Let {Dn} be a sequence of bounded invertible operators on Hilbert space H. It is shown that the collection of operators T for which the norm-limit limDnTD n exists is an algebra. Furthermore, some sufficient conditions on this sequence are established for the corresponding algebra to have a nontrivial invariant subspace. By considering specific sequences of operators several invariant subspace ...

In the more general framework ' shift invariant subspace", the paper obtains a different estimate of sampling in function subspace to our former work, by using the Frame Theory. The derived formula is easy to be calculated, and the estimate is relaxed in some shift invariant subspaces.

We present an invariant subspace attack on the block cipher Midori64, proposed at Asiacrypt 2015. Our analysis shows that Midori64 has a class of 232 weak keys. Under any such key, the cipher can be distinguished with only a single chosen query, and the key can be recovered in 216 time with two chosen queries. As both the distinguisher and the key recovery have very low complexities, we confirm...

The n-variable PARITY function is computable (by a well-known recursive construction) by AC0 formulas of depth d+ 1 and leafsize n·2dn . These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0, 1}, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2d(n−1) lower...