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In the paper we obtain some new upper bounds for exponential sums overmultiplicative subgroups G of F^*_p having sizes in the range [p^{c_1},p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As anapplication we prove that in symmetric case G is always an additive basis oforder five, provided by |G| > p^{1/2} log^{1/3} p. Also the method allows us togive a new upper bound for Heilbronn's exponential sum.

On exponential sums over multiplicative subgroups of medium size