ATPG for very large scale integrated circuit designs is an important problem in industry. With the advent of SOC designs, testing and verification of the core-based designs become a challenging problem. This paper presents an algebraic test generation algorithm with unspecified variable assignments. Given a stuck at fault of the circuit with unspecified signals, the proposed algorithm uses a new encoding scheme for unspecified variable assignments, and solves the Boolean satisfiability formula representing the Boolean difference to obtain a test pattern. Experimental results demonstrate the efficiency and feasibility of the proposed algorithm. With the advent of SOC designs, testing and verification of the core-based designs become a challenging problem. This paper presents an algebraic test generation algorithm with unspecified variable assignments. Given a stuck at fault of the circuit with unspecified signals, the proposed algorithm uses a new encoding scheme for unspecified variable assignments, and solves the Boolean difference to obtain a test pattern. Experimental results demonstrate the efficiency and feasibility of the proposed. algorithm.