Algebraic cryptanalysis of symmetric ciphers: modeling and multistep solving strategies

This paper presents a systematized study of algebraic cryptanalysis for symmetric ciphers. It explains how encryption algorithms can be expressed as systems of polynomial equations over finite fields and analyzed using algebraic and logical solvers. The study beginsby detailing block and stream ciphers through the lens of difference algebra, interpreting state updates and key additions as polynomial mappings. It then outlines the main solving strategies used in algebraic cryptanalysis, starting from Gröbner-basis and SAT approaches and progressing to hybrid solving. It next focuses on multistep and oracle-based strategies, which adjust variable guessing according to the hardness of each instance. These strategies extend classical hybrid attacks and make the solving process more efficient. Four case studies on Bluetooth E0, Bivium, Trivium, and Aradi illustrate these ideas. Each example shows how algebraic modeling and adaptive solving reveal structural properties and solvability thresholds in the cipher design. The results show that multistep reasoning reduces computational effort compared with static hybrids and provides a unified way to understand algebraic complexity in both stream and block ciphers.