Lemma 3. (Termination) For each run, every process pi ∈ Correct of the sys- tem HASf [L, ∅, n] eventually decides some value. ∈
Lemma 3. 5.3. Let C/Q be a hyperelliptic curve of genus g ≥ 4, with a rational Weierstrass point, geometrically simple Jacobian with r ≤ 1, good reduction at 3, which satisfies condition (†). Let P1, P2 ∈ C(Q) be conjugate quadratic points, with P1, P2 ∈ CF3 (F9) \ CF3 (F3), and P3 ∈ C(Q) a rational point. If n(ΛC, P1) = 1, there are at most 26 ordered triples (Q1, Q2, Q3) of conjugate cubic points in DP1 × DP2 × DP3 . If n(ΛC, P1) = 0, there are no such triples.
Lemma 3. .8. The series L(s, ψ) converges to a positive real number at s = 1.
Lemma 3. 2.1. (Convergence of private empirical marginal distribution). limn→∞ F˜n(t) = limn→∞ Fˆn(t) = F (t) almost surely, where F˜n(t) is the empirical CDF based on the
Lemma 3. 3.4 For every A ⊂ [n] \ S with |A| ≥ 16n/s2, we have s4 2 eC˜GS (A) ≥ 100n|A| , (3.14) where the edges are counted with multiplicity. The proof of Lemma 3.3.4 will be given in Section 3.3.2. ˜ In view of Lemma 3.3.4, if the maximum multiplicity of an edge in CGS is at most r then the graph CGS satisfies the conditions of Lemma 3.3.2 with β = s4 100rn and δ = 16/s2. Consequently, we are interested in bounding the multi- ˜ plicity of the edges of CGS. For any z ∈ Z, let S 2 RS(z) = {a1, a2}, {b1, b2} ∈ 2 : z = a1 + a2 − b1 − b2 . (3.15) ˜ By construction, the multiplicity of a pair {x, y} in the graph CGS is given by RS(x − y) = RS(y − x). We define z∈Z R∗S = max RS(z)} (3.16) and show that R∗S = Θ(s) in the following proposition.
Lemma 3. 4.1 For 1 m ≤ o(n1/5), almost surely we have m ≥ F3([n]m) ≥
Lemma 3. 4.2 For any 1 m ≤ n, almost surely we have F3([n]m) = Ω(m1/3). For this proof, it will be convenient to use the model [n]p with p = m/n rather than [n]m (recall Remark 3.1.6). Without loss of generality we assume that 1/p, pn, pn/3 ∈ N. Our strategy here follows that of [31]. In order to show the existence of a Sidon set of order (pn)1/3 in a typical instance of a random set [n]p we will use the following theorem of Xxxx and Xxxxxx [10] (with the statement adapted for our purposes).
Lemma 3. If all honest parties in Cb input b to BA(1), and no honest party in C1−b inputs
Lemma 3. 3.8. Let d > 1 be a positive integer and let C/Q be a hyperelliptic curve of genus g > d, with a rational Weierstrass point, geometrically simple Jacobian with r ≤ 1, good reduction at a prime p > d2 + 3, and which satisfies condition (†). Let P1, · · · , Pd ∈ Cd(Q) be a conjugate d-tuple with well-behaved uniformizers
Lemma 3. 4.1. Let C/Q be a hyperelliptic curve of genus g ≥ 3, with a rational Weierstrass point, geometrically simple Jacobian with r ≤ 1, and let p be an odd prime of good reduction for C. Let P1, P2 ∈ C(Q) be either two rational points or a pair of conjugate quadratic points, with well-behaved uniformizers zP1 , zP2 . Let (Q1, Q2) be a pair of unexpected conjugate quadratic points with the same reduction as (P1, P2). Then {(Q1, Q2)} is a zero-dimensional component of (C2)ΛC ∩ (B1 (P1, zP ) × X0 (X0, zP )).
