Code and ciphers: Julius Caesar, the Enigma and the internet

216
appendix
So, for example, the points (1,�5) are integer points of the curve
Y 2 �X 3 �2X�3 (mod 19).
From any one or two points on the curve another may be constructed by
using the tangent at the single point or the chord joining the two points.
This tangent or chord meets the curve in a third point which must have
rational co-ordinates and these rationals are convertible into integers in
GF( p), the Galois field (mod p). So, for example, for the curve above with
p�19, the equation of the tangent at the point (1, 5) is
2Y�X�9
and we find that this tangent meets the curve again at the point where
X��7/4. This is equivalent to an integer value in GF(19); since 4 is the
denominator of this fraction we must first find the integer n such that
4n�1 (mod 19).
This gives n�5 since 20�1�19�1; hence �7/4�(�7)�5��35�3
(mod 19) and so the fraction �7/4 is equivalent to the integer 3 in GF(19).
This gives 3 as the integer value of X and the corresponding value of Y,
obtained from the tangent above, is 6. Since
Y 2 �36 and X 3 �2X�3�27�6�3�36
we have verified that the points (3,�6) lie on the curve above. (We only
need to show that they lie on the curve in GF(19); in fact they lie on the
curve (mod p) for all p, but that is a fluke; this will not normally be the case.)
Thus another integer point is found on the curve. Since all arithmetic
is (mod p) there are only a finite number of possible points (X, Y) with
integer values. It follows therefore that the construction method that
gives new points must eventually terminate. If we start with a particular
(integer) point Q(X, Y) on the curve we can generate a finite set, �Q�, of
points which we denote by 2Q, 3Q, 4Q,... etc. (these are not to be confused
with the points (2X, 2Y) etc). For example, starting with the point Q (1, 5)
on the curve above we have just found the point, 2Q, generated from the
tangent at Q. Continuing in this way we find that we are led to the points
2Q�(3,�6), 4Q�(10,�4), 8Q�(12,�8) and so on
(for another example see [13.9]).
If we are given a point R(X�, Y�) and asked to find if there is an integer n
such that R�nQ, a point within the set �Q�, we will have a very difficult