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 d1/2, 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 N0.292, 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.


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