We employ the methods of statistical physics to study the performance of
Gallager type error-correcting codes. In this approach, the transmitted
codeword comprises Boolean sums of the original message bits selected by two
randomly-constructed sparse matrices. We show that a broad range of these codes
potentially saturate Shannon's bound but are limited due to the decoding
dynamics used. Other codes show sub-optimal performance but are not restricted
by the decoding dynamics. We show how these codes may also be employed as a
practical public-key cryptosystem and are of competitive performance to modern
cyptographical methods.Comment: 6 page