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Notes, Chapter 14

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14 El Gamal and elliptic curve cryptography

The El Gamal cryptosystem is based on the DDH assumption and is
provably secure under the assumption of the latter. Just like DH-key
exchange it is parametrised by a group (generating algorithm) G(n) and
works as follows:

Gen(n) :
  (G,q,g) = G(n)
  x <- Z_q
  h <- g^x
  pk = (G,q,g,h)
  sk = (G,q,g,x)
  output (pk,sk)

Enc_(G,q,g,h)(m) :
  check that m:G
  y <- Z_q
  c = (g^y,h^y*m)
  output c

Dec_(G,q,g,g,x)(c1,c2) :
  m = c2 / c1^x
  output m

Let us first see that this scheme is correct: If c1=g^y and c2=h^y*m
where h=g^x then c2 / c1^x= g^(xy) * m / g^(xy) = m

We also notice that messages and ciphertexts are actually elements of
the group G which must therefore by somehow encoded as binary
strings. This can be done in various ways for the various groups, see
KL for details.

Theorem: If the DDH assumptions holds for a group (generating
algorithm) G() then the El Gamal cryptosystem basd on G() is
semantically secure.

Proof: Let A be an adversary against the El Gamal system and let its
success probability be 1/2+eps(n). We must show that eps(n) is
negligible.  We construct from A an adversary A' as in the DDH

Suppose that A' receives g^x, g^y, u where u is either g^(xy) or g^z
for random z and A' is to decide which of the two.

We forward (G,q,g,g^x) as "public key" to A and wait for it to output
m0,m1:G. We choose a random bit b and build c = (g^y,u*m_b) and let b'
be the bit that A returns.

If u=g^(xy) then A is played with according to the rules of the El
Gamal cryptosystem and therefore b'=b with probability 1/2+eps(n). If
u=g^z for random z then u*m0 is also uniformly random (since x|->x*m0
is a bijection G->G) and therefore the probability that b'=b is
exactly 1/2. The difference between the two probabilities is precisely
eps(n) which under the DDH assumption will be negligible. QED

Elliptic curve cryptography
- - - - - - - - - - - - - -

As already mentioned, elliptic curves furnish a large reservoir of
cyclic groups for which taking discrete logarithms, CDH, and DDH are
believed to be hard. With existing algorithms, solving these problems
on elliptic curve groups is consistently harder than doing so for
groups of invertible elements mod p or in a finite field.

Recall that an elliptic curve over some field F of characteristic =/=
2,3 is defined by two constants a, b and comprises the points (x,y)
that solve the equation

                        y^2 = x^3 + ax + b

Here, one assumes that the *discriminant* 4a^3+27b^2 =/= 0 for
otherwise the curve might be *singular*, i.e. contain anomalies like
isolated points, sharp bends ("cusps") or similar. When viewed over
the reals such a curve looks roughly as follows:

                                     |           |
                                     |          /
                                   - | -       /
                                  /  |   \ _ -
                                 |   |
                                 |   |     - _
                                  \  |   /     \ 
                                    -| -        \
                                     |		 |

Depending on the parameters the knob to the left may also become
detached and form a closed loop sitting to the right. In either case
whenever a line intersects the curve in two points it meets it again
in a third point except when the line is parallel to the y-axis in
which case we say that the third point of intersection is at infinity,
here denoted O.

This property also holds for curves over other fields, in particular
finite ones which are of interest for cryptography. There is an
explicit formula for obtaining the third point of intersection. To
wit, if (x1,y1) and (x2,y2) lie on the curve then the line through
them is y=m(x-x1)+y1 where m=(y2-y1)/(x2-x1). Plugging this into the
curve yields

                    (m(x-x1)+y1)^2 = x^3+ax+b

The three solutions of this equation are x=x1, x=x2, x=m^2-x1-x2, so
the last one is the desired third point of intersection.

Now, it is a remarkable fact that the points on the curve together
with O form an abelian group, traditionally written additively, where
the sum of points P and Q is obtained by first taking the third curve
point on the line through P and Q, call it R=(x_R,y_R), and then
reflecting R on the x-axis, i.e. P+Q=(x_R,-y_R).

This, however, only applies when P=/=Q and when P and Q do not have
the same x-coordinate. In the latter case, one has P+Q=O, the special
point at infinity. In the former case, i.e. when P=Q, one must take
the tangent through P and intersect it with the curve.

One can show using implicit differentiation that the coordinates of
that point are given by

       (((3x_1^2+a)/2y_1)^2 - 2x1, -y1+((3x_1^2+a)/2y_1)^2(x1-x2))

and this formula, just as the one above for the general case, also
makes sense for finite fields where there are no "tangents". Yet
another special case is when P or Q is O. One simply puts O+P=P+O=P.

Curves over fields of characteristic 2 or 3 are also possible, but
then both the defining equations and the formulas for addition of
points become slightly different (notice e.g. that in the formula
above we divide by 2). We refer to the literature, but stress that it
is very important not to make mistakes with the implementation of the
addition formulas and that doing so has lead to security flaws.

We have already seen that addition of points is well-defined and
commutative. It is also clear that the inverse of a point (x,y) is
given by (x,-y), i.e. reflection on the x-axis. Associativity of
addition follows from nontrivial results, e.g. the Weierstrass
p-function, but can in principle also be validated "by hand" on the
basis of the above formulas.

For integer n and point P on a curve we use the notation n.P for
P+P+...+P with summands. This is the same as the exponent notation in
multiplicatively written groups. Therefore, the "repeated squaring"
method can be used here as well:

              100.P = 4.P+32.P+64.P = 2.(2.(P + 2.(2.(2.(P+2.P)))))

Now, the plan is to use the groups coming from elliptic curves for
cryptosystems like Diffie-Hellman key exchange or El Gamal. For this
to work, a couple more considerations are in order: we need to encode
plaintexts as points on the group and we need to ascertain that the
group is cyclic or else move to a cyclic subgroup.

In order to find a cyclic group one can rely on one of the publicly
available and recommended curves. Many of them have a prime number of
points and hence are cyclic with any element other than O being a
generator. Alternatively, one can try out random coefficients a and b
until one finds a curve with a prime number of points. To do this, one
needs to invoke Schoof's algorithm which allows one to determine the
number of points on a given elliptic curve.

If one would like to use a curve with a nonprime number, say N, of
points, then one must factor N and it is then important that N has a
large prime factor p, i.e. p>sqrt(N). In this case, by the
representation theorem for finite(ly generated) abelian groups there
is a subgroup of order p and any of its nonzero elements are
generators. Moreover, the remaining subgroups have smaller order.  To
check whether a point P is in the subgroup of order p it suffices to
check that p.P = O. To obtain a point in that subgroup one can pick a
random point P and if Q := N/p.P =/= O then p.Q is in the subgroup,
hence Q can serve as a generator.

For the most, however, one uses one of the already publicised elliptic
curves such as those recommended by the NIST "for government use".
There is no need to choose a "fresh" curve for every application and
on the other hand, some curves are "weak" and should be avoided.

In order to encode plaintexts as points we can proceed as
follows. First, we note that it suffices to store the x-coordinate of
a point and a single bit telling whether the y-coordinate is above or
below the x-axis, i.e., which of the two square roots of x^3 +ax+b are
meant. This is known as point compression. As for the x-coordinates
one fixes a bijection alpha between bitstrings of length l=log(q) and
field elements. Now one puts t=l-k where k is chosen sufficiently
large. Then, to encode message x of length t one searches for a string
z of length k so that alpha(x||z) has a square root y over GF(q),
i.e., is a quadratic residue. At that point, (alpha(x||z),y) can be
taken as the desired encoding of x as a curve point.  For random
padding z the chance for alpha(x||z) to be a quadratic residue is
about 1/2, so after a short number of iterations one is almost sure to
find an appropriate encoding. We also remark that there exist
efficient algorithms for testing whether a given field element is a
quadratic residue and if so to compute its square root.

With all this machinery in place, the Diffie-Hellman and El Gamal
cryptosystems can be used with elliptic curve groups without any
change, bearing in mind, of course, that instead of exponentiation
g^x, we now use multiplication x.P and instead of multiplication we
use addition, so that, e.g. the encryption of message m encoded as a
point M becomes (y.P,y.Q+M) where P is the generator and Q=x.P

We close by remarking that since the early 2000 elliptic curve
cryptography (ECC) has become more and more common. Also the bitcoin
cryptocurrency uses it.

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