97 research outputs found
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Polynomial evaluation over finite fields: new algorithms and complexity bounds
An efficient evaluation method is described for polynomials in finite fields.
Its complexity is shown to be lower than that of standard techniques when the
degree of the polynomial is large enough. Applications to the syndrome
computation in the decoding of Reed-Solomon codes are highlighted.Comment: accepted for publication in Applicable Algebra in Engineering,
Communication and Computing. The final publication will be available at
springerlink.com. DOI: 10.1007/s00200-011-0160-
Polynomial evaluation over finite fields: new algorithms and complexity bounds
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques, when the degree of the polynomial is large enough compared to the field characteristic. Specifically, if n is the degree of the polynomiaI, the asymptotic complexity is shown to be , versus O(n) of classical algorithms. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighte
Complexity of multivariate polynomial evaluation
We describe a method to evaluate multivariate polynomials over a finite field
and discuss its multiplicative complexity
Near-optimal Polynomial for Modulus Reduction Using L2-norm for Approximate Homomorphic Encryption
Since Cheon et al. introduced an approximate homomorphic encryption scheme for complex numbers called Cheon-Kim-Kim-Song (CKKS) scheme, it has been widely used and applied in real-life situations, such as privacy-preserving machine learning.
The polynomial approximation of a modulus reduction is the most difficult part of the bootstrapping for the CKKS scheme.
In this paper, we cast the problem of finding an approximate polynomial for a modulus reduction into an L2-norm minimization problem.
As a result, we find an approximate polynomial for the modulus reduction without using the sine function, which is the upper bound for the approximation of the modulus reduction.
With the proposed method, we can reduce the degree of the polynomial required for an approximate modulus reduction, while also reducing the error compared with the most recent result reported by Han et al. (CT-RSA\u27 20).
Consequently, we can achieve a low-error approximation, such that the maximum error is less than for the size of the message .
By using the proposed method, the constraint of is relaxed as , and thus the level loss in bootstrapping can be reduced.
The solution of the cast problem is determined in an efficient manner without iteration
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