Elliptic Curve DSA

2007-06-12T23:48:55Z

203.206.169.155: Making DSA a link makes it less confusing - I kept thinking ECDSA was being compared with itself.

'''Elliptic Curve DSA (ECDSA)''' is a variant of the [[Digital Signature Algorithm]] (DSA) which operates on [[elliptic curve]] [[group (mathematics)|groups]]. The EC variant provides smaller key sizes for (supposedly) similar security level. On the other hand, the execution time is roughly the same and the signature size is exactly the same: <math>4 t</math>, where <math>t</math> is the security parameter. For example, [[DSA]] with 1024-bit <math>p</math> and 160-bit <math>q</math> and ECDSA over the 160-bit prime field both produce 320-bits signatures and need only few milliseconds [http://www.eskimo.com/~weidai/benchmarks.html] for execution on a 2 GHz Pentium.

==Signature generation algorithm==

Suppose [[Alice and Bob|Alice]] wants to send a signed message to [[Alice and Bob|Bob]]. Initially, the curve parameters <math>(q, FR, a, b, G, n, h)</math> must be agreed upon. Also, Alice must have a key pair suitable for elliptic curve cryptography, consisting of a private key <math>d_A</math> (a randomly selected integer in the interval <math>[1, n-1]</math>) and a public key <math>Q_A</math> (where <math>Q_A = d_A G</math>).

For Alice to sign a message <math>m</math>, she follows these steps:

# Calculate <math>e = \textrm{HASH}(m)</math>, where HASH is a [[cryptographic hash function]], such as [[SHA-1]].
# Select a random integer <math>k</math> from <math>[1, n-1]</math>.
# Calculate <math>r = x_1 \pmod{n}</math>, where <math>(x_1, y_1) = k G</math>. If <math>r = 0</math>, go back to step 2.
# Calculate <math>s = k^{-1}(e + r d_A ) \pmod{n}</math>. If <math>s = 0</math>, go back to step 2.
# The signature is the pair <math>(r, s)</math>.

==Signature verification algorithm==

For Bob to authenticate Alice's signature, he must have a copy of her public key <math>Q_A</math>. He follows these steps:

# Verify that <math>r</math> and <math>s</math> are integers in <math>[1, n-1]</math>. If not, the signature is invalid.
# Calculate <math>e = \textrm{HASH}(m)</math>, where HASH is the same function used in the signature generation.
# Calculate <math>w = s^{-1} \pmod{n}</math>.
# Calculate <math>u_1 = ew \pmod{n}</math> and <math>u_2 = rw \pmod{n}</math>.
# Calculate <math>(x_1, y_1) = u_1 G + u_2 Q_A</math>.
# The signature is valid if <math>x_1 = r \pmod{n}</math>, invalid otherwise.

Note that using Straus's algorithm (also known as Shamir's trick) a sum of two scalar multiplications <math>u_1 G + u_2 Q_A</math> can be calculated faster than with two scalar multiplications.

==References==
* Accredited Standards Committee [http://www.x9.org X9], ''American National Standard X9.62-2005, Public Key Cryptography for the Financial Services Industry, The Elliptic Curve Digital Signature Algorithm (ECDSA)'', November 16, 2005.
* Certicom Research, [http://www.secg.org/download/aid-385/sec1_final.pdf ''Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography''], Version 1.0, September 20, 2000.
* López, J. and Dahab, R. [http://citeseer.ist.psu.edu/333066.html ''An Overview of Elliptic Curve Cryptography''], Technical Report IC-00-10, State University of Campinas, 2000.
* Daniel J. Bernstein, [http://cr.yp.to/papers/pippenger.pdf Pippenger's exponentiation algorithm], 2002.
* Daniel R. L. Brown, ''Generic Groups, Collision Resistance, and ECDSA'', Designs, Codes and Cryptography, '''35''', 119-152, 2005. [http://eprint.iacr.org/2002/026 ePrint version]
* Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart, editors, ''Advances in Elliptic Curve Cryptography'', London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.
* Darrel Hankerson, Alfred Menezes and Scott Vanstone, ''Guide to Elliptic Curve Cryptography, Springer'', Springer, 2004.

==External links==
* [http://csrc.nist.gov/cryptval/dss.htm Digital Signature Standard; includes info on ECDSA]
* [http://www.certicom.com/index.php?action=res,cc&issue=2-2&&article=1 Commercial/Marketing comparison between RSA and ECDSA]

==See also==

* [[Elliptic curve cryptography]]

{{Crypto navbox | public-key}}

[[Category:Asymmetric-key cryptosystems]]

[[es:ECDSA]]
[[fr:Elliptic curve digital signature algorithm]]

Wikipedia - Benutzerbeiträge [de]

Elliptic Curve DSA