A residue numeral system is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if is the product of the moduli, there is, in an interval of length, exactly oneinteger having any given set of modular values. The arithmetic of a residue numeral system is also called multi-modular arithmetic. Multi-modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken into account. Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography.
Definition
A residue numeral system is defined by a set of integers called the moduli, which are generally supposed to be pairwise coprime. Residue number systems have been defined for non-coprime moduli, but are not commonly used because of worse properties. Therefore, they will not be considered in the remainder of this article. An integer is represented in the residue numeral system by the set of its remainders under Euclidean division by the moduli. That is and for every Let be the product of all the. Two integers whose difference is a multiple of have the same representation in the residue numeral system defined by the s. More precisely, the Chinese remainder theorem asserts that each of the different sets of possible residues represents exactly one residue class modulo. That is, each set of residues represents exactly one integer in the interval. In applications where one is also interested with negative integers, it is often more convenient to represent integers belonging to an interval centered at 0. In this case, if is odd, each set of residues represents exactly one integer of absolute value at most.
For adding, subtracting and multiplying numbers represented in a residue number system, it suffices to perform the same modular operation on each pair of residues. More precisely, if is the list of moduli, the sum of the integers and, respectively represented by the residues and is the integer represented by such that for . Subtraction and multiplication are defined similarly. For a succession of operations, it is not necessary to apply the modulo operation at each step. It may be applied at the end of the computation, or, during the computation, for avoiding overflow of hardware operations.
Comparison
If two integers are equal, then all their residues are equal. Conversely, if all residues are equal, then the two integers are equal, or their differences is a multiple of. It follows that testing equality is easy. At the opposite, testing inequalities is difficult and, usually, requires to convert integers to the standard representation. As a consequence, this representation of numbers is not suitable for algorithms using inequality tests, such Euclidean division and Euclidean algorithm.
Division
Division in residue numeral systems is problematic. A paper describing one possible algorithm is available at . On the other hand, if is coprime with then can be easily calculated by where is multiplicative inverse of modulo, and is multiplicative inverse of modulo.
