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Complexity and Performance of Secure Floating-Point Polynomial Evaluation Protocols

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 12973)

Abstract

Secure computation provides cryptographic protocols for collaborative applications with private inputs and outputs. In this paper, we examine a collection of protocols for secure evaluation of polynomials using secure floating-point arithmetic. The main goal is to provide a comparative analysis of their construction, complexity, performance, and tradeoffs in different application settings. The analysis demonstrates the performance gains that can be obtained by evaluating the polynomials using optimized secure multi-operand arithmetic instead of relying on generic constructions based on two-operand arithmetic. It also examines the relations between performance and complexity metrics for different execution environments (LAN, Internet), floating-point precision, and problem sizes. These protocols are part of a framework for secure multiparty computation with fixed-point and floating-point numbers based on Shamir secret sharing and related techniques.

Keywords

  • Secure multiparty computation
  • Secret sharing
  • Secure floating-point arithmetic
  • Polynomial evaluation

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Fig. 1.
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Fig. 4.

Notes

  1. 1.

    \(\mathsf {Div2mP}\) returns \(\bar{c}=\lfloor \bar{a}/2^m \rfloor + u\), where \(u\in \{0,1\}\) and \(u=1\) with probability \(p = \frac{\bar{a} \bmod 2^m}{2^m}\). For example, if \(\bar{a}=46\) and \(m=3\) then \(\bar{a}/2^m = 5.75\); the output is \(\bar{c}=6\) with probability \(p=0.75\) or \(\bar{c}=5\) with probability \(1-p=0.25\).

  2. 2.

    This bound is determined as follows: Let \(\{\bar{v}_i\}_{i=0}^m\) the significands of \(\{\hat{a}_i \hat{x}^i\}_{i=0}^m\) after radix-point alignment, with up to \(\ell + \lceil i / 2^\theta \rceil \) bits, and \(\sigma = \lceil m / 2^\theta \rceil \). \(\mathsf {SumFL}\) computes \(\sum _{i=0}^m \bar{v}_i = \bar{v}_0 + \sum _{k=0}^{\sigma -2} \sum _{t=1}^{2^\theta } \bar{v}_{k 2^\theta + t} + \sum _{t=1}^{m \bmod 2^\theta } \bar{v}_{(\sigma -1) 2^\theta + t}< 2^\ell + 2^{\theta + \ell + 1} \sum _{k=0}^{\sigma -2} 2^{k} + 2^{\ell +\sigma }(m \bmod 2^\theta ) < 2^{\ell + \sigma + \theta +1}\). So the maximum bitlength is \(\ell + \lceil m / 2^\theta \rceil + \theta + 1\) bits.

  3. 3.

    For example, if \(m=64\) and \(\theta = 0\) the modulus grows by 128 bits. If \(\theta = 3\), it grows by 16 bits, at the cost of fully normalizing 7 out of 127 multiplications.

  4. 4.

    For example, if \(m=64\) and \(t=64\) the modulus grows by 62 bits. If \(t=16\) it grows by 14 bits and \(\beta =t'=4\). However, if \(t=16\) then \(\mathsf {ProdFL2}\) needs 6 more rounds and \(\beta (2\ell +4(t-1))\) more interactive primitives.

  5. 5.

    Differences between the measured values in Table 4 and those computed based on Table 2 are due to simplified complexity formulas, implementation tradeoffs between round optimization and modularity, and precomputation optimizations. Table 4 lists between brackets the minimum number of rounds computed using the exact formulas.

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Acknowledgements

Part of this work was supported by POC72/1/2, nr.127454, “SECREDAS Support Project”, contract 7/1.1.3H/6.01.2020, associated to the EUs Horizon 2020 ECSEL Joint Undertaking research project SECREDAS.

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Catrina, O. (2021). Complexity and Performance of Secure Floating-Point Polynomial Evaluation Protocols. In: Bertino, E., Shulman, H., Waidner, M. (eds) Computer Security – ESORICS 2021. ESORICS 2021. Lecture Notes in Computer Science(), vol 12973. Springer, Cham. https://doi.org/10.1007/978-3-030-88428-4_18

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