Elliptic curve cryptosystems
Open accessAbstract
We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF ( 2 n ) . We discuss the question of primitive points on an elliptic curve modulo p , and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.
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The security of SAE lies in the intractable nature of the dot product operation in discrete logarithmic computation; knowing EA and PE makes it computationally intractable to find mA.
“The security of this protocol lies in the intractable nature of the dot product operation in discrete logarithmic computation.By knowing E A and PE, it is computationally intractable to find m A [33].”
From A Comprehensive Attack Flow Model and Security Analysis for Wi-Fi and WPA3 (2018) · cites this paper as [33]
The cryptosystems passage (indexed as Abstract; page unavailable) defines the elliptic-curve discrete logarithm problem as finding x from P and Q = xP, and says breaking the systems discussed requires solving it. This supports the claimed difficulty of recovering a secret scalar from public points. The paper does not address SAE or its variables EA, PE and mA, and describes scalar multiplication rather than a dot product. (LLM, conf=0.86)
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“By knowing E A and PE , it is computationally intractable to find m A [33].”
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- Authors
- Koblitz, N.
- Corresponding author
- Koblitz, Neal
- ORCID
- Koblitz, Neal
- Venue
- Mathematics of Computation
- Publisher
- American Mathematical Society (AMS)
- Published
- 1987
- Volume
- 48
- Issue
- 177
- Pages
- 203-209
- ISSN
- 0025-5718 (ISSN-L)
- 1088-6842
- OpenAlex
- W2036378739
- Cited by
- 5,634
- References
- 18
- Citations by year
- 2026: 165 · 2025: 242 · 2024: 245 · 2023: 222 · 2022: 228 · 2021: 277 · 2020: 291 · 2019: 244 · 2018: 242 · 2017: 263 · 2016: 239 · 2015: 234 · 2014: 260 · 2013: 256 · 2012: 258
- FWCI
- 14.25
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Mathematics of Computation (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Information Systems › Cryptography and Residue Arithmetic
- Physical Sciences › Computer Science › Artificial Intelligence › Cryptography and Data Security
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
- Keywords
- elliptic curve cryptography, public-key cryptography, elliptic curves over finite fields, discrete logarithm problem
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