609 research outputs found
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
Sub-quadratic Decoding of One-point Hermitian Codes
We present the first two sub-quadratic complexity decoding algorithms for
one-point Hermitian codes. The first is based on a fast realisation of the
Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer
algebra for polynomial-ring matrix minimisation. The second is a Power decoding
algorithm: an extension of classical key equation decoding which gives a
probabilistic decoding algorithm up to the Sudan radius. We show how the
resulting key equations can be solved by the same methods from computer
algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity
results, as well as a number of reviewer corrections. 20 page
Arion: Arithmetization-Oriented Permutation and Hashing from Generalized Triangular Dynamical Systems
In this paper we propose the (keyed) permutation Arion and the hash function
ArionHash over for odd and particularly large primes. The design
of Arion is based on the newly introduced Generalized Triangular Dynamical
System (GTDS), which provides a new algebraic framework for constructing
(keyed) permutation using polynomials over a finite field. At round level Arion
is the first design which is instantiated using the new GTDS. We provide
extensive security analysis of our construction including algebraic
cryptanalysis (e.g. interpolation and Groebner basis attacks) that are
particularly decisive in assessing the security of permutations and hash
functions over . From a application perspective, ArionHash is
aimed for efficient implementation in zkSNARK protocols and Zero-Knowledge
proof systems. For this purpose, we exploit that CCZ-equivalence of graphs can
lead to a more efficient implementation of Arithmetization-Oriented primitives.
We compare the efficiency of ArionHash in R1CS and Plonk settings with other
hash functions such as Poseidon, Anemoi and Griffin. For demonstrating the
practical efficiency of ArionHash we implemented it with the zkSNARK libraries
libsnark and Dusk Network Plonk. Our result shows that ArionHash is
significantly faster than Poseidon - a hash function designed for
zero-knowledge proof systems. We also found that an aggressive version of
ArionHash is considerably faster than Anemoi and Griffin in a practical zkSNARK
setting
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