16,875 research outputs found
The iterated Carmichael \lambda-function and the number of cycles of the power generator
Iteration of the modular l-th power function f(x) = x^l (mod n) provides a
common pseudorandom number generator (known as the Blum-Blum-Shub generator
when l=2). The period of this pseudorandom number generator is closely related
to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely
the maximal multiplicative order of any integer modulo n. In this paper, we
show that for almost all n, the size of \lambda(\lambda(n)) is
n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula
for the k-th iterate of \lambda. We deduce that for almost all n, the
psuedorandom number generator described above has at least exp((1+o(1))(log log
n)^2 log log log n) disjoint cycles. In addition, we show that this expression
is accurate for almost all n under the assumption of the Generalized Riemann
Hypothesis for Kummerian fields. We also consider the number of iterations of
\lambda it takes to reduce an integer n to 1, proving that this number is less
than (1+o(1))(log log n)/log 2 infinitely often and speculating that log log n
is the true order of magnitude almost always.Comment: 28 page
A Beta-splitting model for evolutionary trees
In this article, we construct a generalization of the Blum-Fran\c{c}ois
Beta-splitting model for evolutionary trees, which was itself inspired by
Aldous' Beta-splitting model on cladograms. The novelty of our approach allows
for asymmetric shares of diversification rates (or diversification `potential')
between two sister species in an evolutionarily interpretable manner, as well
as the addition of extinction to the model in a natural way. We describe the
incremental evolutionary construction of a tree with n leaves by splitting or
freezing extant lineages through the Generating, Organizing and Deleting
processes. We then give the probability of any (binary rooted) tree under this
model with no extinction, at several resolutions: ranked planar trees giving
asymmetric roles to the first and second offspring species of a given species
and keeping track of the order of the speciation events occurring during the
creation of the tree, unranked planar trees, ranked non-planar trees and
finally (unranked non-planar) trees. We also describe a continuous-time
equivalent of the Generating, Organizing and Deleting processes where tree
topology and branch-lengths are jointly modeled and provide code in
SageMath/python for these algorithms.Comment: 23 pages, 3 figures, 1 tabl
On formal verification of arithmetic-based cryptographic primitives
Cryptographic primitives are fundamental for information security: they are
used as basic components for cryptographic protocols or public-key
cryptosystems. In many cases, their security proofs consist in showing that
they are reducible to computationally hard problems. Those reductions can be
subtle and tedious, and thus not easily checkable. On top of the proof
assistant Coq, we had implemented in previous work a toolbox for writing and
checking game-based security proofs of cryptographic primitives. In this paper
we describe its extension with number-theoretic capabilities so that it is now
possible to write and check arithmetic-based cryptographic primitives in our
toolbox. We illustrate our work by machine checking the game-based proofs of
unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and
semantic security of the public-key cryptographic scheme of Goldwasser and
Micali.Comment: 13 page
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