Abstract
This paper presents faster inversion-free point addition formulas for the curve \(y (1+ax^2) = cx (1+dy^2)\). The proposed formulas improve the point doubling operation count record (I, M, S, D, a are arithmetic operations over a field. I: inversion, M: multiplication, S: squaring, D: multiplication by a curve constant, a: addition/subtraction) from \(6\mathbf{{M}}+ 5\mathbf{{S}}\) to \(8\mathbf{{M}}\) and mixed addition operation count record from \(10\mathbf{{M}}\) to \(8\mathbf{{M}}\). Both sets of formulas are shown to be 4-way parallel, leading to an effective cost of \(2\mathbf{{M}}\) per either of the group operations.
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Notes
Here, “small” refers to any distinguished element for which multiplications with other field elements are significantly faster than the usual multiplication of two arbitrary elements in \(\mathbb {K}\).
The singularities of E is ommitted here for simplicity. One can work with desingularization of E in \(\mathbb {P}^1\times {}\mathbb {P}^1\) to get a technically better statement.
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The authors thank the anonymous reviewers for their suggestions and comments.
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Communicated by A. Enge.
This project is funded by Yasar University Scientific Research Project SRP-024.
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Orhon, N.G., Hisil, H. Speeding up Huff form of elliptic curves. Des. Codes Cryptogr. 86, 2807–2823 (2018). https://doi.org/10.1007/s10623-018-0475-4
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DOI: https://doi.org/10.1007/s10623-018-0475-4
Keywords
- Elliptic curves
- 2-Isogeny
- Efficient
- Scalar multiplication
- Huff curves
- Inversion-free point addition
- Parallel computation