The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.
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The original form of the theorem, contained in a third-century AD book Sun Zi suanjing (孙子算经 The Mathematical Classic by Sun Zi) by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao, the Shushu Jiuzhang (數書九章 Mathematical Treatise in Nine Sections) is a statement about simultaneous congruences (see modular arithmetic).
Suppose n1, n2, …, nk are positive integers which are pairwise coprime. Then, for any given integers a1,a2, …, ak, there exists an integer x solving the system of simultaneous congruences
Furthermore, all solutions x to this system are congruent modulo the product N = n1n2…nk.
Hence for all
, if and only if
.
Sometimes, the simultaneous congruences can be solved even if the ni's are not pairwise coprime. A solution x exists if and only if:
All solutions x are then congruent modulo the least common multiple of the ni.
Versions of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202).
This algorithm only treats the situations where the 's are coprime. The method of successive substitution can often yield solutions to simultaneous congruences, even when the moduli are not pairwise coprime.
Suppose, as above, that a solution is needed to the system of congruences:
Again, to begin, the product is defined. Then a solution x can be found as follows.
For each i the integers and
are coprime. Using the extended Euclidean algorithm we can find integers
and
such that
. Then, choosing the label
, the above expression becomes:
Consider . The above equation guarantees that its remainder, when divided by
, must be 1. On the other hand, since it is formed as
, the presence of
guarantees that it's evenly divisible by any
so long as
.
Because of this, combined with the multiplication rules allowed in congruences, one solution to the system of simultaneous congruences is:
For example, consider the problem of finding an integer x such that
Using the extended Euclidean algorithm for 3 and 4×5 = 20, we find (−13) × 3 + 2 × 20 = 1, i.e. e1 = 40. Using the Euclidean algorithm for 4 and 3×5 = 15, we get (−11) × 4 + 3 × 15 = 1. Hence, e2 = 45. Finally, using the Euclidean algorithm for 5 and 3×4 = 12, we get 5 × 5 + (−2) × 12 = 1, meaning e3 = −24. A solution x is therefore 2 × 40 + 3 × 45 + 1 × (−24) = 191. All other solutions are congruent to 191 modulo 60, (3 × 4 × 5 = 60) which means that they are all congruent to 11 modulo 60.
NOTE: There are multiple implementations of the extended Euclidean algorithm which will yield different sets of ,
, and
. These sets however will produce the same solution i.e. 11 modulo 60.
For a principal ideal domain R the Chinese remainder theorem takes the following form: If u1, ..., uk are elements of R which are pairwise coprime, and u denotes the product u1...uk, then the quotient ring R/uR and the product ring R/u1R × ⋯ × R/ukR are isomorphic via the isomorphism
such that
This isomorphism is unique; the inverse isomorphism can be constructed as follows. For each i, the elements ui and u/ui are coprime, and therefore there exist elements r and s in R with
Set ei = s u/ui. Then the inverse of f is the map
such that
Note that this statement is a straightforward generalization of the above theorem about integer congruences: the ring Z of integers is a principal ideal domain, the surjectivity of the map f shows that every system of congruences of the form
can be solved for x, and the injectivity of the map f shows that all the solutions x are congruent modulo u.
The general form of the Chinese remainder theorem, which implies all the statements given above, can be formulated for commutative rings and ideals. If R is a commutative ring and I1, ..., Ik are ideals of R which are pairwise coprime (meaning that Ii + Ij = R whenever i ≠ j), then the product I of these ideals is equal to their intersection, and the quotient ring R/I is isomorphic to the product ring R/I1 x R/I2 x ... x R/Ik via the isomorphism
such that
In the RSA algorithm calculations are made modulo , where
is a product of two large prime numbers
and
. 1024-, 2048- or 4096-bit integers
are commonly used, making calculations in
very time-consuming. By the Chinese Remainder Theorem, however, these calculations can be done in the isomorphic ring
instead. Since
and
are normally of about the same size, that is about
, calculations in the latter representation are much faster. Note that RSA algorithm implementations using this isomorphism are more susceptible to fault injection attacks.
The Chinese Remainder Theorem may also be used to construct an elegant Gödel numbering for sequences, which is needed to prove Gödel's incompleteness theorems.
The Chinese remainder theorem does not hold in the non-commutative case. Consider the ring of non-commutative real polynomials in
and
. Let
be the principal two-sided ideal generated by
and
the principal two-sided ideal generated by
Then
but
Proof:
Observe that is formed by all polynomials with an
in every term and that every polynomial in
vanishes under the substitution
. Consider the polynomial
. Clearly
. Define a term in
as an element of the multiplicative monoid of
generated by
and
. Define the degree of a term as the usual degree of the term after the substitution
. On the other hand, suppose
. Observe that a term in
of maximum degree depends on
otherwise
under the substitution
can not vanish. The same happens then for an element
. Observe that the last
, from left to right, in a term of maximum degree in an element of
is preceded by more than one
. (We are counting here all the preceding
s. e.g. in
the last
is preceded by
s.) This proves that
since that last
in a term of maximum degree (
) is preceded by only one
. Hence
.
On the other hand, it is true in general that implies
. To see this, note that
, while the opposite inclusion is obvious. Also, we have in general that, provided
are pairwise coprime two-sided ideals in
, the natural map
is an isomorphism. Note that can be replaced by a sum over all orderings of
of their product (or just a sum over enough orderings, using inductively that
for coprime ideals
).