The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices

. Let Λ be an n -dimensional lattice. For any n -dimensional vector c and positive real number s , let D s, c and D Λ ,s, c denote the continuous Gaussian distribution and the discrete Gaussian distribution over Λ , respectively. In this paper, we establish the exact relationship between the second and fourth moments centered around c of the discrete Gaussian distribution D Λ ,s, c and those of the continuous Gaussian distribution D s, c , respectively. Tis provides a quantization form of the result obtained by Micciancio and Regev on the second and fourth moments of discrete Gaussian distribution. Using the relationship, we also derive an uncertainty principle for Gaussian functions, which extend the result of Zheng, Zhao, and Xu. Our proof is based on combination of the idea of Micciancio and Regev and the idea of Zheng, Zhao, and Xu, where the main tool is high-dimensional Fourier transform.


Introduction
An n-dimensional lattice Λ ⊆ R n is an additive subgroup of R n generated by n linearly independent vectors b 1 , . . ., b n ∈ R n .Te dual lattice  Λ of Λ is defned to be where 〈x, y〉 denotes the canonical inner product of x and y (see, for example, [1,2]).Given parameters s > 0 and c ∈ R n , let ρ s,c (x) � e − π (x− c)/s ‖ ‖ 2 with x being any n-dimensional vector.As in [3,4], the probability density function of the continuous Gaussian distribution around c with parameter s is defned by D s,c (x) � ρ s,c (x)/s n for any x ∈ R n .By direct calculation, one can obtain that the second and fourth central moments of the random variable subject to D s,c are (ns 2 /2π) and (3ns 4 /4π 2 ), respectively.Te discrete Gaussian distribution over Λ is defned for any x ∈ Λ by where ρ s,c (S) �  x∈S ρ s,c (x) for any countable subset S ⊆ R n (see [3][4][5] for a more in-depth discussion on D Λ,s,c ).
In [6], Banaszczyk initiated the study of discrete Gaussian distributions and established several measure inequalities to prove transference theorems for lattices.In [7,8], Banaszczyk obtained more measure inequalities for convex bodies, which are very useful in the study of latticebased cryptography.In [3,4], Micciancio and Regev introduced an important lattice parameter named as smoothing parameter, which is defned for any ϵ > 0 and any lattice Λ by Subsequently, Micciancio and Regev [3,4] proved that D Λ,s,c has very good statistical properties if s is large enough relatively to η ϵ (Λ).For more detailed background information and development on the study of smoothing parameter and discrete Gaussian distribution, we refer readers to the important papers [3,4,[9][10][11][12][13][14][15][16].

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In this paper, we mainly focus on the statistical properties of discrete Gaussian distributions.Our motivation is to provide a quantization form of the result obtained by Micciancio and Regev [3,4] on the second and fourth moments of discrete Gaussian distribution D Λ,s,c .In particular, we shall establish the exact relationship between the second and fourth moments centered around c of discrete Gaussian distribution D Λ,s,c and the second and fourth moments centered around c of the continuous Gaussian distribution D s,c .Using the relationship, we can also derive an uncertainty principle for Gaussian functions ρ s,c on lattices.As in [1,5], for a complex function f on R n , the Fourier transform of f is defned by  f(y) �  R n f(x)e − 2πi〈x,y〉 dx.Uncertainty principle plays an important role in harmonic analysis, quantum mechanics, and time-frequency analysis.Te classical n-dimensional Heisenberg uncertainty principle for continuous Fourier transform with respect to a rapidly decreasing function f: R n ⟶ C can be stated as the following inequality (see [17,18]): Te uncertainty principle tells us that the quantities 2 dt dy both cannot be too small.Tat is, a function f and its Fourier transform  f both cannot be essentially localized.
Te celebrated Donoho-Stark uncertainty principle on the cyclic group Z N [19] states that, for a function where supp(f) � j ∈ Z N : f(j) ≠ 0   and S | | denotes the cardinality of a set S. If the group becomes G � Z/pZ for some prime p, Tao [20] then improved the above estimate as For simplicity, we write ρ s,0 (x) by ρ s (x) for any s > 0. Motivated by the technique developed by Banaszczyk in [6], Zheng et al. proved the following simple form of uncertainty principle for Gaussian functions ρ s on lattices by using Fourier analysis in their important work [16].
Theorem 2. Let Λ be an n-dimensional lattice and s > 0. We have Notice that the quantity (ns 2 /2π) in Teorem 2 is just the second moment of the continuous Gaussian distribution D s,0 .Tat is, Teorem 2 gives the exact diference between the second moment of D Λ,s,0 and the second moment of D s,0 .In addition, Teorem 2 also establishes the precise relationship between the second moment of the discrete Gaussian distribution D Λ,s,0 over the lattice Λ and the second moment of discrete Gaussian distribution D  Λ,1/s,0 over the dual lattice  Λ of Naturally, one expects to determine the exact diferences between the second and fourth moments centered around c of the discrete Gaussian distribution D Λ,s,c and those of the continuous Gaussian distribution D s,c and to establish an uncertainty principle for general Gaussian functions ρ s,c on lattices.Combining the idea of the proof of Lemma 4.2 in [3,4] by Micciancio and Regev with the technique developed by Zheng et al. in [16], we prove two equalities connecting the second and fourth moments centered around c of the discrete Gaussian distribution D Λ,s,c and the second and fourth moments centered around c of the continuous Gaussian distribution D s,c , respectively.Our main tool is highdimensional Fourier transform.In addition, we establish an uncertainty principle for Gaussian functions ρ s,c on lattices.We are now in a position to state our main result.Theorem 3.For any n-dimensional lattice Λ, vector c ∈ R n , and real s > 0, we have and 2

Journal of Mathematics
Obviously, when c � 0, equality (11) in Teorem 3 becomes Zheng-Zhao-Xu Teorem (see Teorem 1 of [16]).But we cannot call (11) the uncertainty principle since  ρ s,c (x) � s n ρ 1/s (x)e − 2πi〈x,c〉 may take complex values.Let Re(  ρ s,c (y)) denote the real part of  ρ s,c (y).With the help of the facts that  ρ s,c (y)         ⩾ Re(  ρ s,c (y)) and  ρ s,c (  Λ) � det(Λ)ρ s,c (Λ) > 0, one can easily deduce the following result by (11), which can be called the uncertainty principle for Gaussian functions over lattices.Tis extends the uncertainty principle obtained by Zheng et al. in [16] from the Gaussian function ρ s,0 to the general Gaussian function ρ s,c .Theorem 4. For any n-dimensional lattice Λ, vector c ∈ R n , and real s > 0, we have Equality (11) in Teorem 3 implies that the exact difference between the second moment centered around c of the discrete Gaussian distribution D Λ,s,c and the second moment centered around c of the continuous Gaussian distribution D s,c is just equal to s 4 Equality (12) in Teorem 3 implies that the exact diference between the fourth moment centered around c of the discrete Gaussian distribution D Λ,s,c and the fourth moment centered around c of the continuous Gaussian distribution D s,c is just equal to 3s 6

Proof of Theorem 4
In the following, n is a fxed positive integer.We frst give the following useful properties of high-dimensional Fourier transform.
Lemma 5 (see Lemmas 1.1.2and 1.2.1 of [5]).Suppose that f(x) and g(x) are two complex functions defned on R n .We have the following: (ii) For any vectors c, x, and s > 0, we have For the purpose of establishing the relationship between the second and fourth moments centered around c of the discrete Gaussian distribution D Λ,s,c and those of the continuous Gaussian distribution D s,c , we need the following Poisson summation formula which has been widely used in the theory of lattices.Lemma 6 (see Theorem 2.3 of [1]).Let Λ be an n-dimensional lattice, and let f: R n ⟶ C be a function which satisfes the following conditions (V1), (V2), and (V3): Ten, we have where vol(R n /Λ) denotes the volume of the fundamental parallelepiped of Λ.
Let ξ be a random vector subject to the discrete Gaussian distribution D Λ,s,c .From Teorem 1, we know that the mean value of the random vector ξ subject to D Λ,s,c is very close to c.However, we still cannot determine the exact value of Exp ξ∼D Λ,s,c (ξ) to this day.Terefore, to prove Teorem 3, we cannot estimate the second and fourth moments of the discrete Gaussian distribution D Λ,s,c by direct computation such as the continuous Gaussian distribution.Inspired by the idea of Micciancio and Regev in Lemma 4.2 of [3,4] and the idea of Zheng, Zhao, and Xu in their Teorem 1 of [16], we can now give the proof of Teorem 3 as follows.
Proof of Teorem 3. Let u ∈ R n be a unit vector.We consider the following function: Calculating the frst derivative, the second derivative, and the fourth derivative of the function F(t) directly, we get and Let f(x) � e 2πit〈x− c,u〉 ρ s,c (x) � e − 2πit〈c,u〉 ρ s,c (x)e 2πi〈x,tu〉 .It is easy to check that f(x) satisfes the conditions (V1), (V2), and (V3) in Lemma 6.Hence, by Lemmas 5 and 6, we obtain Journal of Mathematics Now, computing the frst derivative, the second derivative, and the fourth derivative of F(t) according to the above expression, we derive from u ‖ ‖ � 1 that (24) Now, applying Lemma 5 (ii) and dividing both sides of (24) by 4π 2 , we get Similarly, taking u to be e j for 1 ⩽ j ⩽ n in ( 19) and ( 22) successively, and adding them up, respectively, we consequently obtain by setting t � 0 that 16π 4 Dividing both sides of (26) by 16π 4 , we obtain as desired.Tis completes the proof of Teorem 3.

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Applying the same method as in the proof of Teorem 3, we can also obtain the exact diference between the higher moments of discrete Gaussian distribution and those of continuous Gaussian distribution.But the explicit form would be more complex and the analysis will become quite cumbersome.For more general argument concerning the estimate of pth moment of discrete Gaussian distribution, the readers can refer to [13].Applying Teorems 1 and 3, we have the following result.
Corollary .Let Λ be an n-dimensional lattice and ϵ ∈ (0, 1).Let s be a real number greater than or equal to 2η ϵ (Λ).Ten, for any vector c ∈ R n , we have