آشنایی با رمزنگاری خم های بیضوی

نوع مقاله : مقاله علمی - ترویجی

نویسنده

دانشگاه کاشان، دانشکده علوم ریاضی

چکیده

بخش بزرگی از رمزنگاری در سال های اخیر به رمزنگاری خم های بیضوی اختصاص یافته است. خم های بیضوی دسته ای از خم های جبری با ساختار گروه هستند. رمزنگاری خم های بیضوی یک روش رمزنگاری کلید عمومی مبتنی بر نظریۀ خم های بیضوی است که با استفاده از ویژگی های خم های بیضوی به جای روش های قبلی مانند تجزیه به حاصل ضرب اعداد اول، امنیت بالاتری را با طول کلید کوتاهتر فراهم می کند. این بخش از رمزنگاری در توافق و تبادل کلید، امضای رقمی، تجزیۀ اعداد بزرگ، آزمون اول بودن و ... کاربرد دارد. در این مقاله، رمزنگاری بر اساس خم های بیضوی را مرور و کاربردهایی از آن را تشریح می کنیم. در پایان نیز برتریِ استفاده از خم های بیضوی را به طور خلاصه بیان می کنیم.

کلیدواژه‌ها

موضوعات


[1] Brown, D. R. L., The exact security of ECDSA, preprint, 2000.
[2] Demytko, N., A new elliptic curve based analogue of RSA, In: Tor Helleseth (ed.), Advances
in Cryptology-Eurocrypt, 93, Lofthus, Norway, Springer-Verlag, 40–49, 1994.
[3] Diffie, W., Hellman, M., New directions in cryptography, IEEE Transactions on Information
Theory, 22 (1976), no.6, 644–654.
[4] ElGamal, T., A public-key cryptosystem and a signature scheme based on the discrete logarithm,
IEEE Transactions of Information Theory, 31 (1985), no. 4, 469–472.
[5] El Mahassni, E., Nguyen, P. Q., Shparlinski, I. E., The insecuriry of Nyberg-Rueppel and other
DSA-like signature schemes with partially known nonces, Workshop on Lattices and Cryptography, Boston, MA., 2001.
[6] Frey, G., Muller, M., Ruck, G. H., The Tate pairing and discrete logarithm applied to elliptic
curve cryptosystems, IEEE Trans. Inform. Theory, 45 (1998), 1717–1719.
[7] Imai, H., Zheng, Y., Efficient signcryption schemes on elliptic curves, IFIP/SEC 98, the 14th
Interantional Information Security conference, Vienna and Budapest, 1998.
[8] Joye, M., Quisquater, J. J., Takagi, T., How to choose secret parameters for RSA-type cryptosystems over elliptic curves, Technical Report TI-35/97, Technische Universitat Darmstadt,
1997.
[9] Koblitz, N., Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203–209.
[10] Koblitz, N., Hyperelliptic cryptosystems, Journal of Cryptology, 1 (1989), 139–150.
[11] Koyama, K., Kuwakado, K., Security of RSA-type cryptosystems over elliptic curves against
the Hastad attack, Electronics Letters, 30 (1994), no.22, 1834–1844.
[12] Koyama, K., Fast RSA-type schemes based on singular cubic curves y^2+axy = x ٣
(mod n),
In: Saint-Malo, France, Louis C. Guillou & Jean-Jacques Quisquater (eds.), Advances in
Cryptology-Eurocrypt 95, Springer-Verlag, 329–340, 1995.
[13] Kurosawa, K., Okada, K., Tsujii, S., Low exponent attack against elliptic curve RSA, LNCS
917, Advances in Cryptology-Asiacrypt, 94 (1995), 376–383.
[14] Law, L., Menezes, A. J., Qu, M., Solinas, J., Vanstone, S. A., An efficient protocol for authenticated key agreement, Technical Report CORR 98–05, University of Waterloo, Ontario,
Canada, March, 1998.
[15] Massey, J. L., Omura, J. K., Method and apparatus for maintaining the privacy of digital messages conveyed by public transmission, U.S. Patent, 4 (1986), 567–600.
[16] Menezes, A. J., Qu, M., Vanstone, S. A., Some new key agreement protocols providing mutual
implicit authentication, Selected Areas in Cryptology-SAC., 95, 22–32.
[17] Miller, V. S., Uses of elliptic curves in cryptography, In: Hugh C. Williams (ed.), Advances
in Cryptology-CRYPTO, 85 (218), Lecture Notes in Computer Science, Berlin, 417–426, 1986.
[18] National Institute of Standards and Technology (NIST), Secure hash standard. Federal Information Processing Standard, FIPS-180-1, 1995.
[19] National Institute of Standards and Technology (NIST), Announcing Request for Candidate
Algorithm Nominations for a New Cryptographic Hash Algorithm (SHA-3) Family. Federal
Register, 27(212): 62212–62220, 2007.
[20] Nyberg, K., Rueppel, R. A., Message recovery for signature schemes based on discrete logarithm problem, Designs, Codes and Cryptography, 7 (1996), 61–81.
[21] Rabin, M. O., Digitalized Signatures and Public Key Functions as Intractable as Factorisation,
Massachusetts Institute of Technology, 1979.
[22] Rivest, R. L., Shamir, A., Adleman, L., A Method for Obtaining Digital Signatures and PublicKey Cryptosystems, Communications of the ACM., 21 (1978), no.2, 120–126.
[23] Rubin, K., Silverberg, A., Torus-based cryptography, In: Dan Boneh (ed.), Advances in
Cryptology-CRYPTO, 3(2729) of Lecture Notes in Computer Science, Springer-Verlag, 349–
365, 2003.
[24] Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106,
Springer-Verlag, New York, 1986.
[25] Smart, N., The discrete logarithm problem on elliptic curves of trace one, HP-LABS Technical
Report (Number HPL-97-128), preprint, 1997.
[26] Waterhouse, E., Abelian varieties over finite fields, Ann. Sci., Ecole Normale Superieure, 2
(1969), 521–560.
[27] Zheng, Y., Shortened digital signature, signcryption and compact and unforgeable key agreement schemes, submitted to IEEE P1363a-Standard Specifications for Public-Key Cryptography: Additional techniques, 1998.