Prime Numbers

9.7 Exercises 523
9.23. How general can be the initialization of x in Algorithm 9.2.11?
9.24. Write out a (very) simple algorithm that uses Algorithm 9.2.11 to
determine whether a given integer N is a square. Note that there are much
more efficient ways of approaching the problem, for example first ruling out
the square property modulo some small primes [Cohen 2000].
9.25. Implement Algorithm 9.2.13 within a Lucas–Lehmer test, to prove or
disprove primality of various Mersenne numbers 2 q − 1. Note that with the
special form mod reduction, one does not even need general multiplication for
Lucas–Lehmer tests; just squaring will do.
9.26. Prove that Algorithm 9.2.13 works; that is, it terminates with the
correct returned result.
9.27. Work out an algorithm for fast mod operation with respect to moduli
of the form
p =2 a +2 b + ···+1,
where the existing exponents (binary-bit positions) a,b,... are sparse; i.e.,
a small fraction of the bits of p are 1’s. Work out also a generalization in
which minus signs are allowed, e.g., p =2 a ± 2 b ±···±1, with the existing
exponents still being sparse. You may find the relevant papers [Solinas 1999]
and [Johnson et al. 2001] of interest in this regard.
9.28. Some computations, such as the Fermat number transform (FNT)
and other number-theoretical transforms, require multiplication by powers of
two. On the basis of Theorem 9.2.12, work out an algorithm that for modulus
N =2 m +1, quickly evaluates (x2 r )modN for x ∈ [0,N−1] and any (positive
or negative) integer r. What is desired is an algorithm that quickly performs
the carry adjustments to which the theorem refers, rendering all bits of the
desired residue in standard, nonnegative form (unless, of course, one prefers
to stay with a balanced representation or some other paradigm that allows
negative digits).
9.29. Work out the symbolic powering relation of the type (9.16), but for
the scheme of Algorithm 9.3.1.
9.30. Prove that Algorithm 7.2.4 works. It helps to track through small
examples, such as n = 00112, for which m = 10012 (and so we have
intentionally padded n to have four bits). Compare the complexity with that of
a trivial modification, suitable for elliptic curve arithmetic, to the “left-right”
ladder, Algorithm 9.3.1, to determine whether there is any real advantage in
the “add-subtract” paradigm.
9.31. For the binary gcd and extended binary algorithms, show how to
enhance performance by removing some of the operations when, say, y is
prime and we wish to calculate x −1 mod y. The key is to note that after
the [Initialize] step of each algorithm, knowledge that y is odd allows the
removal of some of the internal variables. In this way, end up with an inversion