QUANTUM-RESISTANT ELLIPTIC CURVE-BASED SECURE KEY EXCHANGE

THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 536 QUANTUM-RESISTANT ELLIPTIC CURVE-BASED SECURE KEY EXCHANGE 1D.T. Muhamediyeva, 1Tagaev Farkhad Abduvahabovich 1National Research University “Tashkent Institute of Irrigation and Agricultural Mechanization Engineers” Tashkent, Uzbekistan https://doi.org/10.5281/zenodo.17801930 Abstract. In this paper, a secure public key exchange algorithm is proposed, combining classical elliptic curve cryptography (ECC) and quantum algorithms. The algorithm is implemented using Python and the Qiskit library. Public keys are generated by point multiplication on ECC, and quantum superposition and isogeny mapping simulations are performed using Qiskit. The proposed approach can be stable against quantum attacks and can serve as the basis for future quantum-resistant cryptosystems. Keywords: Quantum security, elliptic curve, cryptography, public key exchange, quantum computing. Аннотация. В данной статье предлагается безопасный алгоритм обмена открытыми ключами, сочетающий классическую криптографию на эллиптических кривых (ECC) и квантовые алгоритмы. Алгоритм реализован с использованием Python и библиотеки Qiskit. Открытые ключи генерируются путём умножения точек на ECC, а моделирование квантовой суперпозиции и изогенического отображения выполняется с использованием Qiskit. Предложенный подход может быть устойчив к квантовым атакам и может служить основой для будущих квантово-устойчивых криптосистем. Ключевые слова: Квантовая безопасность, эллиптическая кривая, криптография, обмен открытыми ключами, квантовые вычисления. 1. Introduction Classical cryptographic systems, especially algorithms based on elliptic curves, are widely used in modern security systems. However, Shor's algorithm and other quantum algorithms pose a threat to systems such as RSA and ECC in the future. Therefore, active research is being conducted on quantum-resistant (post-quantum) algorithms. Cryptographic algorithms, which are the basis of modern information security systems, are often based on classical mathematical problems such as factorization or discrete logarithms. However, with the advent of quantum computers, these basic problems can be effectively solved using quantum algorithms such as Shor's algorithm. This poses a serious threat to the security of existing classical cryptosystems [1-3]. Therefore, quantum-resistant cryptography (Post-Quantum Cryptography, PQC) has emerged as a relevant direction. Its main task is to develop cryptosystems that can withstand quantum computers in the future. One such approach is isogeny-based elliptic curve cryptography (ECC), which allows for the quantum-secure reworking of algorithms such as Diffie-Hellman. In particular, algorithms such as SIKE (Supersingular Isogeny Key Encapsulation) are showing great promise in this regard [4-7]. In this paper, we model a small-area (𝑝 = 17) ECC-based secret and public key exchange algorithm using classical and quantum simulations. The goal is to illustrate isogeny-based ECC THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 537 with a simple example, to implement quantum elements using Qiskit, and to demonstrate text encryption and decryption using XOR with real code. The results of the research may serve as a theoretical basis for future quantum-secure communication systems [8-10]. 2. Methods Elliptic curve cryptography (ECC) is based on the following general elliptic curve equation: ( ) 23 : modE y x ax b p= + + . Here: ,p a b F — field elements, p — prime modulus, 2 ( , ) p x y F — points on the curve. Adding points on an elliptic curve can be divided into two main cases: Adding dissimilar points (P ≠ Q): If ( ) ( ) 1 1 2 2 , , , ,P x y Q x y== then: 21 21 2 3 1 2 3 1 3 1 mod , mod , ( ) mod . yy p xx x x x p y x x y p    − =− = − − = − − Adding the same point twice ( PQ= i.e. doubling): 2 1 1 2 31 3 1 3 1 3mod , 2 2 mod , ( ) mod . xa p y x x p y x x y p    + = =− = − − Each user (Alice and Bob) will have private and public keys as follows: Private keys: , A B p d d Z . Public keys: , . AA BB Q d P Q d P = = Here is P — the generation point (primary point). Alice calculates the public key as follows: () A A B A B S d Q d d P=  =   . And Bob: () B B A B A S d Q d d P=  =   . As a result: A B A B S S S d d P= = =   . This key can only be calculated by Alice and Bob. For an outside observer, calculating this value is practically impossible due to the Elliptic Curve Discrete Logarithm Problem (ECDLP). Using the qubit, a superposition state is created: THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 538 1 | 0 (| 0 |1 ) 2 H  ⎯⎯→ +  . CNOT a link is created through: 0,1 CNOT This represents the idea of quantum entanglement in a simple quantum communication model: 1 | (| 00 |11 ) 2   = +  . This Bell state models a reliable quantum channel between two parties. The results are simulated using AerSimulator. The RSA (Rivest-Shamir-Adleman) algorithm is a public-key encryption algorithm that essentially encrypts and decrypts data using two keys (public and private). The mathematical basis of the algorithm consists of the following steps: 3.1. Key generation 1. Two large prime numbers are chosen: ,pqP , where P is the set of all prime numbers. 2. The modulus is defined as: n p q= . This module is used in n encryption and decryption formulas. 3. The Euler function is: ( ) ( 1)( 1)n p q  = − − . 4. The public key exponent e is chosen: 1 ( ), gcd( , ( )) 1e n e n    = , that is, it ()n  must be a common prime to the number e . 5. The secret key d is: 1mod ( )d e n  −  , where d is the modular inverse of e . Result: “Public key: ( , )en ” “Private key: ( , )dn ” The given text M is encrypted using the following formula: mod e C M n= , where C is the ciphertext. The encrypted text C is reconstructed as follows: mod d M C n= , where M is the original plaintext that was reconstructed. Algorithm steps 1. Select private keys: A d and B d — are the private keys of Alice and Bob, respectively. 2. Calculate public keys: , A A B B Q d P Q d P=  =  . 3. Calculate public key: · , · A A B B B A S d Q S d Q== . THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 539 If AB SS= , the key exchange is successful. A 3-qubit quantum register is created using a Qiskit. It models the superposition state using a Hadamard gate and the isogeny mapping using a CNOT gate. This step allows the key to be exchanged via a quantum channel. 3. Results According to the results, Alice and Bob have the same public key. Using symmetric encryption based on XOR, the text is successfully encrypted and decrypted using the key. Alice's public key: (9, 16). Bob's public key: (0, 6). Alice's public key: (10, 11). Bob's public key: (10, 11). Qiskit Output: {'011': 540, '000': 484}. Alice and Bob have the same public key: (10, 11). In the study, the RSA algorithm was tested in the Python programming language. The following demonstrates the operation of the RSA algorithm based on prime numbers: Selected prime numbers: 17, 23pq== . Calculated modulus: 391n p q=  = . Euler function: ( ) ( 1)( 1) 16 22 352n p q  = − − =  = . Selected public key coefficient: 3e= . Private key: 235d= (i.e., 3 235 mod352 1= ). Given text (plaintext): 89M= . Ciphertext: 3 mod 89 mod391 47 e C M n= = = . Decrypted text: 235 mod 47 mod391 89 d M C n= = = . This demonstrated that the RSA algorithm performed encryption ( ) , ne and decryption ( ) d correctly, operating on the principles of a public key and a private key algorithm. 4. Conclusion The article proposes a secure public key exchange algorithm combining the foundations of classical elliptic curve cryptography (ECC) and quantum computing. In the proposed approach, operations on points were performed on ECC, public keys were generated, and isogeny mapping elements were simulated using quantum registers. Quantum superposition states were created using the Qiskit library, and quantum components were connected to classical key exchange. The experimental results showed that using this approach, it is possible to create a key exchange mechanism that not only meets classical security requirements, but also is more stable against THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 540 potential quantum attacks. The simplicity and efficiency of the ECC algorithm also confirmed that it is a convenient tool for integrating it with quantum. The results of this work will serve as a basis for in-depth study and development of quantum-resistant cryptosystems, post-quantum security protocols, and cryptographic systems based on isogeny in the future. This article analyzes the mathematical basis of the RSA algorithm, key generation, encryption and decryption formulas. A model was also created and tested in the Python programming language. The experimental results showed that the RSA algorithm performs its function accurately and efficiently. The advantages of the algorithm: Works on the basis of arithmetic operations on a large modulus to ensure security. Allows for secure exchange in public key communication. In order to combat the threats of quantum computing, it is recommended to supplement the RSA algorithm with alternatives based on post-quantum algorithms. REFERENCES 1. Koblitz, N. (1987). Elliptic Curve Cryptosystems. Mathematics of Computation, 48(177), 203–209. https://doi.org/10.2307/2007884 2. Miller, V. S. (1985). Use of Elliptic Curves in Cryptography. In Advances in Cryptology — CRYPTO ’85 Proceedings (pp. 417–426). Springer. 3. Bernstein, D. J., Lange, T. (2017). Post-quantum cryptography. Nature, 549(7671), 188–194. https://doi.org/10.1038/nature23461 4. Jao, D., De Feo, L. (2011). Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. 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