Carry-free addition and borrow-free subtraction on the redundant modified signed-digit number are adopted for lensless polarization-encoded optical shadow casting (POSC). First, a new recoding truth table, which has a symmetrical complemented relationship between the input minterms (entries that produce an output of 1 or -1) and their corresponding output bits, is developed for the redundant signed-digit numbers. Then, the carry-free POSC adder/subtracter is designed by determining the encoding of the redundant input bits for generating the sum/difference output bits and the source light-emitting-diode pattern. Two design methods are presented: direct implementation and truth-table partitioning. It is shown that the number of pixel subcells in the input encoding of the POSC system can be significantly reduced using the new symmetrical recoding algorithm for the modified signed-digit number representation.