Definition 8. Let A be a register automaton and w ∈ Ls(A). Then we write symb(w) for the unique symbolic run δ of A with strace(δ)= w. A
Definition 8. We say a TFHE scheme is simulation secure if there is a proba- bilistic polynomial-time simulator Sim such that for any probabilistic polynomial- time adversary , the following experiments are computationally indistinguish- able: realA,C(1κ, 1t, 1N ) :
Definition 8. 5 (Satisfiability problem). The Boolean constants false and true are represented by 0 and 1. Let x1, x2, . . . , xm be Boolean variables, i.e., xi 0, 1 for each i. Variables and their negations are called literals. A Boolean formula ϕ is satisfiable if and only if there is an assignment to the variables in ϕ that makes the formula true. A Boolean formula ϕ is in conjunctive normal form (CNF, for short) i=1 if and only if ϕ is of the form ϕ(x1, x2, . . . , xm) = .n ki j=1 Xx,jΣ, where the Ai,j j=1 are literals over {x1, x2, . . . , xm}. The disjunctions .ki Ai,j of literals are called the clauses of ϕ. A Boolean formula ϕ is in k-CNF if and only if ϕ is in CNF and each clause of ϕ contains exactly k literals. Define the following two problems: SAT = {ϕ | ϕ is a satisfiable Boolean formula in CNF} ; 3-SAT = {ϕ | ϕ is a satisfiable Boolean formula in 3-CNF} . Example 8.2[Boolean formulas] Consider the following two satisfiable Boolean formulas (see also Exercise 8.2-1): ϕ(w, x, y, z) = (x ∨ y ∨ ¬z) ∧ (x ∨ ¬y ∨ ¬z) ∧ (w ∨ ¬y ∨ z) ∧ (¬w ∨ ¬x ∨ z); ψ(w, x, y, z) = (¬w ∨ x ∨ ¬y ∨ z) ∧ (x ∨ y ∨ ¬z) ∧ (¬w ∨ y ∨ z) ∧ (w ∨ ¬x ∨ ¬z) . Here, ϕ is in 3-CNF, so ϕ is in 3-SAT. However, ψ is not in 3-CNF, since the first clause contains four literals. Thus, ψ is in SAT but not in 3-SAT. Theorem 8.6 states the above-mentioned result of Xxxx.
Definition 8. A pseudorandom function family prf is (t, ε)-secure if for all t- attackers A, Advprf (A) ≤ ε. prf K × K → Y K × K → Y K × K → Y For a function family F : 1 2 , let Fswap : 2 1 be defined by Fswap(k1, k2) = F(k2, k1). A dual Pseudorandom Function Family (dPRF) is a function family dprf : 1 2 such that both dprf and dprfswap are PRFs [6].
Definition 8. .2.3. Let A be a a-adically complete ring for some non-zerodivisor a ∈ A. Then we say that A is integrally perfectoid if A[1/a] is perfectoid. The following Lemma gives a technical variation of the Definition which we will need:
Definition 8. Given a V AF , AR, A, , η , an argument x AR is subjectively ac- ceptable if and only if, x appears in the preferred extension for some specific audiences but not all. An argument x AR is objectively acceptable if and only if, x appears in the preferred extension for every specific audience. An argument which is neither objectively nor subjectively acceptable is said to be indefensible.
Definition 8. For Q 1, the maximal secret-key rate for an adversary with an at least Q times better channel, denoted by S+(Q), is defined as S+(Q) := sup QA,QB≥Q sup sA,sB,sE ρ(sA,sE )=QA ρ(sB,sE )=QB S(sA, sB, sE), where S(sA, sB, sE) denotes the secret-key rate of the satellite setting when Alice, Bob, and Eve have error probabilities sA, sB, and sE, respectively.
