Homomorphism

Full Title or Meme

A Homomorphism is a concept from algebra that describes a structure-preserving map between two algebraic structures of the same type.

Context

• The word Homomorphism comes from the Ancient Greek language: "ὁμός" (homos) meaning "same" and "μορφή" (morphe) meaning "form" or "shape".
• Essentially, a homomorphism is a map between two sets equipped with the same structure, such that it preserves the operations of the structures. In other words, if we have an operation (usually a binary operation) defined on both sets, the homomorphism ensures that this operation behaves consistently across the two sets.
• Here are some examples:
  • A **semigroup homomorphism** is a map between semigroups that preserves the semigroup operation.
  • A **monoid homomorphism** is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid.
  • A **group homomorphism** is a map between groups that preserves the group operation, including mapping the identity element of the first group to the identity element of the second group.

Homomorphic Encryption

An interesting application of Homomorphisms in the field of cryptography called homomorphic encryption. It allows computations to be performed on encrypted data without first decrypting it. The result of these computations remains encrypted, and when decrypted, it produces the same output as if the operations were performed on the unencrypted data.