Aside non-associative scheme A non-associative key agreement scheme is any scheme k that is not asso- ciative. / In particular, Diffie–Xxxxxxx key agreement k (Definition 2.2.1) is not as- sociative. To see this, first k1(a, b) = k3(a, b) = ba = ab = k2(b, a) = k4(b, a), for most a and b. But if k were associative, then it would be the subscheme of a multiplicative scheme K, and would have k1(a, b) = K1(a, b) = K2(a, b) = k2(a, b), a contradiction. − − ∈ { } For a smaller non-associative scheme, consider rock, scissors, paper key agreement again. Represent it as k1(a, b) = a and k2(b, c) = c and k3(a, b) = k4(a, b) = (a b) mod 3, where all a, b, c Z/3 = 0, 1, 2 , making each ki a binary operation on Z/3. Recall that this is a key agreement scheme because k3(a, k2(b, c)) = k3(a, c) = (a c) mod 3 = k4(a, c) = k4(k1(a, b), c). Suppose that k is a subscheme of a multiplicative scheme K. Then all the sessions of k must be sessions of K, so k1(a, b) = K1(a, b) and k2(b, c) = K2(b, c). But since K is multiplicative, K is symbiotic with K1 = K2. Clearly, k1 k cannot be a subscheme of K. k2, so
Aside reduction to a category Category theory is an abstraction that unifies many notions across modern al- gebra. Perhaps, it is then unsurprising that key agreement can be abstracted into category theory. Recall that category has objects and morphisms between objects. Mor- phisms compose associatively. (There is also a unique identity morphism form each object to itself, but we shall not need that.) Write Mor(O, P ), for the set of morphisms between two objects O, P . Write f ◦ g ∈ Mor(O, P ) for the composition of morphisms, f ∈ Mor(P, Q) and g ∈ Mor(O, Q). (This convention means the objects have opposite left-to-right ordering from the morphisms in a product.) First, we construct a key agreement scheme from any category, and any four objects O1, O2, O3, O4 in the category. Define a key agreement scheme k as follows: k1 : Mor(O4, O3) × Mor(O3, O2) → Mor(O4, O2) : [a, b] '→ a ◦ b k2 : Mor(O3, O2) × Mor(O2, O1) → Mor(O3, O1) : [b, c] '→ b ◦ c k3 : Mor(O4, O3) × Mor(O3, O1) → Mor(O4, O1) : [a, e] '→ a ◦ e
Aside faulty schemes Faulty key agreement is not studied in this report, but is defined in this section, for completeness and clarification. Faulty key agreement relaxes the condition on k in a probabilistic key agreement scheme.
Aside packed associated semigroups ⊆ If key agreement k is associated with semigroup S, define a subset Sk | | S, the set of sessional elements, by including each entry of each session [a, b, c, d, e, f ] of k, as mapped to elements of an associated semigroup S, via the equivalence of k to a subscheme of the multiplicative scheme kS. The sessionality of S associated with k is the cardinality Sk of the set of sessional elements. A semigroup S is a packed associated semigroup of k, if its sessionality is minimal among all associated semigroups of k.
Aside instance-verifiability of divulgers A divulger L1 is instance-verifiable meaning that the correctness of any valid instance of the output L1(d, b) can be verified without knowledge of any secrets, by checking that k1(L1(d, b)) = d.
Aside. In retrospect, it may seem strange that the hypothesis of spec-head agreement has been entertained at all. Fact is that it was the standard analysis for quite a few years. Counter-evidence was explained away by further assumptions. Agreement without movement Expletives: Some languages, among them English, have a construction where SpecT is not occupied by an argument but by an semantically empty element, an expletive (it, there, cf. (8-b)). Nevertheless, T agrees with some vP-internal argument (indicated by in (16)), and not with the expletive, see (15-a-d). This required additional (ad hoc) assumptions under the hypothesis of spec-head agreement. (15) a. There arrive-s a train.
Aside the email Mr xxx xxx Xxxxx and Xxxxxxxxx sent to Ren Capes more or less concluded the relation between EGDI and PanGeo. Any portals that want to be sustained should become integrated into 1G-E, which should be maintained by EGS.
Aside the taking of minutes of the meetings should be spread across the consortium members. Xx Xxxxxxxxx makes the point that Xx Xxxx took minutes at the previous meetings in Brussels as well. All agree that this task should be shared. Item 9 – 3D workshop 24/25 October, Germany We have been invited by Xxxxxx Xxxxxxxxx (Regional Survey of Bavaria) to a workshop on 3D Geology. It is debated whether we should we attend as EGDI-Scope partners to exchange information and make a presentation. Most of the workshop is in German however and focused on German issues. There will only be one or two other presentations in English. Xx Xxx will be attending. It is decided to present EGDI-Scope if possible during the workshop and particularly the relation to 3D modelling. Item 10 – WP4 Xx Xxxxxx reviews WP4 developments. Mr xxx xxx Xxxxx raises two questions on the four schemes presented: - Should we use different schemes for different branches of the EGDI or should we have one or two only? It could be good to have different schemes tailored to the different branches of the EGDI. This is something we probably cannot answer now but should be kept in mind as things develop. - We should start simple but we also need to describe different stages and not just the end picture. It is suggested to look at other projects which are not portals as well. For example GEUS have worked on projects from which, while the data is confidential, we can get ideas how these projects operate in terms of how useful they are to the end users such as oil companies etc. We should pose this question to all the surveys. Perhaps they can give us some good ideas for the architecture of the EGDI in terms of user interfaces.
Aside packed associated semigroups ⊆ If key agreement k is associated with semigroup S, define a subset Sk | | S, the set of sessional elements, by including each entry of each session [a, b, c, d, e, f ] of k, as mapped to elements of an associated semigroup S, via the equivalence of k to a subscheme of the multiplicative scheme kS. The sessionality of S associated with k is the cardinality Sk of the set of sessional elements. A semigroup S is a packed associated semigroup of k, if its sessionality is minimal among all associated semigroups of k. For example, for Diffie–Xxxxxxx agreement mod p, the associated semi- group from the proof Lemma 2.20.1 has sessionality of 6p − 6, while the associated semigroup from the proof of Lemma 2.18.1 has sessionality 2p − 2. Therefore, the former semigroup is not a packed associated semigroup of k, because the latter has lower sessionality.
Aside. In principle, the spec-head configuration could also come into existence via external Merge. Here, we concentrate on Movement (internal Merge). Spec-head agreement Further motivation (Kayne 1989): In French/Italian, past-participle agreement with the object (with respect to gender and number) does not arise if the object remains in the position where it is merged (13-a)/(14-a). Only if the object moves (e.g., because it is a clitic) does past-participle agreement become possible (13-b)/(14-b).
