Homomorphic Cryptographic Algorithms for Edge Computing

Security: A Mathematical Model Review

R. S. S. Raju Battula and Lubna Ansari

Department of Computer Engineering & Applications, Mangalayatan University, Aligarh, Uttar Pradesh, India

Keywords: Homomorphic Encryption, Edge Computing Security, Algebraic Cryptography, Mathematical Models,

Lattice‑Based Cryptography, Ring Learning with Errors, Cryptographic Algorithms, Security Optimization,

Privacy‑Preserving Computation, Distributed Systems Security.

Abstract: Edge computing has been one of the transformative paradigms in distributed systems, requiring security

mechanisms to be robust for the protection of data at the periphery of a network. This review is hence an all-

embracing discussion on the mathematical basis of homomorphic cryptographic algorithms and their

application in edge computing scenarios. It is an in-depth investigation into the ways algebraic structures,

especially polynomial rings and lattice-based constructions, can be used to devise secure homomorphic

encryption schemes. The review comprises recent advances in homomorphic encryption (FHE) and semi-

homomorphic encryption (SHE), emphasis on mathematical models, high computational efficiency, and

practical challenges of implementing fully homomorphic encryption within resource-constrained Edge

environments. Based on a detailed analysis of Ring Learning with Errors (RLWE) and related mathematical

problems, we shall evaluate the security parameters and performance metrics that are critical for edge

computing applications. Our results show that although the currently available homomorphic encryption

schemes offer strong security guarantees, they suffer from important challenges in computational overhead

and resource optimization. We present a novel framework for assessing the trade-off between security strength

and computational efficiency, which is particularly relevant in the context of edge computing constraints.

Further on, we are discussing emerging mathematical approaches to noise management and parameter

optimization, which also gives some insight into further research directions. This review therefore contributes

to the field by synthesizing current mathematical approaches, identification of critical challenges, and a

proposal of possible solutions for moving forward with homomorphic cryptography applications in edge

computing.

1 INTRODUCTION

The rapid evolution of distributed computing systems

has ushered in the era of edge computing,

fundamentally transforming how data is processed

and secured in modern digital infrastructures. Among

these emergent paradigms, edge computing is

becoming increasingly crucial for handling the

increasing demands of Internet of Things (IoT)

devices, those applications that require real-time

processing, and most importantly, latency-sensitive

services. However, this shift toward decentralized

computing introduces complex security challenges in

protecting those pieces of sensitive data that are

processed at edge nodes. Traditional encryption

methods have the capability to secure data both at rest

and in transit but do not allow computation on

encrypted data, which sometimes needs decryption

processes in edge nodes that are potentially

vulnerable.

Homomorphic cryptography comes as a

revolutionary solution to these challenges because

homomorphic cryptography will allow direct

computation on encrypted data without the need for

decryption. First conceived by Rivets, Adelman, and

Detrusors in 1978, this capability had only been a

theoretical concept until Gentry proved in 2009 the

first fully homomorphic encryption scheme.

Homomorphic cryptography is important for edge

computing environments because it satisfies the

critical needs of maintaining privacy of data but at the

same time allows for distributed computation. These

cryptographic systems are based on advanced

algebraic structures, including polynomial rings,

534

Battula, R. S. S. R. and Ansari, L.

Homomorphic Cryptographic Algorithms for Edge Computing Security: A Mathematical Model Review.

DOI: 10.5220/0013916200004919

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 1st International Conference on Research and Development in Information, Communication, and Computing Technologies (ICRDICCT‘25 2025) - Volume 4, pages

534-540

ISBN: 978-989-758-777-1

lattice-based constructions, and learning with errors

problems, that altogether form the theoretical basis

for secure computation on encrypted data.

The recent advances in homomorphic

cryptography have been more focused on the

optimization of these mathematical models to be

implemented in resource-constrained edge

environments. These include techniques for noise

management, methods to efficiently choose the

parameters, and improved algorithms conceived for

specific applications of edge computing. However,

considerable challenges are the trade-offs between

security requirements with respect to efficiency in

computations since the processing power, memory

constraints, and bandwidth limitations are much

lower than those found in more powerful hosts.

Homomorphic cryptography for edge computing

needs to overcome these limitations while providing

solid security guarantees and support for real-time

processing.

This broad review analyses the mathematical

fundamentals underpinning homomorphic

cryptographic algorithms and their integration into

edge computing. We study theoretical contributions

regarding algebraic structures, practical strategies to

implement such designs, and methods to optimize the

performance. The current paper draws from a

comprehensive review of pertinent peer-reviewed

articles published from 2009 up to 2024 to include

seminal work alongside more recent efforts,

reviewing the current status of homomorphic

cryptography within the edge computing setting as

well as its promising areas of future development. It

is concerned with the mathematical models that allow

a safe computation over encrypted data, the resistance

of such models against implementation in resource-

constrained environments, and the trade-offs between

strength of security and efficiency in computation.

2 LITERATURE SURVEY

(Craig Gentry's, 2009) paper "A Fully Homomorphic

Encryption Scheme" introduces a ground-breaking

FHE method using ideal lattices and noisy encoding,

enabling computations on encrypted data. It is based

on ring learning with errors hardness and it employs

bootstrapping to support the evaluation of arbitrary

circuits on ciphertexts, so FHE seems feasible.

Zhang et al. (2023) survey challenges to security

edge computing, focusing on data breach, insider

attacks, and denial-of-service. Solutions such as

encryption and access control are overviewed,

although they stress on holistic approaches; the future

work is proposed also.

The additive and multiplicative operations

combined for edge computing use a homomorphic

encryption scheme without much overhead: Wang

and Song (2022). This implies its efficiency,

especially when carried out on resource-constrained

devices in comparison with previously proposed

schemes.

In a 2021 paper titled "Lattice-Based FHE as

Secure as PKE" by Brake ski and Vaikuntanathan,

this breakthrough in lattice-based schemes for

homomorphic encryption provably attains the same

level of security as public-key encryption schemes. It

is based on the hardness of the learning with errors

problem and is secure against attacks trying to break

the encryption.

Smith (2022) discusses mathematical models for

secure edge computing focusing on the areas of data

confidentiality and integrity. He provides a

framework with cryptography, game theory, and

information theory, hereby indicated open research

directions.

The research work gives insight into the

feasibility of homomorphic encryption in IoT

applications and various avenues where future

research is needed for improvement in the

performance and efficiency of this process.

Anderson, M., et al. (2023).

Park and Kim (2022) propose an optimization

technique to enhance homomorphic encryption

performance on resource-limited edge devices. Their

approach eliminates unnecessary computational

overhead and reduces memory usage with proof of

experimentation, promoting efficient encryption for

edge computing applications.

Wilson et al. survey security models on edge

computing in 2023 with respect to threats like data

breaches, insider attacks, and DoS. They mention

cryptographic techniques, access control, and

intrusion detection and highlight research directions

in this area.

Yang and Thompson present an efficient fully

homomorphic encryption (FHE) implementation for

resource-constrained environments in 2023. This

reduces the computational overhead and memory

usage. Experiments validate the effectiveness of this

approach for such environments.

3 MATHEMATICAL MODELS

Cryptographic algorithms are adopted based on some

mathematical models such as number theories,

Homomorphic Cryptographic Algorithms for Edge Computing Security: A Mathematical Model Review

535

algebra and combinatoric for ensuring its

confidentiality, integrity and authenticity as data

encryption depends upon ciphers and key exchanges,

integrity may be realized as source and integrity

authentication through Digital Signatures in the

model related to number theorem, message

authenticity code in case of algebra while information

theory presents the hash models and other similar in

kind. Digital certificates, public-key infrastructure,

and biometric authentication will ensure authenticity

through mathematical models such as number theory,

algebra, and combinatorics in verifying the identity of

devices or entities and authenticity of data. This will

ensure integrity of data as well as devices that process

such data. The mathematics behind these models will

allow safe processing and forwarding of sensitive

information at the edge nodes.

3.1 Algebraic Structures

Homographic cryptography relies on progressive

algebraic structures, by which homomorphic

procedures can straight be accepted out on the

ciphertexts. In simple words, homographic

encryption essentially operates over rings and fields

that provide some homomorphic properties. Assume

R to be a ring, and E as an encryption algorithm. A

homomorphic scheme over the messages m₁, m₂ ∈ R

should satisfy:

𝐸m₁ ⊕ Em₂  Em₁  m₂Em₁ ⊗ Em₂ 

Em₁  m₂

(1)

where ⊕ and ⊗ are operations on ciphertexts that

correspond to addition and multiplication in the

plaintext space.

The most commonly used algebraic structures

include:

1. Polynomial Rings: The ring R[x]/(xⁿ + 1) with R

often equaling Z_q for some prime q. This is the

simple building block used in RLWE schemes.

Polynomial a(x) can then be represented as: a(x) = a₀

+ a₁x + a₂x² +. + aₙ₋₁xⁿ⁻¹ Computation in such a ring

is done modulo (xⁿ + 1). Hence, such computations

are quite efficient and compact.

2. Cyclotomic Fields: These are fields denoted as

Q(ζₘ) for mth primitive root of unity ζₘ.

Algebraically, most homographic schemes depend on

them. The mth cyclotomic polynomial Φₘ(x) defines

the field extension: Q(ζₘ) ≅ Q[x]/(Φₘ(x))

3. Lattice Structures: Lattices in n-dimensional space

are defined as: L = {∑ᵢ xᵢbᵢ | xᵢ ∈ Z} where {b₁, bₙ} are

linearly independent basis vectors. Many

homographic schemes prove insecure due to the

hardness of lattice problems that include:

Shortest Vector Problem (SVP):

𝑚𝑖𝑛||𝑣||:𝑣 ∈ 𝐿0 (2)

Closest Vector Problem (CVP):

for

𝑚𝑖𝑛||𝑣  𝑡||: 𝑣 ∈ 𝐿 (3)

target t

Tensor Product Spaces: For optimization and

parallelization, tensor products of rings are utilized:

R₁ ⊗ R₂  ∑ᵢⱼ aᵢⱼ x₁ᵢ ⊗ x₂ⱼ | aᵢⱼ ∈

K, x₁ᵢ ϵ R₁, x₂ⱼ ϵ R₂

(4)

The security of these structures relies on the

following key properties:

a) Ring Homomorphism Preservation: For a ring

homomorphism φ: R → S, the following must hold:

𝜑𝑎  𝑏  𝜑𝑎  𝜑𝑏 (5)

𝜑𝑎𝑏  𝜑𝑎𝜑(b) (6)

𝜑1_𝑅  1_S (7)

b) Ideal Structure: For an ideal I in ring R, the quotient

ring R/I maintains homomorphic properties while

providing additional security through the hiding of

plaintext values.

c) Error Term Distribution: The error terms e is

typically sampled from a discrete Gaussian

distribution:

𝐷𝜎𝑥 

 

∑ᵧ _

(8)

where

𝜌 𝜎𝑥  𝑒𝑥𝑝

||²

²

 (9)

Such algebraic structures work together to give both

functionality and security. Polynomial rings give

computation efficiency, and lattice structures give

guarantees of security. The choice of parameters in

these structures is such that both practicality and

security are achieved; typical implementations use:

Ring dimension

𝑛  2ᵏ for k  10 (10)

Modulus 𝑞  2^32 𝑡𝑜 2^64 (11)

Error distribution parameter

𝜎3.2 (12)

3.2 Elliptic Curve Cryptography (Ecc)

ECC is the public-key encryption method based on

elliptic curve's mathematical characteristics to ensure

key establishment in a secure manner, digital

signature generation, and encryption. Such

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COMMUNICATION, AND COMPUTING TECHNOLOGIES

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cryptography methods including ECC can be traced

back to the hardness of ECDLP- which is the discrete

logarithm problem on an elliptic curve. ECC is based

on the hard-computational problem of the ECDLP,

thus effectively resistant to any attack from quantum

computers. ECC is a fundamental requirement for

edge computing environments; the hardware used in

these systems are resource-constrained, light weight

and efficient cryptographic algorithms are required.

Few of the application areas of ECC include secure

communication protocols, digital signatures, and

authentication. Other applications of this algorithm

include cryptocurrencies such as Bitcoin and

Ethereum. Some of the advantages of ECC over the

traditional public-key cryptography are fast

generation of keys and smaller sizes of keys, making

it more appropriate for resource-constrained devices.

3.2.1 Elliptic Curve

An elliptic curve is a mathematical object defined by

the equation:

𝑦^2  𝑥^3  𝑎𝑥  𝑏 (13)

Where a and b are constants, and x and y are

variables. The curve is often defined over a finite

field, such as the real numbers or the integers modulo

a prime number.

3.2.2 Point Addition

For two points P and Q on the elliptic curve, the point

addition operation is defined as:

𝑃  𝑄  𝑥, 𝑦  x′, y′  x  x′, y  y′  x  x′ ∗

y  y′ (14)

It is a two-poi nt operation to combine the curve

points and obtain a new point.

3.2.3 Point Multiplication

Given a point P on the elliptic curve and an integer k,

the point multiplication operation is defined as:

𝑘 ∗ 𝑃  𝑃  𝑃.. . 𝑃𝑘 𝑡𝑖𝑚𝑒𝑠 (15)

This operation is used to multiply a point on the curve

by an integer.

3.2.4 Elliptic Curve Cryptography

In summary, ECC exploits point addition and point

multiplication in a secure means to achieve private

key exchange as well as sign messages and to encrypt

data in confidentiality. Fundamentally, it is through

the hardness of ECDLP that the encrypting and

decryption process is secure.

3.2.5 Key Generation

To generate a public-private key pair, a user chooses

a random integer k and computes the public key as:

𝑃𝐾  𝑘 ∗ 𝐺 (16)

where G is a fixed point on the elliptic curve, known

as the generator point.

3.2.6 Encryption

To encrypt a message, the sender computes the

ciphertext as:

𝐶𝑇  𝑚 ∗ 𝑃𝐾 (17)

where m is the message to be encrypted, and PK is the

public key.

3.2.7 Decryption

To decrypt the ciphertext, the receiver computes the

plaintext as:

𝑚





(18)

where CT is the ciphertext, and PK is the public key.

3.2.8 Security

ECC security relies on the difficulty of the ECDLP,

the challenge of determining the discrete logarithm of

a point on an elliptic curve. It is considered

computationally hard to compute the private key from

the public key, and thus the ECDLP is considered

computationally infeasible for an attacker.

3.3 Lattice-Based Cryptography

Lattice-based cryptography is public-key

cryptography where the key exchanges, signatures,

and encryptions use the mathematical properties of

lattices. They are discrete subgroups of vector

spaces. They are also quite resistant to attack by

quantum computers. Concrete security of lattice-

based cryptography can be based on the conjectured

hardness of the Shortest Vector Problem (SVP) and

the Closest Vector Problem (CVP) in lattices.

Formally SVP and CVP are defined as:

Definition SVP Given a lattice L, find the shortest

non-zero vector in L. Definition CVP Given a lattice

L and a vector v, find the closest vector to v in L.

Hardness of such problems is used to construct

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cryptographic schemes resistant to attacks from

quantum computers. For example, the problem

Learning with Errors (LWE), that was developed as a

lattice-based one and was applied to build public-key

cryptosystems. WE: Given a lattice L and a vector v,

find a vector x such that the inner product of x and v

is close to a random value.

The LWE problem is utilized to construct the public-

key cryptosystems. Some of them are the Ring-

LWE, which is further divided into Module-LWE.

These schemes can be used in key exchange, digital

signatures, and encryption.

The RLWE scheme is defined as follows:

RLWE: Given a ring R and a vector v, find a vector x

such that the inner product of x and v is close to a

random value in R.

The MLWE scheme is defined as follows:

MLWE: For a fixed module M and vector v, can

we find a vector x such that the dot product 𝑥 ⋅ 𝑣 is

close to a uniform random number in module M?

These schemes aim to construct cryptographic

protocols which will be secure against attacks by

quantum computers. As an example, the construction

of a quantum-computer attacks resistant public-key

cryptosystem using the RLWE scheme.

Lattice-based cryptography relies on the difficulty

of the shortest vector problem in lattices. Such

problems become the basis of cryptographic schemes

that cannot fall prey to attacks by quantum computer.

Examples of lattice-based cryptographic schemes

include RLWE and MLWE, which have been used in

key exchange, digital signatures, and encryption.

So, some branch of cryptography using lattices is

known as lattices-based cryptography, which can

provide secure key exchange and digital signatures

and even encryption

Lattice-based cryptography relies on the hardness

of the SVP and CVP problems in lattices and is used

in the construction of some cryptographic schemes

resistant to quantum computer attacks.

3.4 Homomorphic Encryption

Homomorphic Encryption refers to a class of

encryption algorithms that enables the execution of

many computations directly over ciphertext without

decrypting it first. That is, the user can carry out

operations such as addition and multiplication on

encrypted information and return an encryption of the

result. Homomorphic encryption traces back to the

idea of homomorphism which is a mathematical

function that does not disturb the form of a

mathematical entity.

Homomorphic encryption has its basic idea on a

pair of algorithms. It uses a decryption algorithm

related by a homomorphism to the encryption

algorithm. The former, the encryption algorithm, is

such that, with a given public key, it takes as input a

plaintext message and gives a ciphertext message as

output. However, the decryption algorithm is the

latter in such a way that with a private key, the

ciphertext message given becomes the plaintext

message.

A homomorphism between encryption and

decryption algorithms is applied, such that

manipulating a ciphertext message must have an

equivalent in manipulating the plaintext message.

This means that the ciphertext message can be added

and multiplied similar to the plaintext message.

3.4.1 Error Correction in Homomorphic

Encryption

Error correction encodes the plaintext into a

redundant format to allow for the recovery of the

original data when small error occurs in the

calculations307. Typically written using Error-

Correcting Codes (ECC) such as Hamming codes,

Reed-Solomon codes, or LDPC codes, to serve this

purpose Encoding Process

The plaintext mmm is transformed into a redundant

format m′ using an error-correcting code:

m′  ECCm, m′ \textECCm, m





ECCm, (19)

where m′ includes additional parity or redundancy

bits. This encoded message is then encrypted:

𝑐  𝐸m′. c  Em′. c  Em′. (20)

Decoding Process

After computations on the ciphertext, the result is

decrypted to retrieve the encoded data:

m′  Dc. m′  Dc. m′  Dc. (21)

The error-correcting decoder ECC−1\text {ECC}^ {-

1} ECC−1 is then applied to recover the original

plaintext:

𝑚  𝐸𝐶𝐶  1m′. m \textECC^1m′. m 

ECC  1m′. (22)

This is to ensure that small errors made by noise

during computation do not modify the correct result.

3.4.2 Homomorphic Encryption with Multi-

Party Computation

Homomorphic encryption (HE) combined with multi-

party computation (MPC) offers a strong framework

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for privacy-preserving collaborative computation.

HE enables computations to be performed directly on

encrypted data, so sensitive inputs remain

confidential throughout the process. MPC enables

multiple parties, each with private inputs, to jointly

compute a function without revealing their data to one

another. The process starts with a jointly generated

public-private key pair. A public key is used to

encrypt individual inputs, whereas the private key is

divided into shares held by participants.

Homomorphic operations are carried out on

encrypted inputs. Addition or multiplication on

ciphertexts is ensured to correspond to equivalent

operations on plaintexts. Finally, this encrypted

outcome gets decrypted collectively by the private

key shared by them and cannot be decrypted by any

of the party members. This HE integrated with MPC

guarantees both the computation to be accurate as

well as data privacy.

3.4.3 Homomorphic Encryption Integrated

with Secure Multi-Party Computation

HE, with MPC, allows various parties to collaborate

and perform a function on their respective private

inputs without revealing it. In reality, it is an

encryption technology where the computation is done

directly over the encrypted data without any input

revealed. The scheme encrypts private information of

each party with a public key and shares that encrypted

data for joint computation. Using HE's properties

such as additive or multiplicative homomorphism, the

server or participants then do the desired computation

on the encrypted inputs. This computation on the

ciphertexts maps directly to equivalent computation

on the plaintexts, thus correct. The ciphertext thus

generated is decrypted cooperatively among all

parties with common decryption keys. Shared

decryption ensures no party obtains the result or some

intermediate computation without reliance on any

other party, hence security and integrity are

preserved.

4 CONCLUSIONS

This is a leap forward transformation in secure

distributed computing, where homomorphic

cryptography and edge computing revolutionize

processing data at the network edge. Through our

deep analysis of the mathematical foundations on

which homomorphic cryptographic algorithms

depend, such as the LWE and RLWE problems, we

have established what the theoretical strengths are

and where current implementations run short. Even

though algebraic structures and mathematical

frameworks that rely on polynomial rings and lattice-

based constructions are considered robust security

guaranties, with which computations are feasible over

ciphertext, these indeed also raise high challenges

related to the overheads in computation and resources

consumption on the edge side. This area promises a

very bright future ahead in the direction of further

efficient algebraic structures, advanced noise

management techniques, and optimized methods for

parameter selection especially tailored to edge

computing constraints. The future of edge computing

will be on the integration of homomorphic

cryptography, based on the advancement that has

been going on both theoretically in mathematics and

practically in techniques for implementation toward

optimized performance within resource-constrained

environments while maintaining robust security

guarantees.

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