Prime Numbers

8.8 Research problems 437
element of Qn black, and all nonzero elements white. Imagine further that this
object is the full infinite-dimensional Qn matrix, but compressed into a finite
planar square, so that we get, if you will, a kind of “snowflake” with many
holes of black within a fabric of white. Now, argue that for prime modulus p,
so that the mod matrix is Qp, the fractal dimension of the “snowflake” object
is given by
δ =
ln(p(p +1)/2)
.
ln p
Technically, this is a “box dimension,” and for this and other dimension
definitions one source is [Crandall 1994b] and references therein. (Hint: The
basic method for getting δ is to count how many nonzero elements there
are in an upper-left p k × p k submatrix of Qp, and see how this scales with
the submatrix size p 2k .) Thus for example, the Pascal triangle modulo 2
has dimension δ = (ln3)/(ln 2) and the triangle modulo 3 has dimension
δ =(ln6)/(ln 3). The case p = 2 here gives the famous Sierpiński gasket, a
well-studied object in the theory of fractals. It is sometimes said that such a
“gasket” amounts to “more than a line but less than the plane.” Clarify this
vague statement in quantitative terms, by looking at the numerical magnitude
of the dimension δ.
Extensions to this fractal-dimension exercise abound. For example, one
finds that for prime p, in the upper-left p × p submatrix of Qp, thenumber
of nonzero elements is always a triangular number. (A triangular number is
a number of the form 1 + 2 + ...+ n = n(n +1)/2.) Question is, for what
composite n does the upper-left n × n submatrix have a triangular number
of nonzero elements? And here is an evidently tough question: What is the
fractal dimension if we consider the object in “gray-scale,” that is, instead
of white/black pixels that make up the gasket object, we calculate δ using
proper weight of an element of Qp not as binary but as its actual residue in
[0,p− 1]?
8.27. In the field of elliptic curve cryptography (ECC) it is important to be
able to construct elliptic curves of prime order. Describe how to adapt the
Schoof method, Algorithm 7.5.6, so that it “sieves” curve orders, looking for
such a prime order. In other words, curve parameters a, b would be chosen
randomly, say, and small primes L would be used to “knock out” a candidate
curveassoonasp+1−t is ascertained as composite. Assuming that the Schoof
algorithm has running time O ln k
p , estimate the complexity of this sieving
scheme as applied to finding just one elliptic curve of prime order. Incidentally,
it may not be efficient overall to use maximal prime powers L =2 a , 3 b , etc.
(even though as we explained these do work in the Schoof algorithm) for such
a sieve. Explain why that is. Note that some of the complexity issues herein
are foreshadowed in Exercise 7.29 and related exercises of that chapter.
If one did implement a “Schoof sieve” to find a curve of prime order, the
following example would be useful in testing the software:
p =2 113 − 133, a = −3, b = 10018.