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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In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).
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.
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties.
Probabilistic proof systems (eg PCPs and IPs) have had a tremendous impact on theoretical computer science,
as well as on real-world secure systems. They underlie delegation of computation protocols a
The goal of the course is to introduce basic notions from public key cryptography (PKC) as well as basic number-theoretic methods and algorithms for cryptanalysis of protocols and schemes based on PKC