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the automorphism $(-1)$ acting on it.
Kummer varieties can be seen as a higher dimensional generalisation of
the $x$-coordinate representation of a point of an elliptic curve
given by its Weierstrass model. Although there is no group law on the
set of points of a Kummer variety, there remains enough arithmetic
to enable the computation of exponentiations via a
Montgomery ladder based on differential additions.
In this paper, we explain that the arithmetic of a Kummer variety
is much richer than
usually thought. We describe a set of composition laws
which exhaust this arithmetic and show that these
laws may turn out to be useful in order to improve certain
algorithms. We explain how to compute efficiently these laws in the model of
Kummer varieties provided by level $2$ theta functions. We also
explain how to recover the full group law of the abelian variety
with a representation almost as compact and in many cases as efficient as
the level $2$ theta functions model of Kummer varieties.
k-SIS problem. The Boneh-Freeman reduction from SIS to k-SIS suffers from an exponential loss in k. We improve and extend it to an LWE to k-LWE reduction with a polynomial loss in k, by relying on a new technique involving trapdoors for random integer kernel lattices. Based on this hardness result, we present the first algebraic construction of a traitor tracing scheme whose security relies on the worst-case hardness of standard lattice problems. The proposed LWE traitor tracing is almost as efficient as the LWE encryption. Further, it achieves public traceability, i.e., allows the authority to delegate the tracing capability to untrusted parties. To this aim, we introduce the notion of projective sampling family in which each sampling function is keyed and, with a projection of the key on a well chosen space, one can simulate the sampling function in a computationally indistinguishable way. The construction of a projective sampling family from k-LWE allows us to achieve public traceability, by publishing the projected keys of the users. We believe that the new lattice tools and the projective sampling
family are quite general that they may have applications in other areas.
worst-case complexity of approximating the Shortest Vector Problem in ideal lattices within polynomial
factors. The distinguishing feature of our scheme is that it achieves short signatures (consisting of a
single lattice vector), and relatively short public keys (consisting of O(log n) vectors.) Previous lattice
schemes in the standard model with similarly short signatures, due to Boyen (PKC 2010) and Micciancio
and Peikert (Eurocrypt 2012), had substantially longer public keys consisting of Ω(n) vectors (even when
implemented with ideal lattices). We also present a variant of our scheme that further reduces the public
key size to just O(log log n) vectors and allows for a tighther security proof by making the signer stateful.
and Patey proposed two biometric authentication schemes between
a prover and a verifier where the verifier has biometric
data of the users in plain form. The protocols are based on secure
computation of Hamming distance in the two-party setting. Their
first scheme uses Oblivious Transfer (OT) and provides security
in the semi-honest model. The other scheme uses Committed
Oblivious Transfer (COT) and is claimed to provide full security
in the malicious case.
In this paper, we show that their protocol against malicious
adversaries is not actually secure. We propose a generic attack
where the Hamming distance can be minimized without knowledge
of the real input of the user. Namely, any attacker can
impersonate any legitimate user without prior knowledge. We
propose an enhanced version of their protocol where this attack
is eliminated. We provide a simulation based proof of the security
of our modified protocol. In addition, for efficiency concerns, the
modified version also utilizes Verifiable Oblivious Transfer (VOT)
instead of COT. The use of VOT does not reduce the security of
the protocol but improves the efficiency significantly.