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Concept
L-notation
Summary
L-notation is an asymptotic notation analogous to big-O notation, denoted as for a bound variable tending to infinity. Like big-O notation, it is usually used to roughly convey the rate of growth of a function, such as the computational complexity of a particular algorithm.
It is defined as
where c is a positive constant, and is a constant .
L-notation is used mostly in computational number theory, to express the complexity of algorithms for difficult number theory problems, e.g. sieves for integer factorization and methods for solving discrete logarithms. The benefit of this notation is that it simplifies the analysis of these algorithms. The expresses the dominant term, and the takes care of everything smaller.
When is 0, then
is a polylogarithmic function (a polynomial function of ln n);
When is 1 then
is a fully exponential function of ln n (and thereby polynomial in n).
If is between 0 and 1, the function is subexponential of ln n (and superpolynomial).
Many general-purpose integer factorization algorithms have subexponential time complexities. The best is the general number field sieve, which has an expected running time of
for . The best such algorithm prior to the number field sieve was the quadratic sieve which has running time
For the elliptic curve discrete logarithm problem, the fastest general purpose algorithm is the baby-step giant-step algorithm, which has a running time on the order of the square-root of the group order n. In L-notation this would be
The existence of the AKS primality test, which runs in polynomial time, means that the time complexity for primality testing is known to be at most
where c has been proven to be at most 6.
L-notation has been defined in various forms throughout the literature. The first use of it came from Carl Pomerance in his paper "Analysis and comparison of some integer factoring algorithms". This form had only the parameter: the in the formula was for the algorithms he was analyzing. Pomerance had been using the letter (or lower case ) in this and previous papers for formulae that involved many logarithms.
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