Teorija brojeva u kriptografiji

Andrej Dujella:

Teorija brojeva u kriptografiji - Seminarske teme

1. Williamsove varijante Rabinovog kriptosustava (Goran Kovačević, 31.5.2006.)

H.C. Williams: A modification of the RSA public-key encryption procedure, IEEE Transaction on Information Theory 26 (1980), 726-729.
H.C. Williams: Some public-key crypto-functions as intractable as factorization, Advances in Cryptology - CRYPTO '84, Springer-Verlag, 1985, pp. 66-70.
H.C. Williams: An M³ public-key encryption scheme, Advances in Cryptology - CRYPTO '85, Springer-Verlag, 1986, pp. 358-368.
K. Kurosawa, T. Itoh, M. Takeuchi: Public key cryptosystem using a reciprocal number with the same intractability as factoring a large number, Cryptologia 12 (1988), 223-233.

2. NTRU kriptosustav

J. Hoffstein, J. Pipher, J.H. Silverman: NTRU: A ring-based public key cryptosystem, Algoritmic Number Theory - ANTS III, LNCS 1423, Springer-Verlag, 1998, pp. 267-288.
J. Hoffstein, J.H. Silverman: Optimizations for NTRU, Public Key Cryptography and Computational Number Theory, de Gruyter, 2001, pp. 77-88.
D. Coppersmith, A. Shamir: Lattice attacks on NTRU, Advances in Cryptology - EUROCRYPT '97, LNCS 1233, Springer-Verlag, 1997, pp. 52-61.
J.A. Proos: Imperfect decryption and an attack on the NTRU encryption scheme, Technical report, University of Waterloo, 2003.

3. McElieceov kriptosustav (Marcel Maretić, 14.4.2004.)

D.R. Stinson: Cryptography. Theory and Practice, CRC Press, Boca Raton, 1995, pp. 193-198.
W. Patterson: Mathematical Cryptology for Computer Scientists and Mathematicians, Rowman & Littlefield, Totowa, 1987, pp. 76-90.
J. van Tilburg: Security-analysis of a class of cryptosystems based on linear error-correcting codes, PhD dissertation, Technical University Eindhoven, 1994.
J. van Tilburg; On the McEliece public-key cryptosystem, Advances in Cryptology - CRYPTO '88, Springer-Verlag, 1990, pp. 119-131.
F. Chabaud: On the security of some cryptosystems based on error-correcting codes, Advances in Cryptology - EUROCRYPT '94, Springer-Verlag, 1995, pp. 131-139.

4. Algoritmi za brzo množenje prirodnih brojeva (Damir Horvat, 13.10.2004.)

R. Crandall, C. Pomerance: Prime Numbers. A Computational Perspective, Springer-Verlag, New York, 2001, pp. 433-467.
D. Knuth: The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Addison-Wesley, Reading, 1981, pp. 294-318.

5. Brojevni sustavi s negativnim znamenkama (Mirsad Puškar, 24.11.2004.)

I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999, pp. 67-72.
F. Morain, J. Olivos: Speeding up the computations on an elliptic curve using addition-subtraction chains, RAIRO, Inform. Theor. Appl. 24 (1990), 531-543.
M. Joye, S.-M. Yen: Optimal left-to-right binary signed-digit recoding, IEEE Transactions on Computers 49 (2000), 740-748.
C. Heuberger, H. Prodinger: On minimal expansions in redundant number systems: Algorithms and quantitative analysis, Computing 66 (2001), 377-393.

6. Distribucija kvocijenata u Euklidovom algoritmu (Ana Jurasić, 15.9.2004.)

D. Knuth: The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Addison-Wesley, Reading, 1981, pp. 356-373.
A. Ya. Khinchin: Continued Fractions, Dover, New York, 1997, pp. 71-94.
A.M. Rockett, P. Szusz: Continued Fractions, World Scientific, Singapore, 1992, pp. 151-167.

7. Duljina perioda u razvoju kvadratnog korijena u verižni razlomak (Vinko Petričević, 10.11.2004.)

A.M. Rockett, P. Szusz: Continued Fractions, World Scientific, Singapore, 1992, pp. 39-52.
J. H. E. Cohn: The length of the period of the simple continued fraction of d^1/2, Pacific J. Math. 71 (1977), 21-32.
H. C. Williams: A numerical investigation into the length of the period of the continued fraction expansion of √d, Math. Comp. 36 (1981), 593-601.
C. D. Patterson, H. C. Williams: Some periodic continued fractions with long periods, Math. Comp. 44 (1985), 523-532.
A. Dujella: Newton's formula and continued fraction expansion of √d, Experiment. Math. 10 (2001), 125-131.

8. Algoritam za prikaz prostih brojeva pomoću kvadratnih formi (Maja Andrić, 23.6.2004.)

R. Crandall, C. Pomerance: Prime Numbers. A Computational Perspective, Springer-Verlag, New York, 2001, pp. 99-101.
F. Morain, J.-L. Nicolas: On Cornacchia's algorithm for solving the diophantine equation u² + dv² = m, Courbes elliptiques et tests de primalite, Universite de Lyon I, 1990.
A. Nitaj: L'algorithme de Cornacchia, Exposition. Math. 13 (1995), 358-365.
R. Schoof: Counting points on elliptic curves over finite fields, Journal de Theorie des Nombres de Bordeaux 7 (1995), 219-254.

9. Mestreova polinomijalna metoda za konstrukciju eliptičkih krivulja velikog ranga (Mirela Jukić, 29.3.2006.)

J.-F. Mestre: Courbes elliptiques de rang ≥ 11 sur Q(t), C. R. Acad. Sci. Paris Ser. I 313 (1991), 139-142.
J.-F. Mestre: Courbes elliptiques de rang ≥ 12 sur Q(t), C. R. Acad. Sci. Paris Ser. I 313 (1991), 171-174.
L. Kulesz: Families of elliptic curves of high rank with nontrivial torsion group over Q, Acta Arith. 108 (2003), 339-356.
S. Fermigier: Exemples de courbes elliptiques de grand rang sur Q(t) et sur Q possedant des points d'ordre 2, C. R. Acad. Sci. Paris Ser. I 332 (1996), 949-952. (engleska verzija)
L. Kulesz, C. Stahlke: Elliptic curves of high rank with nontrivial torsion group over Q, Experiment. Math. 10 (2001), 475-480.

10. Lucasovi pseudoprosti brojevi (Maja Karaga, 30.6.2004.)

H. Riesel: Prime Numbers and Computer Methods for Factorization, Birkhauser, Boston, 1994. pp. 107-121.
R. Crandall, C. Pomerance: Prime Numbers. A Computational Perspective, Springer-Verlag, New York, 2001, pp. 130-140.
S.Y. Yan: Number Theory for Computing, Springer-Verlag, Berlin, 2002. pp. 215-221.

11. Atkin-Morainova metoda za dokazivanje prostosti pomoću eliptičkih krivulja (Matija Kazalicki, 2.6.2004.)

A.O.L. Atkin and F. Morain: Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-67.
F. Morain: Building cyclic elliptic curves modulo large primes, Advances in Cryptology - EUROCRYPT 91, LNCS 547, Springer-Verlag, 328-336.
I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999. pp. 149-166.
J. Buchmann, V. Muller: Primality testing, Universitat des Saarlandes, Saarbrucken, 1992.

12. Verižni razlomci u kriptoanalizi RSA kriptosustava (Mirta Mataija, 20.10.2004.)

M.J. Wiener: Cryptanalysis of short RSA secret exponents, IEEE Trans. Inform. Theory, 36 (1990), 553-558.
E.R. Verheul, H.C.A. van Tilborg: Cryptanalysis of ‘less short’ RSA secret exponents, Appl. Algebra Engrg. Comm. Computing 8 (1997), 425–435.
A. Dujella: Continued fractions and RSA with small secret exponent, Tatra Mt. Math. Publ. 29 (2004), 101-112.
B. de Weger: Cryptanalysis of RSA with small prime difference, Appl. Algebra Engrg. Comm. Computing 13 (2002), 17–28.
R.G.E. Pinch: Extending the Wiener attack to RSA-type cryptosystems, Electronics Letters 31 (1995), 1736–1738.
D. Boneh: Twenty years of attacks on the RSA cryptosystem, Notices of AMS, 46 (1999), 203-213.

13. Primjena LLL algoritma u kriptoanalizi RSA kriptosustava (Marjan Praljak, 22.9.2004.)

D. Coppersmith: Small solutions to polynomial equations, and low exponent RSA vulnerabilities, J. Cryptology, 10 (1997), 233-260.
N. Smart: Cryptography. An Introduction, McGraw-Hill, New York, 2002. pp. 300-312.
L. Lovasz: An Algorithmic Theory of Numbers, Graphs and Convexity, SIAM, Philadelphia, 1986.
D. Boneh, G. Durfee: Cryptanalysis of RSA with private key d less than N^0.292, Advances in Cryptology - Proceedings of Eurocrypt '99, Lecture Notes in Comput. Sci. 1952 (1999), 1-11.
M.J. Hinek: Low Public Exponent Partial Key and Low Private Exponent Attacks on Multi-prime RSA, Master's thesis, University of Waterloo, 2002.

14. Sito polja brojeva

A.K. Lenstra, H.W. Lenstra, Jr. (Eds.): The Development of the Number Field Sieve, Springer-Verlag, Berlin, 1993.
R. Crandall, C. Pomerance: Prime Numbers. A Computational Perspective, Springer-Verlag, New York, 2001, pp. 244-267.
H. Cohen: A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993, pp. 495-504.
K. Nakamula: A survey on the Number Field Sieve, in: Number Theory and its Applications, Kluwer, Dordrecht, 1999, pp. 263-272.

Andrej Dujella home page