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Wikipedia / 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 n>Nc,

is negligible, if for every positive polynomial poly(·) there exists an integer Npoly>0 such that for all n>Npoly

History

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):

  1. (Continuous function) is continuous

, we must define the concept of "infinitesimal function":

  1. (Infinitesimal) is infinitesimal

is a positive polynomial) are a superset of negligible functions.

Use in cryptography

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 must be a natural number.

can be any predetermined discrete system metric and corresponding mathematical analysis would illustrate some hidden analytical behaviors of the system.

The reciprocal-of-polynomial formulation is used for the same reason that computational boundedness is defined as polynomial running time: it has mathematical closure properties that make it tractable in the asymptotic setting (see #Closure properties). For example, if an attack succeeds in violating a security condition only with negligible probability, and the attack is repeated a polynomial number of times, the success probability of the overall attack still remains negligible.

In practice one might want to have more concrete functions bounding the adversary's success probability and to choose the security parameter large enough that this probability is smaller than some threshold, say2128.

Closure properties

One of the reasons that negligible functions are used in foundations of complexity-theoretic cryptography is that they obey closure properties. Specifically,

  1. is negligible.

  2. is negligible.

Conversely.

Examples

:

is negligible:

  • Step.

is negligible:

  • Step, making it negligible.

is negligible:

  • Step: This function decays as the logarithm. The decay here is faster than inverse logarithmic or polynomial rates, making it negligible.

:

  • Step, making it non-negligible.

:

Negligible:

:

  • Step quickly, making it negligible.

:

  • Step.

:

  • Step, this function approaches zero faster than polynomial decay, making it negligible.

:

  • Step quickly, so it’s considered negligible.

:

  • Step, this function would approach zero very rapidly, suggesting negligibility.

Non-negligible:

:

  • Step, this function decays very slowly, failing to approach zero quickly enough to be considered negligible.

See also

References