On the Scaled Inverse of $(x^i-x^j)$ modulo Cyclotomic Polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$

Abstract. The scaled inverse of a nonzero element a(x) ∈ Z[x]/f(x), where f(x) is an irreducible polynomial over Z, is the element b(x) ∈ Z[x]/f(x) such that a(x)b(x) = c (mod f(x)) for the smallest possible positive integer scale c. In this paper, we investigate the scaled inverse of (x − x) modulo cyclotomic polynomial of the form Φps(x) or Φpsqt(x), where p, q are primes with p < q and s, t are positive integers. Our main results are that the coefficient size of the scaled inverse of (x − x) is bounded by p−1 with the scale p modulo Φps(x), and is bounded by q−1 with the scale not greater than q modulo Φpsqt(x). Our results have applications in cryptography: zero-knowledge proofs regarding lattice-based cryptosystems. Along the way of proving the theorems, we prove several properties of {x}k∈Z in Z[x]/Φpq(x) which might be of independent interest.