Methods and apparatuses are disclosed for securing cryptosystems against external monitoring attacks by reducing the amount (and signal to noise ratio) of useful information leaked during processing. In general, this is accomplished by implementing critical operations using "branchless" or fixed execution path routines (115, 125) whereby the execution path does not vary in any manner that can reveal new information about the secret key during subsequent operations. More particularly, various embodiments of the invention include: implementing modular exponentiation without key-dependent conditional jumps (125); implementing modular exponentiation with fixed memory access patterns (120); implementing modular multiplication without using leak-prone multiplication-by-one operations (115); and implementing leak-minimization multiplication (and other operations) for elliptic curve cryptosystems (130).