Quantum Circuit Realization and Grover Cryptanalysis of the Hybrid ARX-SPN Cipher GFSPX

Abstract

The security of classical symmetric-key primitives is fundamentally challenged by the emergence of quantum computing, necessitating a rigorous evaluation of their post-quantum resilience. This paper presents a comprehensive quantum circuit realization and Grover cryptanalysis of GFSPX, a lightweight block cipher featuring a 64-bit data block and a 128-bit secret key. GFSPX utilizes a unique hybrid architecture that integrates a 4-branch generalized Feistel structure with both Addition-Rotation-XOR (ARX) and Substitution-Permutation Network (SPN) components. Our quantum implementation optimizes resource distribution by exploiting the inherent reversibility of the Feistel network and employing a compact ripple-carry adder for the ARX layers. The proposed architecture achieves a qubit-optimized footprint of 209 qubits with a baseline quantum cost of 32,498 and a circuit depth of 7,617. To evaluate the cipher's resistance against quantum adversaries, we construct a parallelized Grover oracle using three plaintext-ciphertext pairs to eliminate spurious matches. Our analysis reveals that the total quantum cost of a key-recovery attack on GFSPX is $1.12 \times 2^{159}$ quantum gates. Although this cost falls below the NIST Level 1 security threshold of $2^{170}$, the hybrid ARX-SPN design demonstrates a higher quantum attack resistance among other lightweight designs. These findings provide critical insights into the balance between classical efficiency and quantum resilience in next-generation cryptographic designs for resource-constrained environments.