Elliptic curve cryptosystems
1987/01/01 by Neal Koblitz · 5,024 citations
Computer Science · Mathematics · #Algorithm #Coding theory and cryptography #Computer science #Cryptography and Data Security #Cryptography and Residue Arithmetic #Discrete logarithm #Discrete mathematics #Elliptic curve #Hessian form of an elliptic curve #Mathematics #Public-key cryptography #Pure mathematics #Schoof's algorithm
paper · pdf · doi:10.1090/s0025-5718-1987-0866109-5
published in Mathematics of Computation 48(177), 203-209 (American Mathematical Society)
openalex publication_date 1987/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="GF left-parenthesis 2 Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>GF</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\text GF(2n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We discuss the question of primitive points on an elliptic curve modulo <italic>p</italic>, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.
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