Page 117 - Kaleidoscope Academic Conference Proceedings 2024
P. 117

Innovation and Digital Transformation for a Sustainable World





























                  Figure 3 – Schematic of the proposed quantum-resistant encryption for secure end-to-end communication.



           to the integration of the Quantum key distributions into  this reduction doesn’t imply that these lattice problems
           systems other than the site-to-site VPNs. This issue may  are NP-Hard for linear approximation factors. Even if a
           be resolved by deriving many keys from the original key  quantum solver for LWE worked for parameters very close
           that QKD supplied. For instance, one may utilize a single  to 1, the resulting approximation factor for lattice problems
           QKD key with keys that are generated for every message  would still be greater than   .  Currently, we lack proof
           transmitted between two fixed participants on the same day.  that these approximate lattice problems are NP-Hard for
           It could also go beyond the classical client-server model and  linear approximation factors. Therefore, while a quantum
           has many use cases such as a secure quantum channel for  polynomial-time algorithm for LWE would place LWE
           the military or some esteemed organization. Since we are  within BQP, it wouldn’t directly impact classical hardness
           successfully able to integrate it into the system and establish  or NP-hardness questions. Such an algorithm would still
           secure communication between two parties. There can be  be a significant breakthrough. The approximate versions of
           still some key exchanges and session chatting features which  lattice problems aren’t known to be NP-Hard, and efficient
           we are introducing further.                        algorithms for solving them remain elusive.

           6.1 Computational Complexity for Attacker               7.  CONCLUSION AND FUTURE WORK
           The difficulty of breaking an encryption scheme based on
                                                              The security of this lattice-based encryption scheme relies
           Learning With Errors (LWE) is closely tied to solving
                                                              on the difficulty of solving the Shortest Vector Problem
           the Shortest Vector Problem (SVP) for the lattice defined
                                                              (SVP) in high-dimensional lattices, particularly when random
           by the encryption parameters.  As lattice dimensionality
                                                              dimensions and error terms are introduced. The algorithm
           increases, the computational complexity of solving SVP
                                                              generates a random basis, random dimensions, and random
           grows exponentially, making it extremely challenging for
                                                              errors for each bit, making it computationally infeasible for
           attackers.  Randomly choosing dimensions for each bit
                                                              an attacker to recover the secret key or decrypt the ciphertext
           and adding random error terms during encryption further
                                                              efficiently.  However, the size and storage of ciphertexts
           increase complexity. The hardness of breaking an LWE-based
                                                              and messages impose restrictions on the length of messages
           encryption scheme depends on lattice dimensionality, the
                                                              that can be sent at once. To address this, we are exploring
           modulus used, and the number of random dimensions
                                                              various transformations and researching methods to reduce
           chosen during encryption. Solving SVP in high-dimensional
                                                              the ciphertext size to a logarithmic scale, possibly through
           lattices is believed to be exponentially complex, even for
                                                              binary search algorithms. Additionally, we are investigating
           powerful computers. In a client-server model, LWE-based
                                                              techniques similar to the Number Theoretic Transform (NTT)
           encryption secures encrypted chats stored in the database,
                                                              for optimizing matrix multiplications, which could enhance
           making decryption without keys extremely difficult due
                                                              performance without compromising security.
           to the complexity of solving SVP and the randomness
           introduced.  Brute-forcing through the more than 10 1200
           possible combinations in LWE encryption makes chosen                REFERENCES
           ciphertext attacks (CCA) infeasible. Regev’s reduction shows
           that a quantum solver for the LWE problem with certain  [1] Oded Regev, "The Learning with Errors Problem"
           parameters can solve approximate versions of lattice problems  in Blavatnik School of Computer Science, Tel Aviv
           like SVP and SIVP over lattices of dimension   . However,  University.
                                                           – 73 –
   112   113   114   115   116   117   118   119   120   121   122