Abstract
The concept of asymmetric cryptographic algorithms introduced by W. Diffie and M. Hellman in 1976 is considered. The problems of factorization of large integers and finding discrete logarithms for elements of finite large-order groups are presented. It is proved that asymmetric cryptographic algorithms based on the problem of finding a discrete logarithm for points of an elliptic curve over a finite field should be used in modern information technology.
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REFERENCES
W. Diffie and M. Hellman, “New directions in cryptography”, IEEE Trans. Inform. Theory, 22, 644-654 (1976).
T. Elgamal, “A public key cryptosystem and a signature scheme based on discrete logarithms”, IEEE Trans. Inform. Theory, 31, 469-472 (1985).
C. P. Schnorr, “Efficient identification and signatures for smart cards”, in: Advances in Cryptology, Crypto'89, Lect. Notes Comp. Sci., 435, Springer-Verlag, Berlin (1990), pp. 239-252.
K. Nyberg and R. Rueppel, “A new signature scheme based on the DSA giving message recovery”, in: Trans. 1st ACM Conf. Computer and Comm. Security, S. l, ACM Press (1993), pp. 56-61.
R. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems”, Comm. ACM, 21, 120-126 (1978).
S. Lehman, “Factoring large integers”, Math. Comp., 28, 637-646 (1974).
A.K. Lenstra and H.W. Lenstra, The Development of the Number Field Sieve, Springer-Verlag, Berlin (1993).
D.M. Gordon, “Discrete logarithms in GF (p) using the number field sieve”, SIAM J. Comput., 6, 124-138 (1993).
D. Coppersmit, “Fast evaluation of logarithms in finite fields of characteristic 2”, IEEE Trans. Inform. Theory, 30, 587-594 (1984).
R. Schoof, “Elliptic curves over finite fields and the computation of square roots mod p”, Math. Comput., 44, 483-494 (1985).
V.S. Miller, “Use of elliptic curves in cryptography”, in: Advances in Cryptology, Crypto'85; Lect. Notes Comput. Sci., Springer-Verlag, Berlin, 218 (1986), pp. 417-426.
N. Koblitz, “Miracles of the height function — a golden shield protecting ECC”, Workshop on Elliptic Curve Cryptography ECC-2000 (Essen), Essen (Germany) (2000).
J. Silverman and J. Suzuki, “Elliptic curve discrete logarithms and the index calculus”, in: Advances in Cryptology, Asiacrypt 98; Lect. Notes Comp. Sci., Springer-Verlag, Berlin, 1514 (1998), pp. 110-125.
J. Silverman, “The xedni-calculus and the elliptic curve discrete logarithm problem”, Providence (RI), (Prepr. / Dep. Math. Brown Univ.) (1998).
M. Jacobson, N. Koblitz, J. Silverman et al., “Analysis of the xedni-calculus attack”, Providence (RI), (Prepr. / Dep. Math. Brown Univ.) (1998).
A. Menezes, T. Okamoto, and S. Vanstone, “Reducing elliptic curve logarithms to a finite field”, IEEE Trans. Inform. Theory, 39, 1639-1646 (1993).
P. Gaudry, F. Hess, and N. Smart, “Constructive and destructive facets of Weil descent on elliptic curves”, Prepr. http://ultralix.polytechnique.fr (2000).
N. Koblitz, “Elliptic curve cryptosystems”, Math. Comput., 48, 203-209 (1987).
V. F. Sinyavskii, “Cryptography systems based on elliptic curves”, in: Proc. Conf. Odessa-1977, Odessa, UNIIRT (1997), pp. 37-40.
A. I. Kochubinskii, “Elliptic curves in cryptography”, Safety of Information, 2, 18-31 (2000).
A. I. Kochubinskii, “Principles of construction of the project of state standard of a digital signature”, Safety of Information, 1, 20-25 (2001).
A. I. Kochubinskii, “Perspectives of application of hyperelliptic curves in cryptography”, in: Information Safety in Information-Telecommunication Systems, 5th Intern. Sci.-Pract. Conf., Interlink, Kiev (2002).
V. Zadiraka and O. Oleksik, “Computer arithmetic of multidigit numbers”, Nauk. Vydannya, Kiev (2003).
L. A. Zavadskaya and A. M. Fal', “Cryptographically strong generators of pseudorandom sequences”, Safety of Information, 1, 7-11 (1997).
S. V. Kapustin and A. M. Fal', “Technology of using asymmetric algorithms”, Corporate Systems, 4, 62-65 (1999).
Directive 1999/03/EC of the European Parliament and of the Council on a Community framework of electronic signatures.
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Kovalenko, I.N., Kochubinskii, A.I. Asymmetric Cryptographic Algorithms. Cybernetics and Systems Analysis 39, 549–554 (2003). https://doi.org/10.1023/B:CASA.0000003504.91987.d9
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DOI: https://doi.org/10.1023/B:CASA.0000003504.91987.d9