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Depth Optimized Efficient Homomorphic Sorting

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Progress in Cryptology -- LATINCRYPT 2015 (LATINCRYPT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9230))

Abstract

We introduce a sorting scheme which is capable of efficiently sorting encrypted data without the secret key. The technique is obtained by focusing on the multiplicative depth of the sorting circuit alongside the more traditional metrics such as number of comparisons and number of iterations. The reduced depth allows much reduced noise growth and thereby makes it possible to select smaller parameter sizes in somewhat homomorphic encryption instantiations resulting in greater efficiency savings. We first consider a number of well known comparison based sorting algorithms as well as some sorting networks, and analyze their circuit implementations with respect to multiplicative depth. In what follows, we introduce a new ranking based sorting scheme and rigorously analyze the multiplicative depth complexity as \(\mathcal {O}(\log (N)+\log (\ell ))\), where N is the size of the array to be sorted and \(\ell \) is the bit size of the array elements. Finally, we simulate our sorting scheme using a leveled/batched instantiation of a SWHE library. Our sorting scheme performs favorably over the analyzed classical sorting algorithms.

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Notes

  1. 1.

    Note that in their implementation Chatterjee et al. [8] perform the comparison using a carry propagate adder based subtraction circuit result in a circuit depth \((N^2-N)(\ell +1)/2\). While the computational complexity of the scheme is low, the \(\mathcal {O}(N^2)\) circuit depth is prohibitive.

  2. 2.

    Note that when there is no ambiguity we will drop the comma, i.e. write \(m_{i,j}^{(\gamma )}\) as \(m_{ij}^{(\gamma )}\) in the indices for brevity.

  3. 3.

    Note that N is not restricted to a power of two.

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Acknowledgments

Funding for this research was in part provided by the US National Science Foundation CNS Awards #1319130.

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Authors and Affiliations

  1. Worcester Polytechnic Institute, Worcester, USA

    Gizem S. Çetin, Yarkın Doröz & Berk Sunar

  2. Sabanci University, Istanbul, Turkey

    Erkay Savaş

Authors
  1. Gizem S. Çetin
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  2. Yarkın Doröz
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  3. Berk Sunar
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  4. Erkay Savaş
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Correspondence to Gizem S. Çetin .

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  1. Microsoft Research, Redmond, Washington, USA

    Kristin Lauter

  2. CINVESTAV-IPN, Mexico City, Distrito Federal, Mexico

    Francisco Rodríguez-Henríquez

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Çetin, G.S., Doröz, Y., Sunar, B., Savaş, E. (2015). Depth Optimized Efficient Homomorphic Sorting. In: Lauter, K., Rodríguez-Henríquez, F. (eds) Progress in Cryptology -- LATINCRYPT 2015. LATINCRYPT 2015. Lecture Notes in Computer Science(), vol 9230. Springer, Cham. https://doi.org/10.1007/978-3-319-22174-8_4

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  • DOI: https://doi.org/10.1007/978-3-319-22174-8_4

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