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In this paper, we present a systematic method for designing single error correcting (SEC) and double error detecting finite field (Galois field) multipliers over GF(2m). The detection and correction are done on-line. We use multiple Parity Predictions to detect and correct errors. Specifically, a structural approach is first presented. The predicted parity bits are derived from the primitive polynomials for generating the fields. Further, a hybrid approach is presented where the multipliers and PP circuits are synthesised, and the decoding and correction circuits are structurally combined to form the complete error correcting designs. Although the article considers only a finite field multiplier, without loss of generality, the technique could be applied to any combinational logic circuit. Our technique, when compared with existing techniques, gives better performance. We show that our SEC multipliers over GF(2m) require about 100% extra hardware, whereas with the traditional SEC techniques, such as the triple-modular redundancy (TMR), this figure is more than 200%. Moreover, the multiple bit error detection and correction using multiple Hamming code were also introduced. [ABSTRACT FROM AUTHOR]