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Efficient Evaluation of Low Degree Multivariate Polynomials in Ring-LWE Homomorphic Encryption Schemes

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Book cover Information Security Practice and Experience (ISPEC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11125))

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Abstract

Homomorphic encryption schemes allow to perform computations over encrypted data. In schemes based on RLWE assumption the plaintext data is a ring polynomial. In many use cases of homomorphic encryption only the \(\text {degree-}0\) coefficient of this polynomial is used to encrypt data. In this context any computation on encrypted data can be performed. It is trickier to perform generic computations when more than one coefficient per ciphertext is used.

In this paper we introduce a method to efficiently evaluate low-degree multivariate polynomials over encrypted data. The main idea is to encode several messages in the coefficients of a plaintext space polynomial. Using ring homomorphism operations and multiplications between ciphertexts, we compute multivariate monomials up to a given degree. Afterwards, using ciphertext additions we evaluate the input multivariate polynomial. We perform extensive experimentations of the proposed evaluation method. As example, evaluating an arbitrary multivariate \(\text {degree-}3\) polynomial with 100 variables over Boolean space takes under 13 s.

This work has been supported in part by the French’s FUI project CRYPTOCOMP and by the European Union’s H2020 research and innovation programme under grant agreement No. 727528 (project KONFIDO).

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Notes

  1. 1.

    The ratio between ciphertext and plaintext message sizes.

  2. 2.

    The number of combinations with repetitions for k degree monomials is \(\frac{\left( n+k-1\right) !}{\left( n-1\right) !\cdot k!}\). Polynomial P has monomials of degree up to d (inclusive). By summing up the number of degree-k monomials one can obtain the expression \(\frac{\left( n+d\right) !}{n!\cdot d!}\).

  3. 3.

    Observe that the term \(\sum _{1\le i\le k}e_{i}\cdot n^{k-i}\) from the \(X\text {-th}\) power cancels out when \(e_{1},\ldots ,e_{k}\) are equal in (4) and (6).

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Authors and Affiliations

  1. CEA, LIST, Point Courrier 172, 91191, Gif-sur-Yvette Cedex, France

    Sergiu Carpov & Oana Stan

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  1. Sergiu Carpov
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  2. Oana Stan
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Correspondence to Sergiu Carpov .

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Editors and Affiliations

  1. University of Aizu, Aizuwakamatsu, Japan

    Chunhua Su

  2. Meiji University, Tokyo, Japan

    Hiroaki Kikuchi

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Carpov, S., Stan, O. (2018). Efficient Evaluation of Low Degree Multivariate Polynomials in Ring-LWE Homomorphic Encryption Schemes. In: Su, C., Kikuchi, H. (eds) Information Security Practice and Experience. ISPEC 2018. Lecture Notes in Computer Science(), vol 11125. Springer, Cham. https://doi.org/10.1007/978-3-319-99807-7_16

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  • DOI: https://doi.org/10.1007/978-3-319-99807-7_16

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