Andrej Dujella:

Algoritmi za eliptičke krivulje - Seminarske teme


1. Edwardsove koordinate za eliptičke krivulje (Andrea Švob, 23.9.2009.)

H. M. Edwards: A normal form for elliptic curves, Bull. Amer. Math. Soc 44 (2007), 393-422.

D. J. Bernstein, T. Lange: Edwards coordinates for elliptic curves

2. Zbrajanje i multiplikacija u Edwardsovim koordinatama; usporedba s drugim koordinatama (Loredana Simčić, 23.9.2009.)

D. J. Bernstein, T. Lange: Faster addition and doubling on elliptic curves, Advances in Cryptology: ASIACRYPT 2007, LNCS 4833, Springer, 2007, pp. 29-50.

D. J. Bernstein, T. Lange: Inverted Edwards coordinates, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-17, LNCS 4851, Springer, 2007. pp. 20-27.

D. J. Bernstein, P. Birkner, M. Joye, T. Lange, Ch. Peters: Twisted Edwards curves Progress in Cryptology - AFRICACRYPT 2008, LNCS 5023, Springer, 2008, pp. 389-405.

W. Castryck, S. Galbraith, R. Rezaeian Farashahi: Efficient arithmetic on elliptic curves using a mixed Edwards-Montgomery representation, Cryptology ePrint Archive: Report 2008/218, 2008.

3. Edwardsove krivulje u karakteristici 2 (Ana Barić, 7.7.2010.)

D. J. Bernstein, T. Lange, R. Rezaeian Farashahi: Binary Edwards curves, Cryptographic Hardware and Embedded Systems – CHES 2008, LNCS 5154 (2008), 244–265.

D. Hankerson, K. Karabina, A. Menezes: Analyzing the Galbraith-Lin-Scott Point Multiplication Method for Elliptic Curves over Binary Fields, Cryptology ePrint Archive: Report 2008/334, 2008.

4. Aritmetičko-geometrijska sredina i primjene na eliptičke krivulje (Jakiša Tomić)

J. M. Borwein, P. B. Borwein: Pi and the AGM, John Wiley, New York, 1987.

L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008, Section 9.4.

N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998, Section XIII.2.

5. Konduktor, diskriminanta i Szpirova slutnja (Sanda Bujačić, 5.5.2010.)

A. Nitaj: The ABC Conjecture Home Page

A. Nitaj: Conséquences et aspects expérimentaux des conjectures abc et de Szpiro, Thèse, Caen, 1994.

A. Nitaj: Algorithms for finding good examples for the abc and Szpiro conjectures, Experiment. Math. 3 (1993), 223-230.

A. Granville, T. J. Tucker: It's as easy as abc, Notices Amer. Math. Soc. 49 (2002), 1224-1231.

6. Karakterizacija torzijskih grupa za krivulje oblika y2 = x(x+M)(x+N) (Miljen Mikić, 16.9.2009.)

K. Ono: Euler's concordant forms, Acta Arith. 78 (1996), 101-123.

A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), 87-101.

7. Odnos ranga, konduktora i minimalne diskriminate (Dinko Cicvarić)

N. D. Elkies, M. Watkins: Elliptic curves of large rank and small conductor, Algorithmic Number Theory, Sixth International Symposium, ANTS-VI, LNCS 3076, Springer, 2004, pp. 42-56.

T. Womack: Best known conductors for elliptic curves of given rank

A. Dujella: High rank elliptic curves with prescribed torsion

8. Računanje kanonskih visina na eliptičkim krivuljama (Željko Gregorović, 9.9.2009.)

J. H. Silverman: Computing heights on elliptic curves, Math. Comp. 51 (1988), 339-358.

J. H. Silverman: Computing canonical heights with little (or no) factorization, Math. Comp. 66 (1997), 787-805.

J. E. Cremona: Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1997, Section 3.4.

9. Ocjene za razliku između naivne i kanonske visine na eliptičkoj krivulji (Aleksandar Hatzivelkos, 4.11.2009.)

J. H. Silverman: The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743.

S. Siksek: Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), 1501-1538.

J. E. Cremona, M. Prickett, S. Siksek: Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006), 42-68.

Y. Uchida: The difference between the ordinary height and the canonical height on elliptic curves, J. Number Theory 128 (2008), 263-279.

S. Schmitt, H.G. Zimmer: Elliptic Curves. A Computational Approach, de Gruyter, Berlin, 2003, Section 5.5.

10. LLL-algoritam i primjene

N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998, Chapter V.

H. W. Lenstra: Lattices, in: J.P. Buhler, P. Stevenhagen (Eds.): Algorithmic Number Theory. Lattices, Number Fields, Curves and Cryptography, Cambridge University Press, 2008, Chapter 12.

L. Lovasz: An Algorithmic Theory of Numbers, Graphs and Convexity, SIAM, Philadelphia, 1986.

A. K. Lenstra, H. W. Lenstra Jr, L. Lovasz,: Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534.

N. Smart: Cryptography. An Introduction, McGraw-Hill, New York, 2002, Chapter 17.

11. Efikasno traženje racionalnih točaka na krivuljama oblika Y2 = F(X) (Marija Maksimović, 7.7.2010.)

M. Stoll: Documentation for the RATPOINTS program

M. Stoll: On the average number of rational points on curves of genus 2

12. Lokalna rješivost kvartika

J. E. Cremona: Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1997, Section 3.6.

B. J. Birch, H. P. F. Swinnerton-Dyer: Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7-25.

S. Siksek: Descent on curves of genus one, PhD Thesis, University of Exeter, 1995.

13. Rang eliptičkih krivulja oblika x3 + y3 = k (Nikolina Kovačević)

N. F. Rogers: Elliptic Curves x3 + y3 = k with High Rank, PhD Thesis, Harvard University, 2004.

N. D. Elkies, N. F. Rogers: Elliptic curves x3 + y3 = k of high rank, Proceedings of ANTS-6 (D. Buell, ed.), LNCS 3076 (2004), 184-193.

J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Springer-Verlag, Berlin, 1992, Section V.2.

14. 3-silazak na eliptičkim krivuljama

H. Cohen, F. Pazuki: Elementary 3-descent with a 3-isogeny

H. Cohen: Number Theory. Volume I: Tools and Diophantine Equations, Springer Verlag, Berlin, 2007. Section 8.4.

E. Schaefer, M. Stoll: How to do a p-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), 1209-1231.

J.E. Cremona, T. Fisher, C. O'Neil, D. Simon, M. Stoll: Explicit n-descent on elliptic curves

S. Chang: Implementation of the 3-descent on a Mordell's elliptic curve, Technical report, University of Georgia, 2003.

15. Tate-Šafarevićeva grupa

L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008, Sections 8.7, 8.8, 8.9.

J. W. S. Cassels: Lectures on Elliptic Curves, Cambridge University Press, Cambridge, 1995, Chapter 23.

S. Schmitt, H.G. Zimmer: Elliptic Curves. A Computational Approach, de Gruyter, Berlin, 2003, Section 7.5.

J. Cremona, B. Mazur: Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), 13-28.

16. Eliptičke krivulje nad Q(i) (Filip Najman, 29.4.2009.)

P. G. Brown, T. Thongjunthug: Elliptic curves over Q(i), Austral. Math. Soc. Gaz. 35 (2008), 264–270.

T.Thongjunthug: Elliptic curves over Q(i), Honours thesis, University of New South Wales, 2006.

R. Denomme, G. Savin: Elliptic curve primality tests for Fermat and related primes, J. Number Theory 128 (2008), 2398-2412.

17. Koblitzove krivulje (Ksenija Smoljak, 1.7.2009.)

D. Hankerson, A. Menezes, S. Vanstone: Guide to Elliptic Curve Cryptography, Springer-Verlag, New York, 2004, Section 3.4

H. Cohen, G. Frey (Eds): Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, 2005, Section 15.1.

J. Solinas: Efficient arithmetic on Koblitz curves, Des. Codes Cryptogr. 19 (2000), 195-249.

18. Poboljšanja Shanks-Mestreove metode za brojenje točaka (Dario Oreščanin)

J. E. Cremona, A. V. Sutherland: On a theorem of Mestre and Schoof

R. Schoof: Counting points on elliptic curves over finite fields, J. Theor. Nombres Bordeaux 7 (1995), 219–254.

19. Schoof-Elkies-Atkinov algoritam (Anamarija Perušić, 7.7.2010.)

I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999, Chapter VII.

A. Enge: Elliptic Curves and Their Applications to Cryptography. An Intoduction, Kluwer, 1999, Chapter 5.

R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985). 483-494.

R. Schoof: Counting points on elliptic curves over finite fields, J. Theor. Nombres Bordeaux 7 (1995), 219–254.

20. Pollardova ρ-metoda za diskretni logaritam i faktorizaciju (Goran Kovačević)

L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008, Section 5.2.

N. Smart: Cryptography. An Introduction, McGraw-Hill, 2002, Chapter 12.4.

H. Cohen: A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993, Section 8.5.

21. XEDNI Calculus

J. H. Silverman: The xedni calculus and the elliptic curve discrete logarithm problem, Des. Codes Cryptogr. 20 (2000), 5-40.

H. J. Kim, J. H. Cheon, S. G. Hahn: On remarks of lifting problems for elliptic curves, Adv. Stud. Contemp. Math. (Pusan) 2 (2000), 21-36.

M. J. Jacobson, N. Koblitz, J. H. Silverman, A. Stein, E. Teske, Analysis of the xedni calculus attack, Des. Codes Cryptogr. 20 (2000), 41-64.

22. MOV napad

L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008, Section 5.3.

I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999, Section V.2.

J. H. Silverman: The Arithmetic of Elliptic Curves, Springer-Verlag, 2009, Section XI.6.

G. Frey, H.-G. Rüuck: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp. 62 (1994), 865-874.

23. Problem diskretnog logaritma za anomalne eliptičke krivulje

I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999, Section V.3.

J. H. Silverman: The Arithmetic of Elliptic Curves, Springer-Verlag, 2009, Section XI.6.

I. A. Semaev: Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p, Math. Comp. 67 (1998), 353-356.

24. Primjena eliptičkih krivulja s velikom torzijskom grupom u faktorizaciji (Ana Laštre, 16.2.2011.)

P. L. Montgomery: Speeding the Pollard and elliptic curve methods of factorization, Math. Comp. 48 (1987), 243-264.

A. O. L. Atkin, F. Morain: Finding suitable curves for the elliptic curve method of factorization, Math. Comp. 60 (1993), 399-405.

A. O. L. Atkin, F. Morain: Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-68.

25. Eliptičke krivulje nad prstenom (Konrad Burnik, 12.5.2010.)

L. C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2008, Section 2.11.

H. W. Lenstra: Elliptic curves and number-theoretic algorithms, In: Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), pages 99–120, Amer. Math. Soc., Providence, 1987.

K. Koyama, U. M. Maurer, T. Okamoto, S. A. Vanstone: New public-key schemes based on elliptic curves over the ring Zn, Advances in Cryptology - Crypto '91, LNCS 576, Springer, 1991, pp. 252-266.

Andrej Dujella home page