Definition 2 Clause Samples

Definition 2. A family of d × d matrices {Ai}l with entries in R is irre- ducible over Rd if there is no non-zero proper linear subspace V of Rd such that SiV ⊂ V for all 1 ™ i ™ l. For 0 < s ™ 1, the ‘irreducibility’ condition formulated in Theorem 2.5.17 is satisfied when our collection of matrices is irreducible, in the sense of Definition 2.5.18, see for instance [Fe]. Moreover, if we set d = 2, then by [FSl] this condition is satisfied if and only if {Ai}l refinement of Theorem 2.5.17. is irreducible. This provides us with the following

Definition 2. A negligible function g : N R approaches zero faster than the reciprocal 1A curve is called supersingular if k ≤ 6 in the exte→nsion field Fqk . of any polynomial. That is, for every k ∈ N there is an integer kc such that g(k) ≤ k−c for all k ≥ kc. Cryptographic protocols require the adversary’s advantage to be insignificant in guess- ing the solution to some problem. For instance, one might say that the adversary’s success probability in recovering a session key is a negligible function of the security parameter. The security parameter, denoted by k in many cases, represents the complexity of the input problem. The value of k is important because it can adjust parameters such as the size of cryptographic groups and key lengths. The larger k is, the more computation is required by the algorithm.

Definition 2. We say that the stochastic process (Xt)t∈[0,T ] is a strong solution to (2.3.1) if and only if P − a.s we have Xt = x + a(Xs, s) ds + b(Xs, s) dWs. The most standard existence and uniqueness result for strong solutions to (2.3.1) is given below.

Definition 2. A privilege is a boolean function of the current agents’ states that is given to a node v. We say a privilege is present at a given time if the function is true at that time. ▇▇▇▇▇▇▇▇ defines a global state as legitimate if it follows the following criterion:

Definition 2. 1. A probability measure Pµ on DΩ[0, ∞) satisfying (2.3) is said to be cone-mixing if, for all θ ∈ (0, 1 π), t→∞ A∈F0, B∈Fθ P (A)>0 sup Pµ(B | A) − Pµ(B) = 0, (2.11)

Definition 2. Given a formula φ, a model M = Q, R, π and a state q Q. We say that φ is true in M at q (denoted M, q = φ) if, and only if (by induction on the structure of φ) M, q |= p ⇐⇒ p ∈ π(q) M, q |= ¬ψ ⇐⇒ not M, q |= ψ M, q |= ψ1 ∧ ψ2 ⇐⇒ both M, q |= ψ1 and M, q |= ψ2 M, q |= ♦ψ ⇐⇒ there is a qj ∈ Q such that (q, qj) ∈ R and M, qj |= ψ ( ) | ( ) | ( ) ∈ → | When a formula φ is true in every state of some model M, we say that the formula is true in that model and denote it M = φ . A formula is said to be valid on a frame Q, R at a state q Q if, and only if, for every possible valuation π : Q 2Π, we have Q, R, π , q = φ . Furthermore, it is said to be valid on a frame (denoted Q, R = φ ) if, and only if, it is valid in every state of the frame. | | When we study a logic, it will be relative to some class of frames or models, S. A formula which is true in every state of every such semantic structure (frame or model) is said to be valid, denoted S = φ . If our structures (models or frames) are exactly all structures of interest, we often denote this =S φ . This gives rise to a particular set of formulas: the tautologies or validities.

Definition 2. Let k ∈ N. A subset Γ ⊂ Rn+1 is called a Ck-hypersurface if for each point x0 ∈ Γ there exists an open set U ⊂ Rn+1 containing x0 and a function φ ∈ Ck(U ) such that U ∩ Γ = {x ∈ U | φ(x) = 0} and ∇φ(x) ƒ= 0 for every x ∈ U ∩ Γ. This allows us to define what it means for a function on Γ to be differentiable.

Definition 2. Suppose Γ(t) is evolving with normal velocity vν . Define the material velocity field v := vν + vτ where vτ is the tangential velocity field. The material derivative of a scalar function f = f (x, t) defined on GT := ∪t∈[0,T ]Γ(t)×{t} is given as ∂•f := ∂f + v · ∇f. We now give a generalisation of integration by parts for a hypersurface Γ, the proof of which is found in ▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇▇▇ [2001].

Definition 2. 4. (GDH assumption) Let g be a generator of a finite cyclic group G and x1, ..., xl, z 0, G 1 be chosen at random with l N