## Efficient Computation of Singular Moduli with Application in Cryptography (2001)

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Venue: | In Fundamentals of Computing Theory, Proceedings of FCT 2001, LNCS 2138 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Baier01efficientcomputation,

author = {Harald Baier},

title = {Efficient Computation of Singular Moduli with Application in Cryptography},

booktitle = {In Fundamentals of Computing Theory, Proceedings of FCT 2001, LNCS 2138},

year = {2001},

pages = {71--82},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. We present an implementation that turns out to be most efficient in practice to compute singular moduli within a fixed floating point precision. First, we show how to efficiently determine the Fourier coefficients of the modular function j and related functions γ2, f2, and η. Comparing several alternative methods for computing singular moduli, we show that in practice the computation via the η-function turns out to be the most efficient one. An important application with respect to cryptography is that we can speed up the generation of cryptographically strong elliptic curves using the Complex Multiplication Approach.

### Citations

256 |
Introduction to Elliptic Curves and Modular Forms
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- 1984
(Show Context)
Citation Context ...ng the representation of j by the normalized Eisenstein series E4 and E6 of weight 4 and 6, respectively: j(τ) = 1728 · E4(τ) 3 E4(τ) 3−E6(τ) 2 . For a definition of the Eisenstein series we refer to =-=[Kob93]-=-. The Fourier coefficients of the Eisenstein series can easily be computed. We refer to [Bai01] for details. Finally making use of Mahler’s equations (3.5) - (3.8) we compute the values of cn up to n ... |

162 | Elliptic curves and primality proving
- Atkin, Morain
(Show Context)
Citation Context ...ct funded by the German Department of Trade and Industry R. Freivalds (Ed.): FCT 2001, LNCS 2138, pp. 71–82, 2001. c○ Springer-VerlagBerlin Heidelberg2001s72 H. Baier via Complex Multiplication (CM) (=-=[AM93]-=-, [LZ94], [BB00]). The general CMmethod requires the computation of a ring class polynomial R, which is in general the most time consuming step of the CM-method. Hence speeding up the computation of R... |

114 |
Primes of the form x 2 + ny 2
- Cox
- 1989
(Show Context)
Citation Context ...r Moduli with Application in Cryptography 73 γ2(τ) = f2(τ) 24 +16 f2(τ) 8 , (2.3) j(τ) =γ2(τ) 3 . (2.4) We fix the discriminant ∆ and set K =Q( √ ∆). The First Main Theorem of Complex Multiplication (=-=[Cox89]-=-, Theorem 11.1) states that all values j(τQ) are conjugated over K and hence have the same minimal polynomial over K : The ring class polynomial R. Thus we can write R = � Q∈C(∆) (X − j(τQ)). It turns... |

45 |
Constructing elliptic curves with given group order over large finite fields. Algorithmic Number Theory
- Lay, Zimmer
- 1994
(Show Context)
Citation Context ...d by the German Department of Trade and Industry R. Freivalds (Ed.): FCT 2001, LNCS 2138, pp. 71–82, 2001. c○ Springer-VerlagBerlin Heidelberg2001s72 H. Baier via Complex Multiplication (CM) ([AM93], =-=[LZ94]-=-, [BB00]). The general CMmethod requires the computation of a ring class polynomial R, which is in general the most time consuming step of the CM-method. Hence speeding up the computation of R yields ... |

22 |
A C++ Library for Computational Number Theory
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(Show Context)
Citation Context ... at 333 MHz and having 512 MByte main memory. Second a Pentium III running Linux 2.2.14 at 850 MHz and having 128 MByte main memory. All algorithms are implemented in C++ using the library LiDIA 2.0 (=-=[LiDIA]-=-) with libI as underlying multiprecision package and the GNU compiler 2.95.2 using the optimization flag O2. Sample tests indicate that running times on the Pentium are about a quarter of the timings ... |

12 | Efficient construction of cryptographically strong elliptic curves
- Baier, Buchmann
(Show Context)
Citation Context ... German Department of Trade and Industry R. Freivalds (Ed.): FCT 2001, LNCS 2138, pp. 71–82, 2001. c○ Springer-VerlagBerlin Heidelberg2001s72 H. Baier via Complex Multiplication (CM) ([AM93], [LZ94], =-=[BB00]-=-). The general CMmethod requires the computation of a ring class polynomial R, which is in general the most time consuming step of the CM-method. Hence speeding up the computation of R yields a signif... |

9 |
On a class of non-linear functional equations connected with modular functions
- Mahler
(Show Context)
Citation Context ...r computing j(τ) we first have to determine the Fourier series of these four functions. For the determination of the Fourier coefficients of j and γ2 we make use of efficient algorithms due to Mahler =-=[Mah76]-=-. Furthermore, in the case of f2 and η we develop efficient formulae in Sect. 3. Efficient computation of j(τ), γ2(τ), and f2(τ) has an important application in cryptography: The generation of cryptog... |

3 |
Traces of Singular Moduli and the Fourier Coefficients of the Elliptic Modular Function j(τ). Volume 19 of Number Theory. Fifth Conf. Canad. Number Theory Assoc
- Kaneko
- 1996
(Show Context)
Citation Context ...nomials R, and we are not aware of any counterexample where F does not yield a right result. We remarkthat there is another efficient method to compute the Fourier coefficients of j due to M. Kaneko (=-=[Kan]-=-), who extends work of D. Zagier. We refer to his paper for details. 3.2 Computing the Fourier Coefficients of γ2 1 − For τ ∈ h one can easily derive the formula γ2(τ) =q 3 · �∞ n=0 gnqn . As in the c... |

1 |
Efficient Computation of Fourier Series and Singular Moduli with Application in Cryptography
- Baier
- 2001
(Show Context)
Citation Context ...pectively: j(τ) = 1728 · E4(τ) 3 E4(τ) 3−E6(τ) 2 . For a definition of the Eisenstein series we refer to [Kob93]. The Fourier coefficients of the Eisenstein series can easily be computed. We refer to =-=[Bai01]-=- for details. Finally making use of Mahler’s equations (3.5) - (3.8) we compute the values of cn up to n = 50000 and store them in a file of size 38.7 MByte. The running time on the Pentium III was 9.... |