The public key of the Oil-Vinegar scheme consists of a set of m quadratic equations in m + n variables over a finite field F_q. Kipnis and Shamir broke the balanced Oil-Vinegar scheme where d = n − m = 0 by finding equivalent keys of the cryptosytem. Later their method was extended by Kipnis et al to attack the unbalanced case where 0 < d < m and d is small with a complexity of O(q^{d−1} m^4). This method uses the matrices associated with the quadratic polynomials in the public key, which needs to be symmetric and invertible. In this paper, we give an optimized search method for Kipnis el al’s attack. Moreover, for the case that the finite field is of characteristic 2, we find the situation becomes very subtle, which, however, was totally neglected in the original work of Kipnis et al. We show that the Kipnis-Shamir method does not work if the field characteristic is 2 and d is a small odd number, and we fix the situation by proposing an alternative method and give an equivalent key recovery attack of complexity O(q^{d+1} m^4). We also prove an important experimental observation by Ding et al for the Kipnis-Shamir attack on balanced Oil-Vinegar schemes in characteristic 2.

No abstract uploaded!

No abstract uploaded!

We consider amalgamated free product II_{1}factors$M$=$M$_{1*$B$}$M$_{2*$B$}… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra$Q$⊂$M$can be unitarily conjugated into one of the$M$_{$i$}’s. We apply this to the case where the$M$_{$i$}’s are w-rigid II_{1}factors, with$B$equal to either$C$, to a Cartan subalgebra$A$in$M$_{$i$}, or to a regular hyperfinite II_{1}subfactor$R$in$M$_{$i$}, to obtain the following type of unique decomposition results, àla Bass–Serre: If$M$= ($N$_{1 * C}N_{2*$C$}…)^{$t$}, for some$t$> 0 and some other similar inclusions of algebras$C$⊂$N$_{$i$}then, after a permutation of indices, ($B$⊂$M$_{$i$}) is inner conjugate to ($C$⊂$N$_{$i$})^{$t$}, for all$i$. Taking$B$=$C$and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $ , with {$t$_{$i$}}_{$i$⩾1}=$S$a given countable subgroup of$R$_{+}^{*}, we obtain continuously many non-stably isomorphic factors$M$with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to$S$. For$B$=$A$, we obtain a new class of factors$M$with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking$B$=$R$, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ , Out($M$) =$K$, for any given separable compact abelian group$K$.

We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan (1964) minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer--Saraceno (1981), imposes the stationarity of an action functional. We characterize both principles in terms of metric and a symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a-posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.