# homomorphism

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Related to homomorphic: Homeomorphic

## homomorphism

[‚hä·mə′mȯr‚fiz·əm]
(botany)
Having perfect flowers consisting of only one type.
(mathematics)
A function between two algebraic systems of the same type which preserves the algebraic operations.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Homomorphism

a concept of mathematics and logic that first appeared in algebra but proved to be very important in understanding the structure and the area of possible applications of other branches of mathematics. The concept of homomorphism applies to a set of objects with prescribed operations (or relations). Thus, a homomorphism (homomorphic mapping) of a group G onto a group H is a mapping that associates to every element G∈G a definite element h∈H (the image of g) and satisfies the requirements that every element of H is the image of some element in G, and the image of the product (sum) of two elements in G is the product (sum) of their images. For example, the mapping that associates to an integer a the remainder when a is divided by a fixed positive integer m is a homomorphism of the group of integers (under addition) onto the group of residues modulo m. (The latter consists of m elements represented by the remainders 0, 1, . . . , m - 1.) The sum of two elements is represented by the sum of the corresponding remainders possibly diminished by m.

## homomorphism

A map f between groups A and B is a homomorphism of A into B if f(a1 * a2) = f(a1) * f(a2) for all a1,a2 in A.

where the *s are the respective group operations.
References in periodicals archive ?
How Homomorphic Encryption Works: HE allows AI computation on encrypted data, enabling data scientists and researchers to gain valuable insights without decrypting or exposing the underlying data or models.
A Study on security and privacy issues using encryption techniques in cloud computing Proposed Work Focus References Intelligent Security and privacy [17] Cryptography Homomorphic Security and privacy [19] encryption Attribute based Privacy [18], [20], [27] Encryption Data Sharing Security and privacy [22], [23], [26] Attribute based Privacy and Storage Management [21] Storage Privacy Preserving Privacy [24] Key Update Method Security [25] Table 5.
The scheme is homomorphic under addition and multiplication.
* In Partially Homomorphic Encryption (PHE), Eval can perform only one type of operation (e.g., either addition or multiplication), though the number of operations performed is unlimited.
[10] proposed proof of ownership and retrievability (PoOR) based on the Merkle hash tree and homomorphic verifiable tags.
The homomorphic encryption algorithm is the data obfuscation algorithm in code obfuscation [18].
Sousa, "Enhancing data parallelism of fully homomorphic encryption," in International Conference on Information Security and Cryptology, Springer, Cham, 2016.
If mutually distrusting countries do not want to share their trajectory information, homomorphic encryption could allow both countries to compute together a result that would provide them a probability of collision "without revealing the details of those satellite trajectories," Archer said.
We first preprocess the face image by using a homomorphic filtering (HF) based illumination normalization algorithm [7].
Asharov proposed the threshold homomorphic encryption scheme to improve efficiency of the privacy protection algorithm [8].
We shall call f([PHI]) the homomorphic image of O under f.

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