Definition 4 Sample Clauses

Definition 4. The code rate of an authentication scheme, denoted by R, is de- fined as the number of bits that can be authenticated by ▇▇▇▇▇ and ▇▇▇ with one bit of their initial, correlated strings. For traditional authentication codes, the code rate R is determined by the length of the source states divided by that of the encoding rules (authentication keys).

Definition 4. 1.1 The grievance shall mean a written complaint by a member of the bargaining unit or the Association that there has been a violation, misinterpretation, or misapplication of any provision(s) of this agreement. The provision(s) grieved shall be so designated.

Definition 4. An MKEM scheme is (t, s)-cpa-secure if for all t-attacker A, AdvMKEM (A) ≤ s (2) Figure 1 shows the flow of a CKA that uses an MKEM (Panel (a)) and also compares (Panel (b)) to a CKA based on a KEM. We require that the MKEM is IND-CPA (similarly to KEM, IND-CCA is not required). Constructing an MKEM scheme is not necessarily simple. Indeed, in Section 4 we show how to transform BIKE1 KEM into BIKE1-MKEM and explain why the same technique cannot be applied to BIKE2/3. Consequently, we work on each case separately, without stating a general security relation between an MKEM and its related KEM (although we believe that equivalence exists).

Definition 4. Given a compression function f : {0, 1}c × {0, 1}c → C and a function split : N≥2 → N, we recursively define the tree hashes TH M : Mn → M, TH C : Cn → C and TH X : Xn → {0, 1}c as TH M (m, ⊥,..., ⊥)= m, TH C(x1,..., xn)= f (TH X (x1,..., xk), TH X (xk+1,..., xn)), TH X = TH M H TH C, where k = split (n). n−k Using these functions, we finally define the variable input length function : M ≤L → {0, 1}c which hashes variable length trees up to length L. Let U , with a size of K blocks, be the smallest tree of which all trees up to length L dom area subtree of. TH ≤L maps an input m1,..., mn, to TH X (x1,..., xK), where x1,..., xK is the tree Extend[U ](Tn), where Tn is the tree of size n with labels of m1,..., mn. This means that xi is equal to either some mj or . We can apply this mapping non-ambiguously as every tree up to length L is a subtree of U . For example, in the ▇▇▇▇▇▇-▇▇▇▇˚▇▇▇ construction every tree of size n is a subtree of the tree of size n′ if n ≤ n′. This means that U is just the normal tree of size L, hence K = L. This is displayed in Fig. 3 for L = 4. Note that this is not the case in general. For example in the construction in Sakura, if L is equal to 2A + 1, then the right branch only contains one leaf, which means that it does not contain the tree for 2A, which is a full binary tree with a right branch of size 2A−1. However, we can still choose the next power of two as a tree that contains the necessary trees, hence K ≤ 2L. This is displayed in Fig. 4 for L = 5. dom We are ready to prove the collapseability of TH ≤L . dom

Definition 4. The free energy of a set of solutions (configurations) is defined as

Definition 4. For a set of data points P and a query point q, RkNN query retrieves every points p ∈ P for which Dist(q,p) ≤ Dist(p,pk) where pk is the kth nearest point to p in P — {q}.

Definition 4. Given a formula Ψ, the set of maximal consistent subsets of ecl(Ψ), is denoted Γ. The soundness of these rules are argued for exactly as for the case of ATL ([40]), except the (S) axiom which is slightly different from the super-additivity axiom we know from ATL. We show the soundness of this axiom.

Definition 4. 1 (Constellation). Given the (non-empty) set of agents A, a non-empty set of roles R and a state independent role assignment ρ, the grand constellation of A is the vector Σ NR where Σr = Ar for all r. Unless the parameters are clear from the context, we denote the grand constellation for A with roles ρ, Σ(A, ρ), Given a coalition A ⊆ A, the constellation of a coalition A is σA ≤ Σ(A, ρ) where for every role r, (σA)r = |Ar|. As mentioned, we require that constellations never fail to designate. We ensure this in two different ways. When specifying the language HATL we assume that the role assignment is state independent. We show completeness for a logic based on this assumption. Towards the end of the chapter, in Example 4.30, to illustrate the difference between coalitions and constellations (hence between agents and positions), we relax this assumption and simply require that the grand constellation is invariant in the model, i.e., that the social structure is constant.

Definition 4. Random set R and the probability spaces (Ω, S, P) and (ΩA, SA, PA)). Fix 0 ≤ pm ≤ 1 for each m ∈ N. We generate a random set R ⊂ N by adding m to R with probability pm, independently for each m. We let (Ω, S, P) be the probability space of the random sets R. More generally, for A ⊂ N, let (ΩA, SA, PA) be the probability space of the random sets R ∩ A. In general, we shall fix absolute constants α > 0 and 0 < δ ≤ 1, and let pm = min{1, αm—1+δ} for all positive integers m. Note that we restrict our probabilities only to the above probabilities, ignoring the case when, say, pm = m—1/2 log m. Covering the remaining cases would not require a new proof technique, but it would be a bit more cumbersome. Readers interested in the details of the construction of the spaces (Ω, S, P) and (ΩA, SA, PA) are encouraged to consult, for example, ▇▇▇▇▇▇▇▇▇▇ and ▇▇▇▇ [21, Theorem 13, page 142]. Using the natural correspondence between subsets of N and 0–1 vectors indexed by N, we may identify (Ω, S, P) with the product of the two-point spaces (Ωm, Sm, Pm) (m ∈ N), where Ωm = {0, 1}, Sm = 2Ωm , and Pm({1}) = pm and Pm({0}) = 1 − pm. Thus, S is the σ-algebra generated by the sets C(m) = {R ⊂ N: m ∈ R} (m ∈ N), (4.1) that is, the smallest family of subsets of N that is closed under complementa- tion, finite intersections, and countable unions that contains the sets in (4.1). Furthermore, P(C(m)) = P(m ∈ R) = pm for all m, and this suffices to define P on every member of S uniquely. Similarly, (ΩA, SA, PA) may be identified with the product of the two-point spaces (Ωm, Sm, Pm) (m ∈ A) above. In what follows, we shall often write P instead of PA, as this will not cause any confusion. We will study how dense Sidon sets are contained in R. We introduce the notion of the growth of a set S ⊂ N. We say that S has lower growth at least h(n) if S[n] ≥ h(n) for every sufficiently large n. We also say that S has upper growth at most h(n) if S[n] ≤ h(n) for every sufficiently large n. Let R be a set of N. We will abbreviate the fact that there exists a Sidon subset S ⊂ R with lower growth at least h(n), by writing lgrIS(R) ≥ h(n). Similarly, ugr∀S(R) ≤ h(n) will mean that all Sidon subsets S ⊂ R have upper growth at most h(n). An abridged version of our results of this paper is the following.