137 research outputs found
The Cycle Structure of LFSR with Arbitrary Characteristic Polynomial over Finite Fields
We determine the cycle structure of linear feedback shift register with
arbitrary monic characteristic polynomial over any finite field. For each
cycle, a method to find a state and a new way to represent the state are
proposed.Comment: An extended abstract containing preliminary results was presented at
SETA 201
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
Pairings in Cryptology: efficiency, security and applications
Abstract
The study of pairings can be considered in so many di�erent ways that it
may not be useless to state in a few words the plan which has been adopted,
and the chief objects at which it has aimed. This is not an attempt to write
the whole history of the pairings in cryptology, or to detail every discovery,
but rather a general presentation motivated by the two main requirements
in cryptology; e�ciency and security.
Starting from the basic underlying mathematics, pairing maps are con-
structed and a major security issue related to the question of the minimal
embedding �eld [12]1 is resolved. This is followed by an exposition on how
to compute e�ciently the �nal exponentiation occurring in the calculation
of a pairing [124]2 and a thorough survey on the security of the discrete log-
arithm problem from both theoretical and implementational perspectives.
These two crucial cryptologic requirements being ful�lled an identity based
encryption scheme taking advantage of pairings [24]3 is introduced. Then,
perceiving the need to hash identities to points on a pairing-friendly elliptic
curve in the more general context of identity based cryptography, a new
technique to efficiently solve this practical issue is exhibited.
Unveiling pairings in cryptology involves a good understanding of both
mathematical and cryptologic principles. Therefore, although �rst pre-
sented from an abstract mathematical viewpoint, pairings are then studied
from a more practical perspective, slowly drifting away toward cryptologic
applications
Lower bounds on the non-Clifford resources for quantum computations
We establish lower-bounds on the number of resource states, also known as
magic states, needed to perform various quantum computing tasks, treating
stabilizer operations as free. Our bounds apply to adaptive computations using
measurements and an arbitrary number of stabilizer ancillas. We consider (1)
resource state conversion, (2) single-qubit unitary synthesis, and (3)
computational tasks.
To prove our resource conversion bounds we introduce two new monotones, the
stabilizer nullity and the dyadic monotone, and make use of the already-known
stabilizer extent. We consider conversions that borrow resource states, known
as catalyst states, and return them at the end of the algorithm. We show that
catalysis is necessary for many conversions and introduce new catalytic
conversions, some of which are close to optimal.
By finding a canonical form for post-selected stabilizer computations, we
show that approximating a single-qubit unitary to within diamond-norm precision
requires at least
-states on average. This is the first lower bound that applies to synthesis
protocols using fall-back, mixing techniques, and where the number of ancillas
used can depend on .
Up to multiplicative factors, we optimally lower bound the number of or
states needed to implement the ubiquitous modular adder and
multiply-controlled- operations. When the probability of Pauli measurement
outcomes is 1/2, some of our bounds become tight to within a small additive
constant.Comment: 62 page
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