Topics of Mathematics in Cryptology Asymptotic Notation

And now you ask: What is the running time of the main program in terms of n? Given the above, this is difficult to answer. f(a) makes a2 multiplications. How much time does a multiplication take? That depends strongly on the multiplication algorithm, let’s call the time needed for a multiplication of length-a integers m(a). So the time spent by f(a) is at most “a2m(a) + additional stuff”. For a ≥ a0, the additional stuff takes less time than the multiplications. So for a ≥ a0, the running time of f(a) is at most 2a2m(a). The running time spent by all calls to f(a) is then at most ∑n3 a=0 time(f(a)). For a ≥ a0 we know a bound on time(f(a)). Thus ∑n3 a=0 time(f(a)) = ∑a0−1 a=0 time(f(a)) + ∑n3 a=a0 time(f(a)) ≤ ∑a0−1 a=0 time(f(a)) + ∑n3 a=a0 2a2m(a) =: A Now we can continue to compute the running time of the main program. It is A steps plus the running time of the “elementary operations”. We don’t really know how long an “elementary operation” takes, except that there is a constant bound e on that time (the precise value of e depends on the machine model). So time(main) ≤ A+ n5e. Altogether a relatively complex analysis, with various parameters (e and m) that depend on the specific machine and implementation details, and with a complex summation