Modular multiplication without trial division

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present a method for multiplying two integers (called <italic>N-residues</italic>) modulo <italic>N</italic> while avoiding division by <italic>N</italic>. <italic>N</italic>-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one <italic>N</italic>. The addition and subtraction algorithms are unchanged.