Let E denote the elliptic curve y2 ≡ x3 + x + 26 mod 127. It can be shown that #E = 131, which is a prime
number. There any non-identity element in E is a generator for (E, +). Suppose the ECDSA is
implemented in E, with A = (2,6) and m = 54.
(a) Compute the public key B = mA.
(b) Compute the signature on a message x if SHA-1 (x) = 10, when k = 75.
(c) Show the computations used to verify the signature constructed in part(b).
Note: See next page for the elliptic curve subgroup which is needed for the exercise.
Variants of the E/Gamal Signature Scheme 297
The signature (94, 97) on the message digest 22 is verified by the following computations:
J- 1 = 97-l mod 101 = 25
e1 = 22 x 25 mod 101 = 45
e2 = 94 x 25 mod 101 = 27
(17045456727 mod 7879) mod 101 = 2518 mod 101 = 94.
When the DSA was proposed in 1991, there were several criticisms put forward.
One complaint was that the selection process by NIST was not public. The
standard was developed by the National Security Agency (NSA) without the input
of U.S. industry. Regardless of the merits of the resulting scheme, many people
resented the “closed-door” approach.
Of the technical criticisms put forward, the most serious was that the size of the
modulus p was fixed initially at 512 bits. Many people suggested that the modulus
size not be fixed, so that larger modulus sizes could be used if desired. In reponse
to these comments, NIST altered the description of the standard so that a variety
of modulus sizes were allowed.
7.4.3 The Elliptic Curve DSA
In 2000, the EIJiptic Curve Digital Signature Algorithm (ECDSA) was approved
as FIPS 186-2. This signature scheme can be viewed as a modification of the
DSA to the setting of elliptic curves. We have two points A and B on an elliptic
curve defined over Z’..p for some prime p. 1 The discrete logarithm m = logA B
is the private key. (This is analogous to the relation (3 = aa mod p in the DSA,
where a is the private key.) The order of A is a large prime number q. Computing
a signature involves first choosing a random value k and computing kA (this is
analogous to the computation of akin the DSA).
Now here is the main difference between the DSA and the ECDSA. In the
DSA, the value ak mod p is reduced modulo q to yield a value I which is the
first component of the signature ( /, J). In the ECDSA, the analogous value is r,
which is the x-co-ordinate of the elliptic curve point kA, reduced modulo q. This
value r is the first component of the signature ( r, s).
Finally, in the ECDSA, the values is computed from r, m, k, and the message
x in exactly the same way as J is computed from I• a, k and the message x in the
We now present the complete description of the ECDSA as Cryptosystem 7 .5.
1 We note that the ECDSA also permits the use of elliptic curves defined over finite fields lF 2 n , but
we will not describe this variation here.
298 Signature Schemes
Cryptosystem 7.5: Elliptic Curve Digital Signature Algorithm
Let p be a large prime and let Ebe an elliptic curve defined over IFp. Let A be
a point on E having prime order q, such that the Discrete Logarithm problem
in (A) is infeasible. Let ‘.P = {O, 1}*, A= 7l.q* x 7l.q*, and define
X = {(p,q,E,A,m,B): B = mA},
where 0 < m < q – 1. The values p, q, E, A and Bare the public key, and m
is the private key.
For J( = (p, q, E, A, m, B), and for a (secret) random number k, 1 < k <
q – 1, define
sigK(x, k) = (r, s),
kA = (u, v)
r = u mod q, and
s = k- 1(SHA-l(x) + mr) mod q.
(If either r = 0 or s = 0, a new random value of k should be chosen.)
For x E { 0, 1} * and r, s E 7l.q *, verification is done by performing the following
w = s -1 mo d q
i = wSHA-l(x) modq
j = wr mod q
verK(x, (r, s)) =true<=? u mod q = r.
We work through a tiny example to illustrate the computations in the ECDSA.
Example 7.5 We will base our example on the same elliptic curve that was used
in Section 6.5.2, namely, y2 = x 3 + x + 6, defined over 7l. 11. The parameters of
the signature scheme are p = 11, q = 13, A= (2, 7), m = 7 and B = (7, 2).
Suppose we have a message x with SHA-l(x) = 4, and Alice wants to sign
Provably Secure Signature Schemes
the message x using the random value k = 3. She will compute
(u,v) = 3 (2, 7) = (8,3)
r = u mod 13 = 8, and
s = 3- 1 (4 + 7 x 8) mod 13 = 7.
Therefore (8, 7) is the signature.
Bob verifies the signature by performing the following computations:
w = 7-l mod 13 = 2
i = 2 x 4 mod 13 = 8
j = 2 x 8 mod 13 = 3
(u, v) = 8A + 3B = (8, 3), and
u mod 13 = 8 = r.
Hence, the signature is verified.
7.5 Provably Secure Signature Schemes
We present some examples of provably secure signature schemes in this section.
First, we describe a construction for a one-time signature scheme based on an
arbitrary one-way (i.e., preimage resistant) function, say f. This scheme can be
proven secure against a key-only attack provided that f is a bijective function.
The second construction is for a signature scheme known as Full Domain Hash.
This signature scheme is provably secure in the random oracle model provided
that it is constructed from a trap-door one-way permutation.
7.5.1 One-time Signatures
In this section, we describe a conceptually simple way to construct a provably secure
one-time signature scheme from a one-way function. (A signature scheme is
a one-time signature scheme if it is secure when only one message is signed. The
signature can be verified an arbitrary number of times, of course.) The description
of the scheme, which is known as the Lamport Signature Scheme, is given in
Cryptosystem 7.6.
Informally, this is how the system works. A message to be signed is a binary ktuple.
Each bit of the message is signed individually. If the ith bit of the message
equals j (where j E {O, 1} ), then the ith element of the signature is the value
Yi,j, which is a preimage of the public key value z;,j. The verification consists
simply of checking that each element in the signature is a preimage of the public



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