Pairing theorem definition

Proof sketch. The implicit and explicit requirements from Definitions 7.1 and 7.31 and the property to be polynomial-time in the interface inputs alone are easy to see. Among the criteria from Theorem 7.34, effectiveness of authentication is easily derived from that in the one-time scheme, and the security for the risk bearer is completely identical to that in the underlying one-time scheme. (Recall that the fact that the signer s entity bases many one-time key pairs on the same prekey makes no formal difference at all in Criterion 2 of Theorem 7.34.)... 

Let us now come to the property of integrability in Liouville s sense. To this end we need the following definition a set flq,..., of independent functions is said to be a complete involution system in case it satisfies = 0 for every pair j, k. A classical theorem due to Liouville is the following ... 

Let us fix I and S. By Lemma 2.43 and Lemma 2.55, we have various natural maps between functors on sheaves arising from the closed structures and the monoidal pairs, involving various J-diagrams of schemes, where J varies subcategories of I. In the sequel, many of the natural maps are referred as the canonical maps or the canonical isomorphisms without any explicit definitions. Many of them are defined in [26] and Chapter 1, and various commutativity theorems are proved there. 

Again we see that the identity map induces identity maps, and that the composition of two cellular maps induces the composition of induced maps. Theorem 3.41 is best when one uses the alternative definition of cellular homology using the relative homology groups. It will follow in that context from the naturality of the associated long exact sequence of a pair. We postpone the precise argument until Subsection 5.2.2. 

Remark 5.12. In full analogy with Definition 5.9 one can define the relative cohomology of a pair (X,A) as the cohomology of the cochain complex Cging(X, A), where C sing( j ) consists of all functions on n-cells that have value 0 on the singular simplices inside A. Theorem 5.7 can be applied to the short exact sequence... 

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