L-notation

L-notation is an asymptotic notation analogous to big-O notation, denoted as $L_{n}[\alpha ,c]$ for a bound variable $n$ tending to infinity. Like big-O notation, it is usually used to roughly convey the computational complexity of a particular algorithm.

L-notation is used mostly in computational number theory, to express the complexity of algorithms for difficult number theory problems, eg. sieves for integer factorization and methods for solving discrete logarithms.

It is defined as

$L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}$,

where c is a positive constant, and $\alpha$ is a constant $0\leq \alpha \leq 1$.

When $\alpha$ is 0, then

$L_{n}[\alpha ,c]=L_{n}[0,c]=e^{(c+o(1))\ln \ln n}=(\ln n)^{c+o(1)}$

is a polynomial function of ln n; when $\alpha$ is 1 then

$L_{n}[\alpha ,c]=L_{n}[1,c]=e^{(c+o(1))\ln n}=n^{c+o(1)}$

is a fully exponential function of ln n (and thereby polynomial in n).

If $\alpha$ is between 0 and 1, the function is subexponential (and superpolynomial).

Examples

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

$L_{n}[1/3,c]=e^{(c+o(1))(\ln n)^{1/3}(\ln \ln n)^{2/3}}$

for $c=(64/9)^{(1/3)}$. The best such algorithm prior to the number field sieve was the quadratic sieve which has running time

$L_{n}[1/2,1]=e^{(1+o(1))(\ln n)^{1/2}(\ln \ln n)^{1/2}}\,$.

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

$L_{n}[1,1/2]=n^{1/2+o(1)}\,$.

The existence of the AKS primality test which runs in polynomial time means that the time complexity for primality testing is known to be

$L_{n}[0,c]=(\ln n)^{c+o(1)}\,$

where c has been proven to be at most 6.