Fault injection attacks are a major threat on the reliability and security of cryptographic systems using finite field arithmetic. This paper explores how arithmetic processes like addition, multiplication and inversion in finite fields are affected by deliberate errors. The paper examines the different fault injection methods such as voltage glitching, clock manipulation, and laser-based fault injection methods to discuss its usefulness in interference with computations. An in-depth analysis of vulnerabilities points to the short-term and long-term faults that might cause leakage of information and key recovery in cryptographic implementations. Also, attack methods, common fault, on symmetric and asymmetric schemes are discussed. Experimental evaluation reveals that there is quantifiable deterioration in computational accuracy and there is higher vulnerability to side-channeling. The results provide an insight into the importance of effective fault detection and mitigation measures to increase the resiliency of cryptographic hardware and software systems.