Abstract
For a long time, number theory has influenced information security
and cryptography. This thesis adds examples of its influence.
The first topic is related with Broadcast Group Key Management (BGKM)
in cryptography. After the Access Control Polynomial (ACP) BGKM scheme was
proposed, people tried to check its basic security properties in
BGKM. They found that it has a weakness in the key hiding
property by finding a counterexample when $p=2$. Here, we give strong
evidence that it has a weakness in its key hiding property for
all sufficiently large primes.
The second topic is a well known integer factoring algorithm SQUFOF,
which stands for SQUare FOrm Factorization, invented by Daniel Shanks.
At present, SQUFOF is the fastest factoring algorithm for numbers between
$10^$ and $10^$. In Gower's thesis, he made conjectures
about the probability distribution of the number of forms that SQUFOF must
examine before finding a proper square form and the number of forms
enqueued during the factorization of $N$. We propose a different
probability distribution (geometric rather than exponential) than did
Gower, and we use Gower's data to support our conclusions.
The third topic is the period of the Bell numbers $B(n)$ modulo a
prime. It was proved by Williams that the minimum
period of the sequence $\{B(n) \bmod ~p\}$, $n=0$, 1, 2, $\ldots$,
divides $N_p=(p^p-1)/(p-1)$.
In fact, the minimum period equals $N_p$ for every prime $p$ for which
this period is known. Several people have conjectured that the minimum period
is always $N_p$. This thesis presents a heuristic argument supporting the
conjecture.
