Development and Implementation of the NTRUEncrypt Public Key

Cryptographic System

Elena A. Revyakina

, Larisa V. Cherckesova

and Elena A. Menshenina

Don State Technical University, Rostov-on-Don, Russia

Admiral Ushakov Maritime State University, Rostov-on-Don, Russia

Keywords: Cryptographic System, NTRUEncrypt, Document Encryption, Information Security.

Abstract: The article describes the development of a software tool for encrypting a text document and provides a number

of formulas that were used in this case. It also describes the process of generating a private and public

encryption key in the software, provides formulas for generating keys, and shows the user interface for

entering the necessary parameters. In addition, the algorithm for decrypting the encrypted text was described

and the output of the program results after decrypting the selected document was shown. A comparative

analysis of the speed of key generation, encryption and data decryption of the implemented NTRUEncrypt

cryptographic system with RSA and ECC is performed.

1 INTRODUCTION

In today's world, technology has reached the point

where a large amount of information is stored in

cyberspace, and thanks to this type of storage, it is

easier for people around the world to access

information than ever before. Until the end of the last

century, collecting information was a difficult and

long task, but now people can access a huge amount

of information in a few minutes, and with the

development of high technologies, more and more

organizations in the world now depend on their

information systems and on the information security

of these systems. Information security means the

protection of information and information systems

from unauthorized access, recording, use, reading,

destruction or modification. Cryptography is used in

information security to protect information from

unauthorized or accidental disclosure, during the

transfer of information, or while the information is in

storage. Cryptography is inextricably linked to the

transmission of data, and in addition to providing user

authorization, it guarantees the integrity of the

transmitted information and confidentiality. It is

obvious that for organizations such as banks, military

departments, government agencies, the use of

https://orcid.org/0000-0003-1577-2671

https://orcid.org/0000-0002-9392-3140

https://orcid.org/0000-0003-4487-3498

cryptographic systems is necessary, and cryptography

is used by various institutions on a daily basis.

A cryptographic system is a system for ensuring

the security of a protected system that uses

cryptographic means and methods. There are two

main types of cryptographic systems: symmetric and

asymmetric. Symmetric key systems require that both

the sender and the receiver have the same key.

This key is used by the sender to encrypt the data

and again by the receiver to decrypt the data. Each

user has both a public and a private key. Messages are

encrypted with one key and can only be decrypted

with another key. With the development of quantum

computers, the risk of hacking such pre-quantum

public-key cryptosystems as RSA (a cryptosystem

named after its three inventors Rivest, Shamir, and

Adleman), elliptic curve cryptosystems (ECC), also

increases.

Therefore, post-quantum encryption systems are

becoming more relevant than ever. The

NTRUEncrypt public key cryptographic system can

remain relevant in the post-quantum era due to the

fact that this system is based on a lattice cryptosystem

and its stability is ensured by the complexity of

finding the shortest lattice vector. Until now, there is

no algorithm that would solve this problem, so the

Revyakina, E., Cherckesova, L. and Menshenina, E.

Development and Implementation of the NTRUEncrypt Public Key Cryptographic System.

DOI: 10.5220/0010617100003170

In Proceedings of the International Scientiﬁc and Practical Conference on Computer and Information Security (INFSEC 2021), pages 41-47

ISBN: 978-989-758-531-9; ISSN: 2184-9862

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

NTRUEncrypt cryptographic system can remain

relevant not only at the moment, but also in the future.

The purpose of this work is to develop and

implement the NTRUEncrypt cryptographic system

with a public key.

2 QUANTUM COMPUTERS

Quantum computers are machines that use the

properties of quantum physics to store data and

perform calculations. This can be extremely useful

for certain tasks, where they can significantly

outperform even supercomputers. Classic computers,

including smartphones and laptops, encode

information in binary "bits", which can have a value

of 0 or 1. In a quantum computer, the basic unit of

memory is the qubit (Wenbo Mao, 2015).

Bouquets are created using physical systems, such

as the spin of an electron or the orientation of a photon

(Uttar Pradesh, 2019). These systems can be

simultaneously arranged in many different

configurations, a property known as quantum

superposition. Qubits can also be inextricably linked

together by a phenomenon called quantum

entanglement. As a result, a series of qubits can

simultaneously represent different things.

For example, eight bits are enough for a classical

computer to represent any number from 0 to 255. But

eight dice is enough for a quantum computer to

represent every number from 0 to 255

simultaneously. (Uttar Pradesh, 2019). A few

hundred entangled qubits will be enough to represent

more numbers than the atoms in the universe. In

situations where there are a large number of possible

combinations, quantum computers can consider them

simultaneously (Ryabko, 2012). Examples include

trying to find the prime factors of a very large

number, or the best route between two places.

However, there may also be many situations

where classical computers will still outperform

quantum computers. The computers of the future may

be a combination of both of these types. At the

moment, quantum computers are very sensitive: heat,

electromagnetic fields and collisions with air

molecules can lead to the loss of the quantum

properties of the qubit. This process, known as

quantum decoherence, causes the system to fail, and

this happens the faster the more particles involved

(Schneier B., 2002).

Creating hardware for a quantum computer is a

difficult task. Qubits are inherently very fragile and

can lose the information encoded in them very

quickly. The main objective is to keep the qubits

completely isolated from the environment, while

providing high-precision monitoring and reading of

the qubit state. In order to effectively separate the

qubits from any noise source and therefore maintain

a longer coherence time, these systems are usually

cooled to extremely low temperatures using liquid

helium, however this results in high operating costs.

There are many different ways to implement

qubits, such as trapped ions, superconducting rings,

and more. Each architecture has its own advantages

and disadvantages, and it is not yet clear which qubit

material is the most scalable(Ishmukhametov, 2011).

To provide a large number of parallel processes,

the quantum computing model uses several relatively

simple rules for converting input information. In

other words, it becomes possible to calculate its

values for all arguments in a single pass of the

function. This function is applied to the input data.

The input is data that is a superposition of all possible

values of the argument, therefore, the function must

accept and process such a sequence. In quantum

models, a function is a Hermitian matrix, that is, one

that has its own Hermitian-conjugate, and the product

of these matrices will eventually give a unit one. The

conjugate matrix can be obtained in two steps, the

first step is the transposition of the original matrix and

the second step is the substitution of their complex-

conjugate elements instead of the original elements.

If you multiply the resulting matrix by the vector of

the quantum register, you get a quantum register, such

that the sum of the squares of the coefficients for

quantum states is equal to one (Cheremushkin, 2009).

2.1 Comparison of RSA,

NTRUEncrypt, and ECC

Cryptosystems

Most of the most important information today is

transmitted via the global Internet, for example:

email, chat messages, video conferences, e-

commerce, and online banking data. The use of

public-key cryptosystems is the main way to protect

such data (Bolotov, 2006).

The most widely used cryptosystem is RSA, based

on the complexity of factorization of large numbers,

the Diffie-Hellman scheme, the digital signature

algorithm (DSA), the security of which is based on

the complexity of solving the discrete logarithm

problem in the field, a family of algorithms based on

elliptic curves. But all of them have certain

disadvantages, the main ones being either relatively

low speed, or relatively low stability at comparable

key and parameter sizes (Diffie-Hellman scheme and

other algorithms based on the discrete logarithm in

INFSEC 2021 - International Scientiﬁc and Practical Conference on Computer and Information Security

the field), or both at the same time (RSA) (Zhdanov

O.N. et al., 2012).

The NTRU algorithm, all operations of which are

performed in the ring of truncated polynomials, was

created to solve this problem. The cryptographic

strength of NT RU is based on the complexity of the

problem of finding a short vector in a given lattice

(Bertels, K., 2018). All cryptographic systems based

on integer factorization, discrete logarithm, and

discrete logarithm problems in the elliptic point group

of curves are potentially vulnerable to the

development of a quantum computer of suitable size,

since algorithms are known for such a computer that

can solve these problems in polynomial time,

depending on the size of the input data. Currently,

there are no quantum algorithms with polynomial

complexity for NT RU.

Only cryptosystems based on algebraic lattices

remain virtually invulnerable to quantum

cryptanalysis. One of the main advantages of the

NTRU algorithm is also the very high speed of

encryption / decryption operations. According to the

developer company NT RU Security Innovation, this

algorithm is up to two hundred times faster than the

elliptic curve and RSA algorithms [12]. Based on the

currently available data, we can draw intermediate

conclusions about the high level of NTRU security,

which is not inferior to the security of algorithms

based on elliptic curves.

However, due to the relative novelty and low

prevalence of asymmetric algorithms of this class,

additional research is needed for possible errors and

critical vulnerabilities that can be used to develop

effective attacks. As a result of the performance

analysis of NT RU, it was found that its performance

is much higher than that of RSA and ECC. NT RU

showed the best results in terms of performance, as it

lends itself very well to parallelization, in contrast to

RSA, attempts to parallelize which practically do not

give a speed increase.

The class of asymmetric algorithms based on

problems on algebraic lattices is significantly

superior to other public-key algorithms in the

aggregate of all indicators. Therefore, it is NT RU that

should replace RSA and ECC in many areas and

become the same generally accepted standard for

asymmetric cryptography.

2.2 Advantages of the NTRUEncrypt

System

The NTRUEncrypt algorithm was developed back in

the mid-90s. In comparison with the RSA

cryptosystem, it was not widely used, since it was first

necessary to improve the cryptographic strength and

performance of this cipher algorithm.

At the moment, all the identified shortcomings

have already been fixed, and in practice it is believed

that NTRUEncrypt works much faster than the RSA

algorithm. When comparing the speeds of NT RU,

RSA, and ECC with the parameters corresponding to

the security level k = 256 bits, NT RU is 4 orders of

magnitude faster than RSA and 3 orders of magnitude

faster than ECC [10]. The graph (Figure 1) clearly

shows how much the NTRUEncrypt cryptosystem

exceeds, in terms of the number of operations per

second, most existing pre-quantum algorithms.

Figure 1: Comparison of NTRU performance with other

algorithms.

In addition, the NTRUEncrypt cryptosystem is

post-quantum, and has a much higher cryptographic

resistance to various attacks. And, as mentioned

above, due to the lack of an algorithm for finding the

shortest lattice vector, the cryptographic stability of

the algorithm is ensured (NTRU CryptoLabs. NTRU

Algorithms, 2020).

3 THE NTRUEncrypt PUBLIC

KEY CRYPTOGRAPHIC

SYSTEM

Approaches to the implementation of the system.

NTRUEncrypt, developed by Hofstein, Piefer,

and Silverman, was first introduced to the public at

the Crypto ’96 supplemental session. At the moment,

the NTRUEncrypt encryption system is an alternative

to encryption schemes based on factorization and

Development and Implementation of the NTRUEncrypt Public Key Cryptographic System

discrete logarithm over finite fields and elliptic

curves, which is marked by its inclusion in the P1363

standard.

NTRUEncrypt is considered a viable solution for

post-quantum public-key encryption. While the

number of attacks and practical improvements of

NTRUEncrypt grew, the theoretical part of

cryptography with proven security, based on the use

of lattices, stopped in development.

The starting point is 1996 and the recognized

transformation of the worst-case scenario into

normal, which gave rise to stable hash functions,

cracking which is as difficult as cracking some

problems over worst-case lattices. In the last couple

of years, a large number of cryptographic schemes

have been introduced that have proven the security of

the error-based learning layer (LWE). This list

includes secure encryption schemes based on

authentication, digital signatures, etc.

The main obstacle to LTE-based cryptography

and the Short integer solution (SIS) is its limited

efficiency. The key in most cases contains a random

matrix, the dimensions of the security parameter are

linear; therefore, the space and time requirements are

at least a second power with respect to the security

parameter.

In 2002, Micciancio was able to convert SIS to

structured matrices, while retaining the worst-case to

normal transformation. The worst-case scenario is the

reduction of general lattice problems to a special

family of cyclic lattices. Micciancio's work led to the

formation of a family of hash functions that are

resistant to preimage attacks.

It is proved that the resulting hash function is

collision-resistant for a given stability of the modified

normal scenario problem; it was called Ideal-SIS. It

provides the same robustness as reducing the worst-

case scenario for typical lattices to a special class of

lattices (called ideal ones). In 2009, Stele proposed a

structured version of LTE, which proved to be as

reliable as Ideal-SIZE (ShanYue Bu. Choosing

Parameters for NTRU, 2020).

In parallel, Lubashevsky, in his independent

work, proposed a variant of LWE on rings, called R-

LEE, whose greater flexibility makes it possible to

create more natural and efficient cryptographic

constructions. Most modern cryptographic

algorithms and protocols rely on the computational

complexity of certain mathematical problems, such as

factorization of the products of two large primes

(RSA) or discrete logarithm over certain groups

(Diffie-Hellman key exchange, El-Gamal encryption

system).

These problems are considered to have no

efficient (polynomial-time) solutions, so any

cryptographic protocol based on them should be at

least as difficult to crack. Since it can be assumed that

any potential adversary has limited computing power,

it can be expected that these protocols will be secure.

However, this is not always the case.

If the opponent has the computing power of a

quantum computer, such algorithmic problems are

easily solvable. RSA and many other popular

protocols, such as Diffie-Hellman key exchange, El

Gamal encryption scheme, DES (data encryption

standard), or ECC, will be broken.

They will need to be replaced with something

more secure, involving more computing power of a

potential enemy. There are several algorithmic

problems that are difficult to solve, both on classical

and quantum computers. There are many different

groups of protocols, such as hash-or code-based

cryptography, multidimensional quadratic equation

cryptography, or lattice-based cryptography

(Fergusson N., Schneier B. 2004).

NTRUEncrypt is usually described as a

polynomial-based cryptosystem involving

convolution products. It can naturally be considered

as a lattice cryptosystem for a certain limited class of

lattices. The cryptosystem has several natural

parameters, and, as with all practical cryptosystems,

it is possible to optimize these parameters to increase

efficiency, while avoiding all known cryptanalytic

attacks. The functionality of the cryptosystem will not

be affected if "decryption failures" occur with a very

small probability in random messages.

3.1 Recommended NTRUEncrypt

Parameters

According to the developers of NTRUEncrypt, this

encryption system is 4 orders of magnitude faster than

RSA and 3 orders of magnitude faster than ECC and

can be used in the "post-quantum" era [10-14].

The NTRUEncrypt public key cryptographic

system is based on a lattice cryptosystem and its

strength is ensured by the difficulty of finding the

"shortest lattice vector". This system is designed for

an alternative to RSA, ECC, as it is more resistant to

attacks on quantum computers. NTRUEncrypt

operates on polynomials of degree at most N-1:





a a



a



Xa



⋯

a



a





NTRUEncrypt parameters:

N – the size of the public key

q, p-mutually prime numbers that are used to reduce

the coefficients of polynomials;

INFSEC 2021 - International Scientiﬁc and Practical Conference on Computer and Information Security

df – the number of coefficients equal to " 1 "and df-

1 equal to" -1 " of the polynomial f;

dg – the number of coefficients equal to " 1 " and " -

1 " for the polynomial g;

dr – the number of coefficients equal to "1" and " -1

"for the" blinding " polynomial r.

To achieve the desired level of protection, we

recommend using the following NTRUEncrypt

system parameters, which are shown in

Table 1:

Table 1: Recommended NTRUEncrypt parameters.

N q p df dg dr Guaranteed

durability

167 128 3 61 20 18 Moderate

level of

durabilit

251 128 3 50 24 16 Standard

level of

durability

503 256 3 216 72 55 Highest

level of

durabilit

3.2 Description of the Algorithm of the

Program

At the first stage of the program, the private and

public encryption keys are formed, which will be used

in the future when encrypting and decrypting

information. This process is shown in Figure 2.

Figure 2: Public key generation scheme.

Next, the user selects the message to be encrypted

using the public key. Figure 3 shows the encryption

execution scheme.

Figure 3: Encryption scheme for the selected message.

After the message has been encrypted, this

software tool allows you to decrypt it using a secret

key. In this case, the encrypted message is

represented as bits and then the bit sequence is

divided into blocks of N bits. After that, each block is

decrypted and all the decrypted blocks are combined

into a common array.

The last step in decrypting the encrypted data is to

break the resulting bit string into characters and

display it on the screen. Figure 4 shows a diagram of

this process.

Figure 4: Scheme of decryption of the selected message.

Development and Implementation of the NTRUEncrypt Public Key Cryptographic System

Thus, based on the generalization of general

information about the NTRUEncrypt cryptographic

system, an algorithm for encryption and decryption

by this system, as well as the formation of a public

and private key, was developed. Below we will

consider the principle of operation of the software

product, which is an object-oriented application with

a simple and intuitive interface.

4 TESTING AND

DEMONSTRATION OF THE

SOFTWARE TOOL

The NTRUEncrypt cryptographic system uses six

constant parameters: N, p, q, df, dg, dr. These

parameters are entered by the user in the "Encryption"

panel, as shown in Figure 5.

Figure 5: Entering parameters and generating private and

public keys.

The number N characterizes the size of the public

key, as well as the length of each data block. The

integers p and q, whose GCD (Greatest Common

Divisor) (p, q) is 1, do not have to be prime.

The p parameter is used to determine the interval

to which all coefficients of the polynomials used in

the cryptographic system should belong. The

parameters df and dg specify the number of units in

the program-generated polynomials f and g. The dr

parameter specifies the number of coefficients equal

to 1 and -1 of the "blinding" polynomial r, which is

used for encryption.

First, the user enters the first five parameters, after

which the formation of the polynomials f and g takes

place. After that, we search for the inverse

polynomials for f and g, but it may happen that some

inverse polynomial will not be found. Then the

program will form another polynomial and try to find

the inverse for it again.

To achieve a sufficient level of cryptographic

security, all parameters must meet the recommended

ones.

Figure 6 shows the stored encryption keys in the

"keys" folder. The public key " key. pub "is needed

for data encryption, and the key" key.priv " is needed

to decrypt the received ciphertext.

Figure 6: Encryption Keys.

Next, the program will encrypt the file, save it as

"encrypted. enc" and output the cipher after clicking

on the "Encrypt" button as shown in Figure 7.

Figure 7: Output of the encrypted file.

The last step of the program is to select the encrypted

document and the private key as shown in Figure 8.

When decrypting, the encrypted text is again

represented as bits. Then the bit sequence is

represented as coefficients, after which each series of

coefficients develops into blocks of N numbers. Each

block is represented as a polynomial, and then

decryption is performed to obtain a sequence of bits

that are divided into bytes. Thus, the resulting text is

the same as the original one.

Figure 8: Selecting the ciphertext and private key.

After clicking on the "Decrypt" button, the

decryption of the selected encrypted file will begin.

5 CONCLUSIONS

In this paper, we implemented a system that provides

a resource for encrypting and decrypting data, the

cryptographic strength of which is explained by the

complexity of solving the problem of lattice theory,

which is the basis of the NTRUEncrypt algorithm. In

the framework of this work, the algorithm of the

software tool was described: the choice of variables

for generating the private and public encryption key,

the algorithm for encrypting and decrypting

information using the generated keys. In addition, the

scheme of the program was built with a description of

INFSEC 2021 - International Scientiﬁc and Practical Conference on Computer and Information Security

each stage of the implementation of the implemented

algorithm and demonstrated the operation of the

software with the output of encrypted and decrypted

messages.

The full implementation of the NTRUEncrypt

cryptographic system was performed in the Python

programming language. It is important to note the fact

that after the creation of a quantum computer, the

problems of fast discrete logarithm and factorization

will soon be solved and such algorithms as RSA,

DSA and other similar ones will become irrelevant.

However, the NTRUEncrypt cryptographic

system will remain relevant, since there is no

algorithm that solves the problem of the shortest

lattice vector, which means that it is quite applicable

in the "post-quantum" era. As a result of the work, a

software implementation of the NTRUEncrypt

algorithm with a user interface was developed, and

the developed software tool was tested with various

parameters.

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Development and Implementation of the NTRUEncrypt Public Key Cryptographic System