ModularInverse
ModularInverse [k,n]
gives the modular inverse of k modulo n.
Details
- ModularInverse is also known as modular multiplicative inverse.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- Typically used in modular arithmetic and cryptography.
- ModularInverse [k,n] gives the number r such that the remainder of the division of r k by n is equal to 1.
- If k and n are not coprime, no modular inverse exists and ModularInverse [k,n] remains unevaluated.
Examples
open allclose allBasic Examples (2)
Compute the inverse of 3 modulo 5 and check the result:
Plot the sequence with a fixed modulus:
Scope (2)
Numerical Evaluation (2)
Compute using integers:
Gaussian integers:
Compute using large integers:
Applications (4)
Basic Applications (2)
Two numbers are modular inverses of each other if their product is 1:
Modular computation of a matrix inverse:
First compute the matrix adjoint:
Then compute the modular inverse of a matrix:
Check that the inverse gives the correct result:
Number Theory (2)
Build an RSA-like toy encryption scheme. Start with the modulus:
Find the universal exponent of the multiplication group modulo n:
Private key:
Public key:
Encrypt a message:
Decrypt it:
Create a random number generator that uses the current time as a seed:
Choose modulus parameters:
Compute 20 random numbers between 0 and 1:
Properties & Relations (6)
ModularInverse is a periodic function:
ExtendedGCD returns modular inverses:
Compute using PowerMod :
The results have the same sign as the modulus:
If and are coprime, then is invertible modulo :
Computing ModularInverse twice yields the original argument:
Possible Issues (1)
For nonzero integers k and n, ModularInverse [k,n] exists if and only if k and n are coprime:
However, 10 and 22 are not coprime:
Interactive Examples (1)
Visualize inverses modulo varying prime numbers:
Neat Examples (2)
Visualize when a number is invertible modulo 12:
Modular inverses of sums of two squares:
See Also
PowerMod Mod Power PowerModList ExtendedGCD PolynomialExtendedGCD MultiplicativeOrder EulerPhi PrimitiveRoot CoprimeQ
Function Repository: FractionMod
Related Guides
Related Links
History
Text
Wolfram Research (2017), ModularInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/ModularInverse.html.
CMS
Wolfram Language. 2017. "ModularInverse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ModularInverse.html.
APA
Wolfram Language. (2017). ModularInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ModularInverse.html
BibTeX
@misc{reference.wolfram_2025_modularinverse, author="Wolfram Research", title="{ModularInverse}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/ModularInverse.html}", note=[Accessed: 27-April-2025 ]}
BibLaTeX
@online{reference.wolfram_2025_modularinverse, organization={Wolfram Research}, title={ModularInverse}, year={2017}, url={https://reference.wolfram.com/language/ref/ModularInverse.html}, note=[Accessed: 27-April-2025 ]}