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H/2m each and 0 with probability 1 − H/m. Let f(X) = ∑ m−1 i=0 f iX i . Then for fixed r and H, m → ∞, we have E[|f(τ)| 2r ] ∼ r!H r . In particular, for H ≥ 2r 2 , we have E[|f(τ)| 2r ] ∣ r!H r − 1 ∣ ≤ 2r2 H + 2r+1 r 2 m . Before proving Theorem 1, we introduce some notation and prove some technical results that will be useful. Recall the “falling factorial” notation: for integers n, k with 0 ≤ k ≤ n, we define n k = ∏ k−1 j=0 (n − j). Lemma 1. For n ≥ k 2 > 0, we have n k − n k ≤ k 2 n k−1 . Proof. We have n k ≥ (n − k) k = n k − ( ( ( k k k kn 1) k−1 + k 2) 2 n k−2 − k 3) 3 n k−3 + − · · · . The lemma follows by verifying that when n ≥ k 2 , in the above binomial expansion, the sum of every consecutive positive/negative pair of terms in non-negative. Lemma 2. For n ≥ 2k 2 > 0, we have n k ≤ 2n k . Proof. This follows immediately from the previous lemma. Next, we recall the notion of the Stirling number of the second kind, which is the number of ways to partition a set of l objects into k non-empty subsets, and is denoted { l k} . We use the following standard result: l∑ { l n k} k = n l . (1) k=1 Finally, we define M 2n to be the number of perfect matchings in the complete graph on 2n vertices, and M n,n to be the number of perfect matchings on the complete bipartite graph on two sets of n vertices. The following facts are easy to establish: and M n,n = n! (2) M 2n ≤ 2 n n!. (3) We now turn to the proof of the theorem. We have f(τ) 2r = f(τ) r f(¯τ) r = ∑ f i1 · · · f i2r · τ i1 · · · τ ir · τ −i r+1 · · · τ −i 2r . i 1 ,...,i 2r We will extend the usual notion of expected values to complex-valued random variables: if U and V are real-valued random variables, then E[U + V i] = E[U] + E[V ]i. The usual rules for sums and products of expectations work equally well. By linearity of expectation, we have E[f(τ) 2r ] = ∑ E[f i1 · · · f i2r ] · τ i1 · · · τ ir · τ −i r+1 · · · τ −i 2r . (4) i 1 ,...,i 2r 38
Here, each index i t runs over the set {0, . . . , m − 1}. In this sum, because of independence and the fact that any odd power of f i has expected value 0, the only terms that contribute a non-zero value are those in which each index value occurs an even number of times, in which case, if there are k distinct values among i 1 , . . . , i 2r , we have E[f i1 · · · f i2r ] = (H/m) k . We want to regroup the terms in (4). To this end, we introduce some notation: for an integer t ∈ {1, . . . , 2r} define w(t) = 1 if t ≤ r, and w(t) = −1 if t > r; for a subset e ⊆ {1, . . . , 2r}, define w(e) = ∑ t∈e w(t). We call w(e) the “weight” of e. Then we have: ∑ E[f(τ) 2r ] = (H/m) ∑ k ′ τ j 1w(e 1 )+···+j k w(e k ) . (5) j 1 ,...,j k P ={e 1 ,...,e k } Here, the outer summation is over all “even” partitions P = {e 1 , . . . , e k } of the set {1, . . . , 2r}, where each element of the partition has an even cardinatilty. The inner summation is over all sequences of indices j 1 , . . . , j k , where each index runs over the set {0, . . . , m − 1}, but where no value in the sequence is repeated — the special summation notation ∑ ′ j 1 ,...,j k emphasizes this restriction. Since |τ| = 1, it is clear that ∑ ∣ E[f(τ)2r ] − (H/m) ∑ k ∣ ≤ ∑ (H/m) k (m k − m k ) (6) P ={e 1 ,...,e k } j 1 ,...,j k τ j 1w(e 1 )+···+j k w(e k ) P ={e 1 ,...,e k } Note that in this inequality the inner sum on the left is over all sequences of indices j 1 , . . . , j k , without the restriction that the indices in the sequence are unique. Our first task is to bound the sum on the right-hand side of (6). Observe that any even partition P = {e 1 , . . . , e k } can be formed by merging the edges of some perfect matching on the complete graph on vertices {1, . . . , 2r}. So we have ∑ ∑ (H/m) k (m k − m k ) ≤ (H/m) k k 2 m k−1 (by Lemma 1) P ={e 1 ,...,e k } Combining this with (6), we have ∑ ∣ E[f(τ)2r ] − P ={e 1 ,...,e k } P ={e 1 ,...,e k } ≤ r2 m ≤ r2 m M 2r ≤ r2 2 r r! m ∑ P ={e 1 ,...,e k } ≤ r2 2 r+1 r! m r∑ k=1 r∑ k=1 H k { r k} H k { r k} H k (by (3)) r∑ k=1 = r2 2 r+1 r! m Hr (by 1). (H/m) k ∑ (partitions formed from matchings) { r k} H k (by Lemma 2) j 1 ,...,j k τ j 1w(e 1 )+···+j k w(e k ) 39 ∣ ≤ r!Hr · 2r+1 r 2 m . (7)
Page 1 and 2: Design and Implementation of a Homo
Page 3 and 4: Organization of This Report We begi
Page 5 and 6: mod q l . A polynomial a ∈ A q is
Page 7 and 8: On subsequent calls with the same p
Page 9 and 10: void mapToSlots(vector& maps, const
Page 11 and 12: 2.6.2 The IndexMap class The class
Page 13 and 14: void addPrimes(const IndexSet& s);
Page 15 and 16: void sampleGaussian(double stdev=3.
Page 17 and 18: ool isOne() const; // does it point
Page 19 and 20: variable s(τ m ) (defined over the
Page 21 and 22: space modulus p), we need to conver
Page 23 and 24: with ˜c 2 = [ ∑ j c′′ j ] qQ
Page 25 and 26: Automorphism. “Raw” automorphis
Page 27 and 28: all the a i ’s are derived. The b
Page 29 and 30: from t in G i , then keySwitchMap[i
Page 31 and 32: void addAllMatrices(FHESecKey& sKey
Page 33 and 34: and X ↦→ X f k−ord(f i ) i an
Page 35 and 36: (since the plaintext values can be
Page 37 and 38: *** Perform homomorphic operations
Page 39: References [1] L. I. Bluestein. A l

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