On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chac\'on proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.


Introduction
Let n be a positive integer and let ζ 1 , . . . , ζ m be the primitive nth roots of unity, where m := ϕ(n) is the Euler's totient function of n. Moreover, let V n be the Vandermonde matrix associated with the nth cyclotomic polynomial, that is, is the Frobenius norm of A and A * is the conjugate transpose of A.
Recently, Blanco-Chacón [1] gave some upper bounds for the condition number of V n . This in order to prove the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some infinite families of cyclotomic number fields (see also [2,4,5]).
Our first result is the following.
Theorem 1.1. For every positive integer n, we have where rad(n) denotes the product of all prime factors of n.
Our second result is a formula for the condition number of V n when n is a prime power or a power of 2 times an odd prime power. Theorem 1.2. If n = p k , where k is a positive integer and p is a prime number, or if n = 2 k p ℓ , where k, ℓ are positive integers and p is an odd prime number, then In particular, Theorem 1.2 improves the upper bound Cond(V n ) ≤ 2(p − 1)ϕ(n) given by Blanco-Chacón in the case in which n = p k is a prime power [1, Theorem 3.9].
Our proofs of Theorems 1.1 and 1.2 are based on the study of the Gram matrix G n := V * n V n . Regarding that, we give also the following result. Theorem 1.3. For every positive integer n, the matrix n G −1 n has integer entries. From a number-theoretic point of view, it might be of some interest trying to describe the entries of n G −1 n explicitely, or at least understand the integer sequence Tr(n G −1 n ) n≥1 (which is related to Cond(V n ) by (3) below). CrypTO, the group of Cryptography and Number Theory of Politecnico di Torino. E. Signorini is supported by Telsy S.p.A.

Proofs
For every positive integer n, the Ramanujan's sums modulo n are defined by for all integers t. It is easy to check that c n (·) is an even periodic function with period n. Moreover, the following formula holds [3,Theorem 272] (1) where µ is the Möbius function and (n, t) denotes the greatest common divisor of n and t. Let G n := V * n V n be the Gram matrix of V n . By the previous considerations, we have In particular, G n is a symmetric Toeplitz matrix with integer entries. Let σ 1 , . . . , σ s be the distinct eigenvalues of G n , which are real and positive, since G n is the Gram matrix of an invertible matrix, and let µ 1 , . . . , µ s be their respective multiplicities. We have Therefore, the study of Cond(V n ) is equivalent to the study of the eigenvalues of G n . The next lemma relates the characteristic polynomials of G n and G rad(n) .
Proof. We know from (2) that G n = c n (i − j) 0≤i,j<m , where we shifted the indices i, j to the interval [0, m) since this does not change the differences i−j and simplifies the next arguments. Write the integers i, j ∈ [0, m) in the form i = hi ′ + i ′′ and j = hj ′ + j ′′ , where i ′ , j ′ ∈ [0, m ′ ) and i ′′ , j ′′ ∈ [0, h) are integers. By (1) we have that c n (i − j) = 0 if and only if h divides i − j (otherwise, n/(n, i − j) is not squarefree), which in turn happens if and only if i ′′ = j ′′ . In such a case, we have (n, i − j) = h(n ′ , i ′ − j ′ ) and, again by (1), it follows that Therefore, we have found that G n consists of m ′ × m ′ diagonal blocks of sizes h × h. Precisely, where ⊗ denotes the Kronecker product. Consequently, the characteristic polynomial of G n is as claimed.
Now we are ready to prove the first result.
We need a couple of preliminary lemmas to the proof of Theorem 1.2.
Lemma 2.2. For every odd positive integer n, the matrices G 2n and G n have the same eigenvalues (with the same multiplicities).
Proof. It is known [3,Theorem 67] that Ramanujan's sums are multiplicative functions respect to their moduli, that is, c ab (t) = c a (t) c b (t) for all coprime positive integers a, b. Moreover, it is easy to check that c 2 (t) = (−1) t . Thus, (2) gives where J is the m × m matrix alternating +1 and −1 on its diagonal and having zeros in all the other entries. Therefore, G n and G 2n are similar and consequently they have the same eigenvalues.
Lemma 2.3. Given two complex numbers a and b, the determinant of the k × k matrix Proof. Subtracting the last row from all the other rows, and then adding to the last column all the other columns, the matrix becomes Laplace expansion along the last column gives the desired result.

2.2.
Proof of Theorem 1.2. First, let us consider n = p k , where k is a positive integer and p is a prime number. It follows from (1) that c p (t) = p − 1 if p divides t, while c p (t) = −1 otherwise. Hence, using Lemma 2.3, we have so that the eigenvalues of G p are p and 1, with respective multiplicities p − 2 and 1. As a consequence, (3) gives and, thanks to Theorem 1.1, we obtain as claimed. Now assume that n = 2 k p ℓ , where k, ℓ are positive integers and p is an odd prime number. From Lemma 2.2 and (3) it follows at once that Cond(V 2p ) = Cond(V p ). Hence, Theorem 1.1 and (4) yield The next lemma is the well known orthogonality relation between the roots of unity.
Recalling that G n = V * n V n , we have G −1 n = V −1 n V −1 n * . Hence, also using Lemma 2.4, the S i,ℓ S j,ℓ , which is an integer.