  Building on work of Kontsevich, we introduce a definition of the entropy of a
finite probability distribution in which the "probabilities" are integers
modulo a prime p. The entropy, too, is an integer mod p. Entropy mod p is shown
to be uniquely characterized by a functional equation identical to the one that
characterizes ordinary Shannon entropy. We also establish a sense in which
certain real entropies have residues mod p, connecting the concepts of entropy
over R and over Z/pZ. Finally, entropy mod p is expressed as a polynomial which
is shown to satisfy several identities, linking into work of Cathelineau,
Elbaz-Vincent and Gangl on polylogarithms.
