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Elliptic curves over finite fields and the computation of square roots mod 𝑝
Open accessAbstract
In this paper we present a deterministic algorithm to compute the number of F q -points of an elliptic curve that is defined over a finite field F q and which is given by a Weierstrass equation. The algorithm takes O ( log 9 q ) elementary operations. As an application we give an algorithm to compute square roots mod p . For fixed x ∈ Z , it takes O ( log 9 p ) elementary operations to compute x mod p .
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Citations
Cited by
Koblitz, N. (1987) · Mathematics of Computation · doi:10.1090/s0025-5718-1987-0866109-5 · Cited by 5,634
The Arithmetic of Elliptic Curves
Silverman, J. H. (1986) · Graduate Texts in Mathematics · doi:10.1007/978-1-4757-1920-8 · Cited by 4,044
The State of Elliptic Curve Cryptography
Koblitz, N., Menezes, A., & Vanstone, S. (2000) · Designs, Codes and Cryptography · doi:10.1023/a:1008354106356 · Cited by 685
The Weil Pairing, and Its Efficient Calculation
Miller, V. S. (2004) · Journal of Cryptology · doi:10.1007/s00145-004-0315-8 · Cited by 578
Quantum algorithms for algebraic problems
Childs, A. M., & van Dam, W. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1 · Cited by 344
Metadata
- Authors
- Schoof, R.
- Venue
- Mathematics of Computation
- Publisher
- American Mathematical Society (AMS)
- Published
- 1985
- Volume
- 44
- Issue
- 170
- Pages
- 483-494
- ISSN
- 0025-5718 (ISSN-L)
- 1088-6842
- OpenAlex
- W2075252632
- Cited by
- 515
- References
- 0
- FWCI
- 34.39
- Type
- journal article
- Language
- en
- Open access
- bronze
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- CoChat. (2026). Citation check: Elliptic curves over finite fields and the computation of square roots mod 𝑝. https://app.cochat.ai/papers/10.1090/s0025-5718-1985-0777280-6
- The paper
- Schoof, R. (1985). Elliptic curves over finite fields and the computation of square roots mod 𝑝. Mathematics of Computation. https://doi.org/10.1090/s0025-5718-1985-0777280-6
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Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
