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The next section provides more information about
the changes we applied to the standard ECC. Then
we talk about the security analysis of our approach.
The paper closes with the description of our future re-
search directions.
2 SHORT ELLIPTIC CURVES
In this section we describe the SEC cryptosystem in
detail.
2.1 Notation
In the rest of our paper we will use the following sym-
bols:
• F
q
- a finite prime field
• E(F
q
) - an elliptic curve over field F
q
• (x,y), where x, y ∈F
q
- a point on the elliptic curve
• O - the point at infinity
• P = (x
P
,y
P
) - a base point on the elliptic curve
• p, where pP = O - order of the point P
• #E = n - order of the elliptic curve
• (k,Q) - a key pair , where the private key k ∈[0, p]
and the public key is Q = kP
• < P > - cyclic group of points generated by P
2.2 SEC based Cryptosystem
As Short Elliptic Curves (SEC) we consider elliptic
curves over 32 to 64-bit finite prime fields. Standard
elliptic curves are based on prime fields with an order
at least twice that long (minimum 128-bits). Short
curves are unsafe, if used with standard ECC proto-
cols. Knowing all the public parameters, finding dis-
crete logarithms, i.e., breaking the security system us-
ing such a small group takes only a few seconds on a
standard PC. SEC based cryptosystem has the follow-
ing features:
• It provides security means for a trusted group of
participants sharing the same SEC parameters.
• It uses standard ECC in a layer which manages
the distribution of SEC parameters (together with
an algorithm for generating the chain of SEC or
the set of SEC with the method for choosing valid
SEC) to the new members of the group. After that
the new group participant is able to manage the
SEC parameters and change them if necessary.
• Encryption and digital signatures within the group
are performed using SEC.
• To improve the security, one SEC setting is valid
for a defined period of time only, after which all
the group members generate the new SEC param-
eters and switch to them. Another option is to
have a precomputed set of SEC and to switch be-
tween them according to some algorithm. Each
group member generates the key pair for the new
SEC. Public key is encrypted using previously
validelliptic curve and distributed to trusted group
members.
• Compression for all transmitted SEC points is
used. It reduces the amount of transmitted data
and additionally improves the security.
2.3 Generation of a SEC
Generation of the new SEC parameters may be done
by each node independently. Each node should get
the same SEC parameters, so the generation process
on each node needs to be synchronized and the new
parameters do not need to be distributed. Only the
public key of each node has to be broadcasted in an
encrypted form using for example, the previous SEC.
The general procedure of generation the SEC param-
eters from scratch is described below.
First, a prime number q has to be chosen to be
the order of the finite field. Then the two integers
a,b ∈ F
q
that are the coefficients in the elliptic curve
equation are to be determined. Next steps are to find
the order of E, to choose a base point and to find its
order.
To prevent from Pohlig-Hellman attack (Blake,
2000) it is necessary that the order of the base point
is the largest possible prime integer. The ideal solu-
tion is to find a point which generates all the points on
the curve E and has a prime order, equal to the curve
order.
Lagrange’s theorem says (Blake, 2000) that the or-
der of an arbitrarily chosen group element P 6= O di-
vides the order of the group. Thus, when a curve has
a prime order n, there are only two possible orders of
points on the curve: 1 and n, and since 1P = P 6= O ,
the only possible order of any point P is n. The
Hasse’s theorem (Koblitz, 1998) brings very useful
information about the interval in which to look for the
curve order and in the above mentioned case it is also
the order of the base point. This theorem says that for
an elliptic curve E over a finite field F
q
the order of
E(F
q
) satisfies
q+ 1−2
√
q ≤ #E(F
q
) ≤ q + 1+ 2
√
q (1)
To find the curve order in this interval we proceed as
follows:
Let P = (x,y) be randomly chosen point on E and let
Q = (q+ 1 + 2
√
q)P (2)
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