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Mathematica > Mathematics and Algorithms > Number Theory > Diophantine Equations >

Built-in Mathematica Symbol

ChineseRemainder

ChineseRemainder[{r₁, r₂, ...}, {m₁, m₂, ...}]
gives the smallest positive x that satisfies all the integer congruences x mod m_i = r_i mod m_i.

The smallest positive integer x that satisfies

and

Find the smallest positive integer giving remainder

when divided by

The smallest positive integer x that satisfies

and

Find the smallest positive integer giving remainder

when divided by

Database encryption and decryption:

Key generation:

Encrypted data:

Decryption:

Define a residue number system:

Numbers and their representation in a residue system:

Multiplying and recovering in the residue system:

Adding and recovering:

Modular computation of a determinant:

Modular determinants:

Recover result:

Shift residue to be symmetric:

Solve congruential equations using Reduce or FindInstance:

Not all congruential equations have a solution:

A solution exists when Mod[r_i, GCD[m₁, m₂, ...]]

Mod[r_j, GCD[m₁, m₂, ...]]: