Negligible function

In mathematics, a negligible function is a function such that for every positive integer c there exists an integer Nc such that for all x > Nc, Equivalently, we may also use the following definition. A function is negligible, if for every positive polynomial poly(·) there exists an integer Npoly > 0 such that for all x > Npoly The concept of negligibility can find its trace back to sound models of analysis. Though the concepts of "continuity" and "infinitesimal" became important in mathematics during Newton and Leibniz's time (1680s), they were not well-defined until the late 1810s. The first reasonably rigorous definition of continuity in mathematical analysis was due to Bernard Bolzano, who wrote in 1817 the modern definition of continuity. Later Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain ): (Continuous function) A function is continuous at if for every , there exists a positive number such that implies This classic definition of continuity can be transformed into the definition of negligibility in a few steps by changing parameters used in the definition. First, in the case with , we must define the concept of "infinitesimal function": (Infinitesimal) A continuous function is infinitesimal (as goes to infinity) if for every there exists such that for all Next, we replace by the functions where or by where is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants can be expressed as with a constant polynomial this shows that negligible functions are a subset of the infinitesimal functions. In complexity-based modern cryptography, a security scheme is provably secure if the probability of security failure (e.g., inverting a one-way function, distinguishing cryptographically strong pseudorandom bits from truly random bits) is negligible in terms of the input = cryptographic key length . Hence comes the definition at the top of the page because key length must be a natural number.