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Volumn 47, Issue 4, 2012, Pages 368-400

Genus 2 point counting over prime fields

Author keywords

Hyperelliptic curves; Point counting; Schoof Pila algorithm

Indexed keywords


EID: 84857368161     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2011.09.003     Document Type: Article
Times cited : (61)

References (47)
  • 1
    • 0035204351 scopus 로고    scopus 로고
    • Counting points on curves and abelian varieties over finite fields
    • Adleman L., Huang M.-D. Counting points on curves and abelian varieties over finite fields. Journal of Symbolic Computation 2001, 32:171-189.
    • (2001) Journal of Symbolic Computation , vol.32 , pp. 171-189
    • Adleman, L.1    Huang, M.-D.2
  • 3
    • 33745965912 scopus 로고    scopus 로고
    • Curve25519: new Diffie-Hellman speed records
    • Springer-Verlag, M. Yung, Y. Dodis, A. Kiayias, T. Malkin (Eds.) Public Key Cryptography - PKC 2006
    • Bernstein D.J. Curve25519: new Diffie-Hellman speed records. Lecture Notes in Comput. Sci. 2006, Vol. 3958:207-228. Springer-Verlag. M. Yung, Y. Dodis, A. Kiayias, T. Malkin (Eds.).
    • (2006) Lecture Notes in Comput. Sci. , vol.3958 , pp. 207-228
    • Bernstein, D.J.1
  • 4
    • 84857364613 scopus 로고    scopus 로고
    • Elliptic vs. hyperelliptic, part 1. Talk given at ECC 2006. Slides available at:
    • Bernstein, D.J., 2006b. Elliptic vs. hyperelliptic, part 1. Talk given at ECC 2006. Slides available at: http://cr.yp.to/talks.html#2006.09.20.
    • (2006)
    • Bernstein, D.J.1
  • 6
    • 21844434668 scopus 로고    scopus 로고
    • Polynomial evaluation and interpolation on special sets of points
    • Bostan A., Schost É Polynomial evaluation and interpolation on special sets of points. Journal of Complexity 2005, 21(4):420-446.
    • (2005) Journal of Complexity , vol.21 , Issue.4 , pp. 420-446
    • Bostan, A.1    Schost É2
  • 9
    • 33748971527 scopus 로고    scopus 로고
    • Change of order for bivariate triangular sets
    • ACM
    • Pascal C., Schost E. Change of order for bivariate triangular sets. ISSAC'06 2006, 277-284. ACM.
    • (2006) ISSAC'06 , pp. 277-284
    • Pascal, C.1    Schost, E.2
  • 10
    • 84941611264 scopus 로고
    • On the analogue of the division polynomials for hyperelliptic curves
    • Cantor D.G. On the analogue of the division polynomials for hyperelliptic curves. Journal für die Reine und Angewandte Mathematik 1994, 447:91-145.
    • (1994) Journal für die Reine und Angewandte Mathematik , vol.447 , pp. 91-145
    • Cantor, D.G.1
  • 11
    • 57949101279 scopus 로고    scopus 로고
    • Compter (rapidement) le nombre de solutions d'équations dans les corps finis
    • Astérisque 317, 39-90, séminaire Bourbaki 968, novembre 2006.
    • Chambert-Loir, A., 2008. Compter (rapidement) le nombre de solutions d'équations dans les corps finis. Astérisque 317, 39-90, séminaire Bourbaki 968, novembre 2006.
    • (2008)
    • Chambert-Loir, A.1
  • 13
    • 84857372712 scopus 로고    scopus 로고
    • Fast construction of irreducible polynomials over finite fields
    • Eprint (submitted for publication).
    • Couveignes, J.-M., Lercier, R., 2009. Fast construction of irreducible polynomials over finite fields. Eprint (submitted for publication). arxiv:0905.1642v2.
    • (2009)
    • Couveignes, J.-M.1    Lercier, R.2
  • 14
    • 27644512413 scopus 로고    scopus 로고
    • Sharp estimates for triangular sets
    • ACM
    • Dahan X., Schost É Sharp estimates for triangular sets. ISSAC'04 2004, 103-110. ACM.
    • (2004) ISSAC'04 , pp. 103-110
    • Dahan, X.1    Schost É2
  • 15
    • 84857360459 scopus 로고    scopus 로고
    • Computing modular correspondences for abelian varieties
    • preprint. URL:
    • Faugère, J.-C., Lubicz, D., Robert, D., 2009. Computing modular correspondences for abelian varieties, preprint. URL: http://hal.archives-ouvertes.fr/hal-00426338/en/.
    • (2009)
    • Faugère, J.-C.1    Lubicz, D.2    Robert, D.3
  • 16
    • 84857363516 scopus 로고    scopus 로고
    • Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic
    • Manuscript
    • Feo, L.D., 2010. Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic. Manuscript, arxiv:1002.2597.
    • (2010)
    • Feo, L.D.1
  • 18
    • 0001605828 scopus 로고
    • Computing Frobenius maps and factoring polynomials
    • Gathen J., Shoup V. Computing Frobenius maps and factoring polynomials. Computational Complexity 1992, 2(3):187-224.
    • (1992) Computational Complexity , vol.2 , Issue.3 , pp. 187-224
    • Gathen, J.1    Shoup, V.2
  • 19
    • 84857362738 scopus 로고    scopus 로고
    • Algorithmes de comptage de points d'une courbe définie sur un corps fini
    • In: Méthodes explicites en théorie des nombres, Astérisque. (in press).
    • Gaudry, P., 2006. Algorithmes de comptage de points d'une courbe définie sur un corps fini. In: Méthodes explicites en théorie des nombres, Astérisque. (in press). http://www.loria.fr/~gaudry/publis/pano.pdf.
    • (2006)
    • Gaudry, P.1
  • 20
    • 49949100083 scopus 로고    scopus 로고
    • Fast genus 2 arithmetic based on Theta functions
    • Gaudry P. Fast genus 2 arithmetic based on Theta functions. Journal of Mathematical Cryptology 2007, 1:243-265.
    • (2007) Journal of Mathematical Cryptology , vol.1 , pp. 243-265
    • Gaudry, P.1
  • 21
    • 84946825131 scopus 로고    scopus 로고
    • Counting points on hyperelliptic curves over finite fields
    • Springer-Verlag, B. W. (Ed.) ANTS-IV
    • Gaudry P., Harley R. Counting points on hyperelliptic curves over finite fields. Lecture Notes in Comput. Sci. 2000, Vol. 1838:313-332. Springer-Verlag. B. W. (Ed.).
    • (2000) Lecture Notes in Comput. Sci. , vol.1838 , pp. 313-332
    • Gaudry, P.1    Harley, R.2
  • 22
    • 35048820123 scopus 로고    scopus 로고
    • Construction of secure random curves of genus 2 over prime fields
    • Springer, Advances in Cryptology, Eurocrypt'04
    • Gaudry P., Schost É Construction of secure random curves of genus 2 over prime fields. Lecture Notes in Computer Science. 2004, Vol. 3027:239-256. Springer.
    • (2004) Lecture Notes in Computer Science. , vol.3027 , pp. 239-256
    • Gaudry, P.1    Schost É2
  • 23
    • 27144477571 scopus 로고    scopus 로고
    • A low-memory parallel version of Matsuo, Chao and Tsujii's algorithm
    • Springer, Algorithmic Number Theory, ANTS 6
    • Gaudry P., Schost É A low-memory parallel version of Matsuo, Chao and Tsujii's algorithm. Lecture Notes in Computer Science. 2004, 3076:208-222. Springer.
    • (2004) Lecture Notes in Computer Science. , vol.3076 , pp. 208-222
    • Gaudry, P.1    Schost É2
  • 24
    • 11244288438 scopus 로고    scopus 로고
    • Modular equations for hyperelliptic curves
    • Gaudry P., Schost É Modular equations for hyperelliptic curves. Mathematics of Computation 2005, 74:429-454.
    • (2005) Mathematics of Computation , vol.74 , pp. 429-454
    • Gaudry, P.1    Schost É2
  • 26
    • 67349272736 scopus 로고    scopus 로고
    • Kedlaya's algorithm in larger characteristic
    • Article ID rnm095, 29 pages
    • Harvey D. Kedlaya's algorithm in larger characteristic. International Mathematics Research Notices 2007, 2007. Article ID rnm095, 29 pages.
    • (2007) International Mathematics Research Notices , vol.2007
    • Harvey, D.1
  • 28
    • 0030674617 scopus 로고    scopus 로고
    • Fast polynomial factorization over high algebraic extensions of finite fields
    • ACM
    • Kaltofen E., Shoup V. Fast polynomial factorization over high algebraic extensions of finite fields. ISSAC'97 1997, 184-188. ACM.
    • (1997) ISSAC'97 , pp. 184-188
    • Kaltofen, E.1    Shoup, V.2
  • 29
    • 0013266180 scopus 로고    scopus 로고
    • Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology
    • Kedlaya K.S. Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. Journal of the Ramanujan Mathematical Society 2001, 16(4):323-338.
    • (2001) Journal of the Ramanujan Mathematical Society , vol.16 , Issue.4 , pp. 323-338
    • Kedlaya, K.S.1
  • 30
    • 57949111528 scopus 로고    scopus 로고
    • Fast modular composition in any characteristic
    • Kedlaya K.S., Umans C. Fast modular composition in any characteristic. FOCS. IEEE 2008, 146-155.
    • (2008) FOCS. IEEE , pp. 146-155
    • Kedlaya, K.S.1    Umans, C.2
  • 32
    • 2442696613 scopus 로고    scopus 로고
    • Deformation theory and the computation of zeta functions
    • Lauder A. Deformation theory and the computation of zeta functions. Proceedings of the London Mathematical Society 2004, 88:565-602.
    • (2004) Proceedings of the London Mathematical Society , vol.88 , pp. 565-602
    • Lauder, A.1
  • 34
    • 0003521788 scopus 로고    scopus 로고
    • Algorithmique des courbes elliptiques dans les corps finis
    • Thèse, École polytechnique.
    • Lercier, R., 1997. Algorithmique des courbes elliptiques dans les corps finis. Thèse, École polytechnique.
    • (1997)
    • Lercier, R.1
  • 35
    • 84857360458 scopus 로고    scopus 로고
    • Computing isogenies between Abelian varieties, preprint, 48 pages. URL:
    • Lubicz, D., Robert, D., 2010. Computing isogenies between Abelian varieties, preprint, 48 pages. URL: http://hal.archives-ouvertes.fr/hal-00446062/en/.
    • (2010)
    • Lubicz, D.1    Robert, D.2
  • 36
    • 84958532498 scopus 로고    scopus 로고
    • An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields
    • Springer-Verlag, C. Fiecker, D. Kohel (Eds.) ANTS-V
    • Matsuo K., Chao J., Tsujii S. An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields. Lecture Notes in Comput. Sci. 2002, Vol. 2369:461-474. Springer-Verlag. C. Fiecker, D. Kohel (Eds.).
    • (2002) Lecture Notes in Comput. Sci. , vol.2369 , pp. 461-474
    • Matsuo, K.1    Chao, J.2    Tsujii, S.3
  • 37
    • 84966216845 scopus 로고
    • Frobenius maps of abelian varieties and finding roots of unity in finite fields
    • Pila J. Frobenius maps of abelian varieties and finding roots of unity in finite fields. Mathematics of Computation 1990, 55(192):745-763.
    • (1990) Mathematics of Computation , vol.55 , Issue.192 , pp. 745-763
    • Pila, J.1
  • 38
    • 84857362722 scopus 로고    scopus 로고
    • Efficient point-counting on genus-2 hyperelliptic curves
    • Ph.D. thesis, University of Illinois at Chicago.
    • Pitcher, N., 2009. Efficient point-counting on genus-2 hyperelliptic curves. Ph.D. thesis, University of Illinois at Chicago.
    • (2009)
    • Pitcher, N.1
  • 39
    • 0030692210 scopus 로고    scopus 로고
    • Asymptotically fast computation of subresultants
    • ACM
    • Reischert D. Asymptotically fast computation of subresultants. ISSAC'97 1997, 233-240. ACM.
    • (1997) ISSAC'97 , pp. 233-240
    • Reischert, D.1
  • 40
    • 0033131339 scopus 로고    scopus 로고
    • Solving zero-dimensional systems through the Rational Univariate Representation
    • Rouillier F. Solving zero-dimensional systems through the Rational Univariate Representation. Applicable Algebra in Engineering, Communication and Computing 1999, 9(5):433-461.
    • (1999) Applicable Algebra in Engineering, Communication and Computing , vol.9 , Issue.5 , pp. 433-461
    • Rouillier, F.1
  • 41
    • 0001100056 scopus 로고    scopus 로고
    • The canonical lift of an ordinary elliptic curve over a finite field and its point counting
    • Satoh T. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. Journal of the Ramanujan Mathematical Society 2000, 15:247-270.
    • (2000) Journal of the Ramanujan Mathematical Society , vol.15 , pp. 247-270
    • Satoh, T.1
  • 42
    • 38149147073 scopus 로고
    • Fast construction of irreducible polynomials over finite fields
    • Shoup V. Fast construction of irreducible polynomials over finite fields. Journal of Symbolic Computation 1994, 17(5):371-391.
    • (1994) Journal of Symbolic Computation , vol.17 , Issue.5 , pp. 371-391
    • Shoup, V.1
  • 43
    • 0001465210 scopus 로고
    • A new polynomial factorization algorithm and its implementation
    • Shoup V. A new polynomial factorization algorithm and its implementation. Journal of Symbolic Computation 1995, 20(4):363-397.
    • (1995) Journal of Symbolic Computation , vol.20 , Issue.4 , pp. 363-397
    • Shoup, V.1
  • 44
    • 84857362720 scopus 로고    scopus 로고
    • 1996-2010. NTL: a library for doing number theory.
    • Shoup, V., 1996-2010. NTL: a library for doing number theory. http://www.shoup.net.
    • Shoup, V.1
  • 45
    • 84855212790 scopus 로고    scopus 로고
    • Complex multiplication of abelian surfaces
    • Ph.D. thesis, Universiteit Leiden.
    • Streng, M., 2010. Complex multiplication of abelian surfaces. Ph.D. thesis, Universiteit Leiden.
    • (2010)
    • Streng, M.1
  • 46
    • 58149236779 scopus 로고    scopus 로고
    • A generic approach to searching for Jacobians
    • Sutherland A.V. A generic approach to searching for Jacobians. Mathematics of Computation 2009, 78:485-507.
    • (2009) Mathematics of Computation , vol.78 , pp. 485-507
    • Sutherland, A.V.1
  • 47
    • 77954216433 scopus 로고
    • Algebraic factoring and rational function integration
    • ACM
    • Trager B.M. Algebraic factoring and rational function integration. SYMSAC 76 1976, 219-226. ACM.
    • (1976) SYMSAC 76 , pp. 219-226
    • Trager, B.M.1


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