A Result on the Distribution of Quadratic Residues with Applications to Elliptic Curve Cryptography
نویسندگان
چکیده
In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly distributed. Using this result we describe an efficient implementation of El-Gamal Cryptosystem. that requires efficient computation of a mapping between plain-texts and the points on the elliptic curve.
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Efficient elliptic curve cryptosystems
Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...
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