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Constructing elliptic curves with a known number of points over a prime field
Number Theory · Submitted 13 Nov 2001
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods.
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The abstract, sentence by sentence
Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H(X) for a fundamental discriminant D. The usual way is to compute H(X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo n directly from the knowledge of H(X) modulo enough small primes. Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. | background 1.00 | 0.03 | 0.03 | asserted 0.95 |
| s1 | In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. | background 1.00 | 0.03 | 0.03 | hedged possibility 0.50 |
| s2 | One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H(X) for a fundamental discriminant D. | background 0.77 | 0.06 | 0.04 | asserted 1.00 |
| s3 | The usual way is to compute H(X) over the integers and then to find the root modulo n. | background 0.95 | 0.05 | 0.03 | asserted 1.00 |
| s4 | We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo n directly from the knowledge of H(X) modulo enough small primes. | contribution 0.60 | 0.93 | 0.46 | asserted 1.00 |
| s5 | Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods. | result 0.71 | 0.89 | 0.58 | hedged possibility 0.99 |
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The paper
Original abstract
Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H(X) for a fundamental discriminant D. The usual way is to compute H(X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo n directly from the knowledge of H(X) modulo enough small primes. Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods.
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