On the statistical properties of Diffie-Hellman distributions

Some doubly exponential sums over Zm

“…, t, provided that t ≥ m 10/11+ε . As in [2,3] we actually study the slightly simpler sums Note that in the above sums (and in several more yet to come) we have suppressed in the notation the dependence on ϑ. This does not mean we are claiming that such sums are the same for all ϑ having the same order t. However in all of our results the bounds obtained are uniform for all such ϑ.…”

Incomplete exponential sums and Diffie–Hellman triples

“…This bound is a generalization of Theorem 8 of [4] which dealt with the case K = t, α = 0. The method of [4] would extend to give this result for arbitrary α but not for general K < t.…”

“…Such a method can be quickly applied in our case, and it leads to a bound which is fairly good when the range for x and y is almost complete. In the more interesting case when the sums are shorter we are able to improve on this result by combining elements in the proofs of [4] and [5] and adding new ingredients, rather than directly applying the statement for the complete sum. In particular, we obtain a new upper bound on the number of solutions of n-term exponential equations which we hope may find several other applications.…”

“…In this section we follow to some extent the ideas in Section 3 of [4] and Section 3 of [5]. However, we consider the more general case of incomplete sums as opposed to summing over the full period t, and for this some new ideas lead to stronger results.…”

“…Let p be a prime and ϑ an integer of order t | p − 1 in the multiplicative group modulo p, that is, ϑ t ≡ 1 (mod p) but ϑ j ≡ 1 (mod p) for 1 ≤ j < t. In this paper we continue the study of the distribution of Diffie-Hellman triples (ϑ x , ϑ y , ϑ xy ) as initiated in [5]; see also [4,8,10]. Here we are interested in the case when the exponents x and y belong to an aligned box inside the square [1, t] 2 , thus we are led to the problem of estimating exponential sums with a linear combination of the entries ϑ x , ϑ y , ϑ xy in such triples.…”