Finite fields
Generating Genus Two Hyperelliptic Curves over Large Characteristic Finite Fields
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
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For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums @f"l(v), @j"l(v), and @j"2"l(v), and the Jacobsthal-Whiteman sums @f"l^n(v) and @f"2"l^n(v), over finite fields F"q such that q=p^@a=1(mod2l). These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n=1, the exact number of F"q"^"n-rational points on the projective hyperelliptic curves aY^2Z^e^-^2=bX^e+cZ^e(abc0) (for e=l,2l), and aY^2Z^l^-^1=X(bX^l+cZ^l) (abc0), defined over such finite fields F"q. As a consequence, we obtain the exact form of the @z-functions for these three classes of curves defined over F"q, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F"q.