Page 112 - Kaleidoscope Academic Conference Proceedings 2024

2024 ITU Kaleidoscope Academic Conference

approach simplifies key management, eliminates the need for be added in the resultant p_t which form the public key.
an additional symmetric key exchange, and reduces potential H function is used in the decryption algorithm where it takes
vulnerabilities, making it suitable for applications where the ciphertext and uses it to provide the absolute difference
overhead and complexity are critical concerns. between the ciphertext matrix value and the value that came
Our algorithm has demonstrated strong performance in both from putting s in those equations in ciphertext with redundant
encryption and decryption. We’ve focused on reducing dimensions and then we check the probability that it is less
ciphertext size and optimizing multiplication complexity, than 4 or not for decoding the bits.
2
aiming for a time complexity of ( log ) rather than ( ).
This ensures both efficiency and security in establishing a 2.2 Gaussian Distribution
highly secure communication channel.
Lastly, we present use cases for our messaging As Regev [1] demonstrates a quantum reduction from
solution, showing its applicability in secure governmental worst-case lattice issues to LWE with Gaussian error, given
communications, financial transactions, and private arbitrarily many LWE samples, the Gaussian distribution is
messaging, where quantum-resistant security is essential. the default error distribution for classical LWE.
Definition. Let D be the Gaussian distribution with
2. PRELIMINARIES standard deviation . The rounded Gaussian distribution
R
function D is defined as follows:

R
2.1 Notations For any ∈ R, the sample from D is obtained by rounding

to the nearest integer and adding a sample from D .
The set of integers is denoted by Z, Z q represents the finite In mathematical notation:
field of cardinality q and its elements are represented by the

integers in the interval [−[ ], ]. ( ) denotes an encoded = ⌊ + + 0.5⌉
2 2
array of n integers like { 1 , 2 , 3 ...., } each integer takes
32
4 bytes each makes the value range of 0 to 2 −1. D denotes where is a sample from D . If X follows a Gaussian
distribution then the rounded Gaussian of will be
the Gaussian distribution and D with standard deviation
−
. Here, ← denotes the set of x belongs from the ∫ −1/2
distributed error terms generated which is denoted by . Pr( Z = ) = ,

Pr( ) denotes the Probability of .B denotes an array of 32 +1/2

bits representing the ASCII value for a character. In the following = 0.
Extendable Output Function (XOF) is a cryptographic Goal. In here we have to find the distribution of Z = Z
primitive that takes a seed as input and produces an output of mod q.
any desired length. It’s like a versatile cryptographic tool that
can generate data according to specified distributions. Let’s Pr( Z = [ ]) = Pr( Z = + ) for some ∈ Z
say we have a seed and we want to use XOF to create
matrices and vectors. We start by initializing XOF with the
seed seed, and then we use it to generate the matrices and ∑︁ ∫ + −1/2 ∫ −1/2
0, ( ) = , ( )
vectors [22]. For example, we can generate a matrix by
+ +1/2 +1/2
calling XOF with a specific seed seed , and similarly, we ∈Z
Where
can generate a vector by calling XOF with a seed seed .
Let seed be the seed used by the XOF, a variants of SHA ∑︁ 1 −( + ) 2
, ( ) : → √ exp
(secure hashing algorithm)-256 and SHA-128 pseudo number 2
2 2
generators in these functions which give us a hash value of ∈Z
a definite length. To generate a matrix , we call XOF with
2.3 LWE definition
seed seed :
← XOF (seed ) Learning With Errors (LWE) is a fundamental problem in
lattice-based cryptography and serves as the basis for many
Similarly, to generate a vector , we call XOF with seed seed :
encryption schemes and cryptographic protocols[1][3][4].
← XOF (seed ) It involves the challenge of distinguishing between noisy
information and random noise in a set of equations. terms .
−
The process is deterministic, meaning that for a given seed, Notation. Let be the rounded gaussian mod q and ← .
XOF will always produce the same output. However, in Definition. > 0 be a modulus and , > 0. Now given
practice, XOF may use random oracles to achieve this m samples of such equations of n dimesnions in the form of

determinism while statistically approximating the desired a i , < a i , s i > +e i where a i is uniform at random in Z and

−
distribution. ← . Find the secret ∈ Z
The G function that when we send the . + and _
combine the function output the results as stored in _ as The LWE can be formulated more compactly as a

256
Í matrix-vector equation:
=0 · ( − ) so that the redundant lattice dimensions
which was chosen randomly from 256 dimensions will not the · + = mod
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