# Article

Full entry | PDF   (0.1 MB)
Keywords:
discrete logarithm; finite fields; cryptography
Summary:
For $a$ generator of the multiplicative group of the field $GF(p,k)$, the discrete logarithm of an element $b$ of the field to the base $a$, $b\ne 0$ is that integer $z:1\le z \le p^k -1$, $b=a^z$. The $p$-ary digits which represent $z$ can be described with extremely simple polynomial forms.
References:
[1] Adleman, L. M.: A subexponential algorithm for the discrete logarithm problem, with applications to cryptography. Proc. 20th IEEE Found. Comp. Sci. Symp. (1979), 55-60.
[2] Diffie, W., Hellman, M. E.: New directions in cryptography. IEEE Trans. Inform. Theory, IT-22 (1976), 644-654. MR 0437208
[3] Odlyzko, A. M.: Discrete logarithms in finite fields and their cryptographic significance. Proc. of the Eurocrypt ’84. Zbl 0594.94016
[4] Pohling, S. C., Hellman, M. E.: An improved algorithm for computing logarithms over $GF(p)$ and its cryptographic significance. IEEE Trans. Inform. Theory, IT-24 (1978), 106-110. MR 0484737
[5] Pollard, S. M.: The fast Fourier transform in a finite field. Mathematics of computation 25 (1971), 365-374. MR 0301966 | Zbl 0221.12015
[6] Wells, A. L.: A polynomial form for logarithms modulo a prime. IEEE Trans.Inform. Theory, IT-30 (1984), 845-846. Zbl 0558.12009
[7] Knuth, D. E.: The art of computer programming. Reading MA III (1969), Addison Wesley. MR 0378456 | Zbl 0191.18001

Partner of