A residue numeral system (RNS) 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 M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
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.
A residue numeral system is defined by a set of k integers called the moduli, which are generally supposed to be pairwise coprime (that is, any two of them have a greatest common divisor equal to one).
Residue number systems have been defined for non-coprime moduli, but are not commonly used because of worse properties.
[1] An integer x is represented in the residue numeral system by the set of its remainders under Euclidean division by the moduli.
Two integers whose difference is a multiple of M have the same representation in the residue numeral system defined by the mis.
More precisely, the Chinese remainder theorem asserts that each of the M different sets of possible residues represents exactly one residue class modulo M. That is, each set of residues represents exactly one integer
For signed numbers, the dynamic range is
is even, generally an extra negative value is represented).
[2] 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 x and y, respectively represented by the residues
such that for i = 1, ..., k (as usual, mod denotes the modulo operation consisting of taking the remainder of the Euclidean division by the right operand).
Subtraction and multiplication are defined similarly.
However, operations such as magnitude comparison, sign computation, overflow detection, scaling, and division are difficult to perform in a residue number system.
At the opposite, testing inequalities (x < y) 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 as Euclidean division and Euclidean algorithm.
Division in residue numeral systems is problematic.
RNS have applications in the field of digital computer arithmetic.
By decomposing in this a large integer into a set of smaller integers, a large calculation can be performed as a series of smaller calculations that can be performed independently and in parallel.