Prime Numbers

8.7 Exercises 431
universal Golden scale, based on a geometry of intervals related in Golden
proportion. The author provides the ratios and dimensions of its intervals
and explains the specific intonation interest of such a cycle of Golden fifths,
unfolding into microtonal coincidences with the first five significant prime
numbers ratio intervals (3:5:7:11:13).
From these and other musicology references it appears that not just the
very smallest primes, rather also some two-digit primes, play a role in music
theory. Who can tell whether larger primes will one day appear in such
investigations, especially given how forcefully the human–machine–algorithm
interactions have emerged in modern times?
8.7 Exercises
8.1. Explain quantitatively what R. Brent meant when he said that to
remember the digits of 65537, you recite the mnemonic
“Fermat prime, maybe the largest.”
Along the same lines, to which factor of which Fermat number does the
following mnemonic of J. Pollard apply?
“I am now entirely persuaded to employ rho method, a handy trick, on
gigantic composite numbers.”
8.2. Over the years many attacks on the RSA cryptosystem have been
developed, some of these attacks being elementary but some involving deep
number-theoretical notions. Analyze one or more RSA attacks as follows:
(1) Say that a security provider wishes to live easily, dishing out the same
modulus N = pq for each of U users. A trusted central authority, say,
establishes for each user u ∈ [1,U] a unique private key Du and public
key (N,Eu). Argue carefully exactly why the entire system is insecure.
(2) Show that Alice could fool (an unsuspecting) Bob into signing a bogus (say
harmful to Bob) message x, in the following sense. Referring to Algorithm
8.1.4, say that Alice chooses a random r and can get Bob to sign and
send back the “random” message x ′ = r EB x mod NB. Show that Alice
can then readily compute an s such that s EB mod NB = x, so that Alice
would possess a signed version of the harmful x.
(3) Here we consider a small-private-exponent attack based on an analysis
in [Wiener 1990]. Consider an RSA modulus N = pq with q