A Solution

xy mod m = (x mod m)(y mod m) mod m

Instead of doing the multiplication first and then mod-ing, we can mod first to keep the number relatively small.

Use this to implement a procedure that computes

be mod m	=	(be-1 × b) mod m	(algebra)
	=	(be-1 mod m)(b mod m) mod m	(above property)
	=	(be-1 mod m)b mod m	(simplify)

We need a procedure (mod-expt base exponent modulus) to replace (expt message signing-exponent) in the sign function that uses this observation to compute

b^e mod m

(b^e-1 mod m)b mod m.

Since this is defined recursively, multiplications will be put "on hold", and after each multiplication, a mod will be performed.