem which states that the sum of weights of perfect
matchings of a planar graph can be computed in polynomial time, and thus is in
VP also. For general graphs this problem is VNP-complete. In this paper we
investigate the expressive power of the above results. We show that the
permanent and hamiltonian polynomials for matrices of bounded treewidth both
are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits
are shown to be equivalent to the sum of weights of perfect matchings of planar
graphs.Comment: 14 page
Affine Determinant Programs: A Framework for Obfuscation and Witness Encryption
An affine determinant program ADP: {0,1}^n → {0,1} is specified by a tuple (A,B_1,...,B_n) of square matrices over F_q and a function Eval: F_q → {0,1}, and evaluated on x \in {0,1}^n by computing Eval(det(A + sum_{i \in [n]} x_i B_i)).
In this work, we suggest ADPs as a new framework for building general-purpose obfuscation and witness encryption. We provide evidence to suggest that constructions following our ADP-based framework may one day yield secure, practically feasible obfuscation.
As a proof-of-concept, we give a candidate ADP-based construction of indistinguishability obfuscation (iO) for all circuits along with a simple witness encryption candidate. We provide cryptanalysis demonstrating that our schemes resist several potential attacks, and leave further cryptanalysis to future work. Lastly, we explore practically feasible applications of our witness encryption candidate, such as public-key encryption with near-optimal key generation
Optimal ancilla-free Clifford+T approximation of z-rotations
We consider the problem of approximating arbitrary single-qubit z-rotations
by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new
probabilistic algorithm for solving this problem optimally, i.e., for finding
the shortest possible circuit whatsoever for the given problem instance. The
algorithm requires a factoring oracle (such as a quantum computer). Even in the
absence of a factoring oracle, the algorithm is still near-optimal under a mild
number-theoretic hypothesis. In this case, the algorithm finds a solution of
T-count m + O(log(log(1/epsilon))), where m is the T-count of the
second-to-optimal solution. In the typical case, this yields circuit
approximations of T-count 3log_2(1/epsilon) + O(log(log(1/epsilon))). Our
algorithm is efficient in practice, and provably efficient under the
above-mentioned number-theoretic hypothesis, in the sense that its expected
runtime is O(polylog(1/epsilon)).Comment: 40 pages. New in v3: added a section on worst-case behavio
Cryptanalysis of a family of self-synchronizing chaotic stream ciphers
Unimodal maps have been broadly used as a base of new encryption strategies.
Recently, a stream cipher has been proposed in the literature, whose keystream
is basically a symbolic sequence of the (one-parameter) logistic map or of the
tent map. In the present work a thorough analysis of the keystream is made
which reveals the existence of some serious security problemsComment: 10 pages, 6 figure
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