Computation of cyclic redundancy checks may use whichever form is more convenient. Parallel update for a byte or a word at a time can also be done explicitly, without a table. This is normally used in high-speed hardware implementations. For each bit an equation is solved after 8 bits have been shifted in. The following tables list the equations for some commonly used polynomials, using following symbols: As the CRC-32 polynomial has a large number of terms, when computing the remainder a byte at a time each bit depends on several bits of the previous iteration. In byte-parallel hardware implementations this calls for either multiple-input
Stochastic computing can take the logical AND of the two streams. The probability of a one in the output stream is formula_5. By observing enough output bits and measuring the frequency of ones, it is possible to estimate formula_5 to arbitrary accuracy. The operation above converts a fairly complicated computation (multiplication of formula_3 and formula_4) into a series of very simple operations (evaluation of formula_9) on random bits. More generally speaking, stochastic computing represents numbers as streams of random bits and reconstructs numbers by calculating frequencies. The computations are performed on the streams and translate complicated operations on formula_3 and formula_4 into