2006 | OriginalPaper | Buchkapitel
Rigorous Bounds on Cryptanalytic Time/Memory Tradeoffs
verfasst von : Elad Barkan, Eli Biham, Adi Shamir
Erschienen in: Advances in Cryptology - CRYPTO 2006
Verlag: Springer Berlin Heidelberg
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In this paper we formalize a general model of cryptanalytic time/memory tradeoffs for the inversion of a random function
f
:{0,1,...,
N
–1} ↦{0,1,...,
N
–1}. The model contains all the known tradeoff techniques as special cases. It is based on a new notion of
stateful random graphs
. The evolution of a path in the stateful random graph depends on a
hidden state
such as the color in the Rainbow scheme or the table number in the classical Hellman scheme. We prove an upper bound on the number of images
y
=
f
(
x
) for which
f
can be inverted, and derive from it a lower bound on the number of hidden states. These bounds hold for an overwhelming majority of the functions
f
, and their proofs are based on a rigorous combinatorial analysis. With some additional natural assumptions on the behavior of the
online phase
of the scheme, we prove a lower bound on its worst-case time complexity
$T=\Omega(\frac{N^2}{M^2 \ln N})$
, where
M
is the memory complexity. Finally, we describe new rainbow-based time/memory/data tradeoffs, and a new method for improving the time complexity of the online phase (by a small factor) by performing a deeper analysis during preprocessing.