Abstract. Substitution boxes (S-boxes) are generally the only non-linear part of a block cipher. Using key-dependent S-boxes rather than static ones might increase the security of a block cipher but it would take too much time to build a good key-dependent S-box from scratch just before ciphering or deciphering. What we can do is to transform an existing S-box, assuming
1) the relevant cryptographic properties of the S-box are preserved;
2) the execution of the transformation is sufficiently fast.

Abstract. Primality proving... Cryptography... As soon as we want to build an elliptic curve with a known order over a Z/p field using the so-called complex multiplication, we have to find a root of a class polynomial. Depending on the degree of this polynomial (and on the size of the prime p), this operation might be very lengthy. More concretely, suppose we have to find a root of H[-12932920](x) (the degree of this polynomial is 832). Suppose now we can compute a factor of degree 13 more quickly than we can compute the whole polynomial H[-D](x) itself. Of course, it would make the task easier...