1,764 research outputs found
A kilobit hidden SNFS discrete logarithm computation
We perform a special number field sieve discrete logarithm computation in a
1024-bit prime field. To our knowledge, this is the first kilobit-sized
discrete logarithm computation ever reported for prime fields. This computation
took a little over two months of calendar time on an academic cluster using the
open-source CADO-NFS software. Our chosen prime looks random, and
has a 160-bit prime factor, in line with recommended parameters for the Digital
Signature Algorithm. However, our p has been trapdoored in such a way that the
special number field sieve can be used to compute discrete logarithms in
, yet detecting that p has this trapdoor seems out of reach.
Twenty-five years ago, there was considerable controversy around the
possibility of back-doored parameters for DSA. Our computations show that
trapdoored primes are entirely feasible with current computing technology. We
also describe special number field sieve discrete log computations carried out
for multiple weak primes found in use in the wild. As can be expected from a
trapdoor mechanism which we say is hard to detect, our research did not reveal
any trapdoored prime in wide use. The only way for a user to defend against a
hypothetical trapdoor of this kind is to require verifiably random primes
Integer Factorization with a Neuromorphic Sieve
The bound to factor large integers is dominated by the computational effort
to discover numbers that are smooth, typically performed by sieving a
polynomial sequence. On a von Neumann architecture, sieving has log-log
amortized time complexity to check each value for smoothness. This work
presents a neuromorphic sieve that achieves a constant time check for
smoothness by exploiting two characteristic properties of neuromorphic
architectures: constant time synaptic integration and massively parallel
computation. The approach is validated by modifying msieve, one of the fastest
publicly available integer factorization implementations, to use the IBM
Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6;
Section III, par. 2) and phase calculation (Section IV, par 2
Solving discrete logarithms on a 170-bit MNT curve by pairing reduction
Pairing based cryptography is in a dangerous position following the
breakthroughs on discrete logarithms computations in finite fields of small
characteristic. Remaining instances are built over finite fields of large
characteristic and their security relies on the fact that the embedding field
of the underlying curve is relatively large. How large is debatable. The aim of
our work is to sustain the claim that the combination of degree 3 embedding and
too small finite fields obviously does not provide enough security. As a
computational example, we solve the DLP on a 170-bit MNT curve, by exploiting
the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS
Mean values with cubic characters
We investigate various mean value problems involving order three primitive
Dirichlet characters. In particular, we obtain an asymptotic formula for the
first moment of central values of the Dirichlet L-functions associated to this
family, with a power saving in the error term. We also obtain a large-sieve
type result for order three (and six) Dirichlet characters.Comment: 22 pages; greatly shortened, simplified and corrected versio
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