Common use of Main Result Clause in Contracts

Main Result. ‌ We will use a notion from Daemen et al. [15], namely that of the multicollision limit function. ∈ Definition 1 (multicollision limit function). Let M, c, r N. Consider the experiment of throwing M balls uniformly at random in 2r bins, and let μ be the maximum number of balls in a single bin. We define the multicollision limit as the smallest natural number x that satisfies 2c Pr (μ > x) ≤ x . We derive the following result on the keyed duplex under leakage. Theorem 1. Let b, c, r, k, u, α, λ ∈ N, with c + r = b, k ≤ b, α ≤ b − k, and $ λ ≤ 2b. Let p ←− perm(b) be a random permutation, and K ←D−K− ({0, 1}k)u a L { { } × { } → { } } random array of keys. Let = L : 0, 1 b 0, 1 b 0, 1 λ be a class of leakage functions. For any distinguisher D quantified as in Sect. 5.1, KD AdvL-naLR(D) νfixN 2νM N 2νM νM (L + Ω)+ νfix −1 (L + Ω) ≤ 2c−(R+1)λ + 2c−(R+1)λ + 2c + 2 2c−Rλ + .M−L−qΣ + (M − L − q)(L + Ω) 2b−λ .M+NΣ + .NΣ 2b . Σ + qIV N + q(M − q) 2H∞(DK )+▇▇▇{c,max{b−α,c}−k}−(R+qδ)λ 2H∞(DK )−qδλ + 2 2H∞(DK ) . In addition, except with probability at most the same bound, the final output states have min-entropy at least b − λ. The proof is given in Sect. 5.4; we first give an interpretation of the bound in Sect. 5.3.