How Long Does it Take to Catch a Wild Kangaroo?

The discrete logarithm problem asks to solve for the exponent x given the generator g of a cyclic group G lid all element h is an element of G such that g(x) = h. We give the first rigorous proof that, Pollard's Kangaroo method finds the discrete logarithm in expected time (3+o(1))root b-a for the worst value of x is an element of [a,b] and (2 + o(1))root b-a, when x is an element of(uar) [a,b]. This matches the, conjectured time complexity and, rare among the analysis of a algorithms based oil Markov chains, even the lead constants 2 and 3 are correct.