Residue number system

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 is the product of the moduli, there is, in an interval of length , exactly one integer 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 (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. 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 formula_5. 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 formula_6 in the interval formula_7. For signed numbers, the dynamic range is formula_8 (when formula_9 is even, generally an extra negative value is represented).
Arithmetic operations.
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 formula_11 and formula_12 is the integer represented by formula_13 such that
for (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.
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.
However, operations such as magnitude comparison, sign computation, overflow detection, scaling, and division are difficult to perform in a residue number system.
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. On the other hand, if formula_15 is coprime with formula_9 (that is formula_17) then
can be easily calculated by
where formula_20 is multiplicative inverse of formula_15 modulo formula_9, and formula_23 is multiplicative inverse of formula_24 modulo formula_5.
Applications.
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.