A Sequence of Polynomials for Approximating Arctangent

Medina, Herbert A.

doi:10.2307/27641866

Cited by 4 publications

(7 citation statements)

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“…It is, therefore, sufficient to approximate arctangent over the interval of [0, 1]. To this end, we use the technique introduced by Medina in [50] and its formal proof in [29]. In particular, [50] defines a sequence of polynomials over the input domain of [0, 1], denoted as h N (x), with the property that…”

Section: Compute ([Zmentioning

confidence: 99%

“…This degree N is logarithmic with respect to the desired precision. Afterwards, the coefficients of h N (x) are computed from the recursive definitions in [50]. We choose this approach over other alternatives such as [33] for its efficiency.…”

Section: Compute ([Zmentioning

confidence: 99%

“…In this approach, the search can be performed using binary search trees whose secure realization is non-trivial or alternatively will involve s steps. This extra computation makes the overall performance of the solution in [33] worse than that of [50]'s approach used here. Another candidate for arctangent evaluation is the well-known Taylor series of arctangent.…”

Section: Compute ([Zmentioning

confidence: 99%

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Secure Fingerprint Alignment and Matching Protocols

Bayatbabolghani,

Blanton,

Aliasgari

et al. 2017

Preprint

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We present three private fingerprint alignment and matching protocols, based on precise and efficient fingerprint recognition algorithms that use minutia points. Our protocols allow two or more semi-honest parties to compare privately-held fingerprints in a secure way such that nothing more than an accurate score of how well the fingerprints match is revealed to output recipients. To the best of our knowledge, this is the first time fingerprint alignment based on minutiae is considered in a secure computation framework. We build secure fingerprint alignment and matching protocols in both the two-party setting using garbled circuits and in the multi-party setting using secret sharing. In addition to providing precise and efficient secure fingerprint comparisons, our contributions include the design of a number of secure sub-protocols for complex operations such as sine/cosine, arctangent, and selection, which are likely to be of independent interest.

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Section: Compute ([Zmentioning

confidence: 99%

Section: Compute ([Zmentioning

confidence: 99%

Section: Compute ([Zmentioning

confidence: 99%

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Secure Fingerprint Alignment and Matching Protocols

Bayatbabolghani,

Blanton,

Aliasgari

et al. 2017

Preprint

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“…So it is of practical importance to use a polynomial approximation that converges more quickly to arctangent. A recent result of Medina's provides such an approximation [7], and this paper describes a formalization of that result in ACL2(r).…”

Section: Introductionmentioning

confidence: 95%

Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent

Gamboa

Cowles

2014

Electron. Proc. Theor. Comput. Sci.

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The verification of many algorithms for calculating transcendental functions is based on polynomial approximations to these functions, often Taylor series approximations. However, computing and verifying approximations to the arctangent function are very challenging problems, in large part because the Taylor series converges very slowly to arctangent-a 57th-degree polynomial is needed to get three decimal places for arctan(0.95). Medina proposed a series of polynomials that approximate arctangent with far faster convergence-a 7th-degree polynomial is all that is needed to get three decimal places for arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness and convergence rate of this sequence of polynomials. The proof is particularly beautiful, in that it uses many results from real analysis. Some of these necessary results were proven in prior work, but some were proven as part of this effort.

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“…was recently studied in [Medina 2006], where it is shown that it can be used to produce polynomial approximations to arctan x on the interval [0, 1] whose error is governed by the size of the rational functions on that interval. In this section, we use these methods to produce polynomial approximations to arctan x on a smaller interval where the size of the integrand is much smaller, and hence the approximations converge much faster.…”

Section: Polynomial Approximations To Arctangentmentioning

confidence: 99%