4,589 research outputs found
An experimental exploration of Marsaglia's xorshift generators, scrambled
Marsaglia proposed recently xorshift generators as a class of very fast,
good-quality pseudorandom number generators. Subsequent analysis by Panneton
and L'Ecuyer has lowered the expectations raised by Marsaglia's paper, showing
several weaknesses of such generators, verified experimentally using the
TestU01 suite. Nonetheless, many of the weaknesses of xorshift generators fade
away if their result is scrambled by a non-linear operation (as originally
suggested by Marsaglia). In this paper we explore the space of possible
generators obtained by multiplying the result of a xorshift generator by a
suitable constant. We sample generators at 100 equispaced points of their state
space and obtain detailed statistics that lead us to choices of parameters that
improve on the current ones. We then explore for the first time the space of
high-dimensional xorshift generators, following another suggestion in
Marsaglia's paper, finding choices of parameters providing periods of length
and . The resulting generators are of extremely
high quality, faster than current similar alternatives, and generate
long-period sequences passing strong statistical tests using only eight logical
operations, one addition and one multiplication by a constant
Realizing arbitrary-precision modular multiplication with a fixed-precision multiplier datapath
Within the context of cryptographic hardware, the term scalability refers to the ability to process operands of any size, regardless of the precision of the underlying data path or registers. In this paper we present a simple yet effective technique for increasing the scalability of a fixed-precision Montgomery multiplier. Our idea is to extend the datapath of a Montgomery multiplier in such a way that it can also perform an ordinary multiplication of two n-bit operands (without modular reduction), yielding a 2n-bit result. This
conventional (nxn->2n)-bit multiplication is then used as a “sub-routine” to realize arbitrary-precision Montgomery multiplication according to standard software algorithms such as Coarsely Integrated Operand Scanning (CIOS). We
show that performing a 2n-bit modular multiplication on an n-bit multiplier can be done in 5n clock cycles, whereby we assume that the n-bit modular multiplication takes n cycles. Extending a Montgomery multiplier for this extra
functionality requires just some minor modifications of the datapath and entails a slight increase in silicon area
Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations
Ideas from Fourier analysis have been used in cryptography for the last three
decades. Akavia, Goldwasser and Safra unified some of these ideas to give a
complete algorithm that finds significant Fourier coefficients of functions on
any finite abelian group. Their algorithm stimulated a lot of interest in the
cryptography community, especially in the context of `bit security'. This
manuscript attempts to be a friendly and comprehensive guide to the tools and
results in this field. The intended readership is cryptographers who have heard
about these tools and seek an understanding of their mechanics and their
usefulness and limitations. A compact overview of the algorithm is presented
with emphasis on the ideas behind it. We show how these ideas can be extended
to a `modulus-switching' variant of the algorithm. We survey some applications
of this algorithm, and explain that several results should be taken in the
right context. In particular, we point out that some of the most important bit
security problems are still open. Our original contributions include: a
discussion of the limitations on the usefulness of these tools; an answer to an
open question about the modular inversion hidden number problem
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