@article {IOPORT.05920424,
    author       = {Cenk, Murat and \"Ozbudak, Ferruh},
    title        = {Multiplication of polynomials modulo $x^{n}$.},
    year         = {2011},
    journal      = {Theoretical Computer Science},
    volume       = {412},
    number       = {29},
    issn         = {0304-3975},
    pages        = {3451-3462},
    publisher    = {Elsevier Science Publishers, Amsterdam},
    doi          = {10.1016/j.tcs.2011.02.031},
    abstract     = {Summary: Let $n,\ell $ be positive integers with $\ell \leq 2n - 1$. Let $\cal R$ be an arbitrary nontrivial ring, not necessarily commutative and not necessarily having a multiplicative identity, and let $\cal R[x]$ be the polynomial ring over $\cal R$. In this paper, we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most $(n - 1)$ modulo $x^{n}$ over $\cal R$. Moreover, we introduce a new complexity notion, the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most $(n - 1)$ modulo $x^{\ell }$ over $\cal R$. This new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most $(n - 1)$ modulo $x^{n}$ over $\cal R$.},
    identifier   = {05920424},
}