An improved cross-checking circuit is provided for use within a Reed-Solomon error correction and cross checking apparatus for performing error correction and cross checking upon a data block within an incoming stream of substantially contiguous data blocks flowing from a source to a destination. The circuit is based upon a distinguished primitive element, alpha.sup.1 (2B (Hex))=x.sup.5 +x.sup.3 +x+1, of a Galois field whose elements are represented by residue classes of binary polynomials modulo p(x)=x.sup.8 +x.sup.4 +x.sup.3 +x.sup.2 +1. The apparatus includes a microcontroller for supervising the flow of the data blocks and for making calculations related to error corrections, and a Galois field syndrome generator and remainder recovery circuit is connected to receive the incoming stream and recover therefrom plural error correction remainder bytes for each block and selectively to hold said bytes in a syndrome latch, the remainder bytes being related to syndrome bytes appended to the data block. The generator and remainder recovery circuit includes the Reed-Solomon cross-checking circuit for recovering cross-checking remainder information related to cross check syndrome information in accordance with the polynomial GXC(x)=x.sup.2 +alpha.sup. 134 x+alpha.sup.1.