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2014-06-25

Name: Elisa Gorla

Topic: Lifting properties from the general hyperplane section of a projective scheme

Category:(no category)

2014-06-23

We consider the notion of a non-interactive key exchange (NIKE). A NIKE scheme allows a party \\(A\\) to compute a common shared key with another party \\(B\\) from \\(B\\)\'s public key and \\(A\\)\'s secret key alone. This computation requires no interaction between \\(A\\) and \\(B\\), a feature which distinguishes NIKE from regular (i.e., interactive) key exchange not only quantitatively, but also qualitatively.

Our first contribution is a formalization of NIKE protocols as ideal

functionalities in the Universal Composability (UC) framework.

As we will argue, existing NIKE definitions (all of which are game-based) do not support a modular analysis either of NIKE schemes themselves, or of the use of NIKE schemes. We provide a simple and natural UC-based NIKE definition that allows for a modular analysis both of NIKE schemes and their use in larger protocols.

We proceed to investigate the properties of our new definition, and in

particular its relation to existing game-based NIKE definitions. We find that

(a) game-based NIKE security is equivalent to UC-based NIKE security

against \\emph{static} corruptions, and

(b) UC-NIKE security against adaptive corruptions cannot be achieved

without additional assumptions (but \\emph{can} be achieved in the random oracle model).

Our results suggest that our UC-based NIKE definition is a useful and simple abstraction of non-interactive key exchange.

Applications of elliptic curve cryptography to anonymity, privacy and

censorship circumvention call for methods to represent uniformly random

points on elliptic curves as uniformly random bit strings, so that, for

example, ECC network traffic can masquerade as random traffic.

At ACM CCS 2013, Bernstein et al. proposed an efficient approach,

called ``Elligator,\'\' to solving this problem for arbitrary elliptic

curve-based cryptographic protocols, based on the use of efficiently

invertible maps to elliptic curves. Unfortunately, such invertible maps

are only known to exist for certain classes of curves, excluding in

particular curves of prime order and curves over binary fields. A variant

of this approach, ``Elligator Squared,\'\' was later proposed by Tibouchi

(FC 2014) supporting not necessarily injective encodings to elliptic

curves (and hence a much larger class of curves), but, although some

rough efficiency estimates were provided, it was not clear how an actual

implementation of that approach would perform in practice.

In this paper, we show that Elligator Squared can indeed be implemented

very efficiently with a suitable choice of curve encodings. More

precisely, we consider the binary curve setting (which was not discussed

in Tibouchi\'s paper), and implement the Elligator Squared bit string

representation algorithm based on a suitably optimized version of the

Shallue--van de Woestijne characteristic 2 encoding, which we show can

be computed using only multiplications, trace and half-trace

computations, and a few inversions.

On the fast binary curve of Oliveira et al. (CHES 2013), our

implementation runs in an average of only 22850 Haswell cycles, making

uniform bit string representations possible for a very reasonable

overhead---much smaller even than Elligator on Edwards curves.

As a side contribution, we also compare implementations of Elligator and

Elligator Squared on a curve supported by Elligator, namely

Curve25519. We find that generating a random point and its uniform

bitstring representation is around 35-40% faster with Elligator for

protocols using a fixed base point (such as static ECDH), but 30-35%

faster with Elligator Squared in the case of a variable base point

(such as ElGamal encryption). Both are significantly slower

than our binary curve implementation.

The GGH Graded Encoding Scheme, based on ideal lattices, is the first plausible approximation to a cryptographic multilinear map. Unfortunately, using the security analysis in the original paper, the scheme requires very large parameters to provide security for its underlying encoding re-randomization process. Our main contributions are to formalize, simplify and improve the efficiency and the security analysis of the re-randomization process in the GGH construction. This results in a new construction that we call GGHLite. In particular, we first lower the size of a standard deviation parameter of the re-randomization process of the original scheme from exponential to polynomial in the security parameter. This first improvement is obtained via a finer security analysis of the

drowning step of re-randomization, in which we apply the

Rényi divergence instead of the conventional statistical distance as a measure of distance between distributions. Our second improvement is to reduce the number of randomizers needed from $\\Omega(n \\log n)$ to $2$, where $n$ is the dimension of the underlying ideal lattices. These two contributions allow us to decrease the bit size of the public parameters from $O(\\lambda^5 \\log \\lambda)$ for the

GGH scheme to $O(\\lambda \\log^2 \\lambda)$ in GGHLite, with respect to the security parameter $\\lambda$ (for a constant multilinearity parameter $\\kappa$).

Related-key attacks (RKAs) concern the security of cryptographic primitives in the situation where the key can be manipulated by the adversary. In the RKA setting, the adversary\'s power is expressed through the class of related-key deriving (RKD) functions which the adversary is restricted to using when modifying keys. Bellare and Kohno (Eurocrypt 2003) first formalised RKAs and pin-pointed the foundational problem of constructing RKA-secure pseudorandom functions (RKA-PRFs). To date there are few constructions for RKA-PRFs under standard assumptions, and it is a major open problem to construct RKA-PRFs for larger classes of RKD functions. We make significant progress on this problem. We first show how to repair the Bellare-Cash framework for constructing RKA-PRFs and extend it to handle the more challenging case of classes of RKD functions that contain claws. We apply this extension to show that a variant of the Naor-Reingold function already considered by Bellare and Cash is an RKA-PRF for a class of affine RKD functions under the DDH assumption, albeit with an exponential-time security reduction. We then develop a second extension of the Bellare-Cash framework, and use it to show that the same Naor-Reingold variant is actually an RKA-PRF for a class of degree $d$ polynomial RKD functions under the stronger decisional $d$-Diffie-Hellman inversion assumption. As a significant technical contribution, our proof of this result avoids the exponential-time security reduction that was inherent in the work of Bellare and Cash and in our first result.