Negligible function

For other uses, see negligible.

In mathematics, a negligible function is a function $\mu(x):\mathbb{N}{\rightarrow}\mathbb{R}$ such that for every positive integer c there exists an integer N_c such that for all x > N_c,

$|\mu(x)|<\frac{1}{x^c}.$

Equivalently, we may also use the following definition. A function $\mu(x):\mathbb{N}{\rightarrow}\mathbb{R}$ is negligible, if for every positive polynomial poly(·) there exists an integer N_poly > 0 such that for all x > N_poly

$|\mu(x)|<\frac{1}{\text{poly}(x)}.$

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 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. Lately Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain $\mathbb{R}$):

(Continuous function) A function $f(x):\mathbb{R}{\rightarrow}\mathbb{R}$ is continuous at x = x₀ if for every ε > 0, there exists a positive number δ > 0 such that | x − x₀ | < δ implies | f(x) − f(x₀) | < ε.

This classic definition of continuity can be transformed into the definition of negligibility in a few steps by changing a parameter used in the definition per step. First, in case $x_0=\infty$ with f(x₀) = 0, we must define the concept of "infinitesimal function":

(Infinitesimal) A continuous function $\mu(x):\mathbb{R}{\rightarrow}\mathbb{R}$ is infinitesimal (as x goes to infinity) if for every ε > 0 there exists $N_{\epsilon}$ such that for all $x>N_{\epsilon}$

$|\mu(x)|<\epsilon\,.$^{[citation needed]}

Next, we replace ε > 0 by the functions 1 / x^c where c > 0 or by 1 / poly(x) where poly(x) is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants epsilon > 0 can be expressed as 1 / poly(x) 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 x = cryptographic key length n. Hence comes the definition at the top of the page because key length n must be a natural number.

Nevertheless, the general notion of negligibility has never said that the system input parameter x must be the key length n. Indeed, x can be any predetermined system metric and corresponding mathematic 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. 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, say 2⁻¹²⁸.

References

• Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3. Fragments available at the author's web site.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  Section 10.6.3: One-way functions, pp.374–376.
• Christos Papadimitriou (1993). Computational Complexity (1st edition ed.). Addison Wesley. ISBN 0-201-53082-1.  Section 12.1: One-way functions, pp.279–298.
• Jean François Colombeau (1984). New Generalized Functions and Multiplication of Distributions. Mathematics Studies 84, North Holland. ISBN 0-444-86830-5. 

See also

• Negligible set
• Colombeau algebra
• Nonstandard numbers
• Gromov's theorem on groups of polynomial growth
• Non-standard calculus