For t, γ∈R with \( 0\leq t\leq 1 \) , the notation \( L_x[t,\gamma] \) is used for any function of x that equals
where logarithms are natural and where o(1) denotes any function of x that goes to 0 as \( x\to\infty \) (see O-notation). This function has the following properties:
\( L_x[t,\gamma]+L_x[t,\delta]=L_x[t,\max(\gamma,\delta)] \) ,
\( L_x[t,\gamma]L_x[t,\delta]=L_x[t,\gamma+\delta] \) ,
\( L_x[t,\gamma]L_x[s,\delta]=L_x[t,\gamma] \) , if \( t>s \) ,
for any fixed k:\( L_x[t,\gamma]^k=L_x[t,k\gamma], \) ,if \( \gamma>0 \) then \( (\log x)^kL_x[t,\gamma]=L_x[t,\gamma] \) .
\( \pi(L_x[t,\gamma])=L_x[t,\gamma] \) where \( \pi(y) \) is the number of primes \( \leq \)y.
When used to indicate runtimes and for γ fixed, \( L_x[t,\gamma] \) for t ranging from 0 to 1 ranges from polynomial time to exponential time in \( \log(x) \) :
runtime
$$ L_x[0,\gamma]=e^{(\gamma+o(1))\log\log x}=(\log...
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© 2005 International Federation for Information Processing
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Lenstra, A.K. (2005). L-Notation. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_237
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DOI: https://doi.org/10.1007/0-387-23483-7_237
Publisher Name: Springer, Boston, MA
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