# Elliptic Curve DSA

Elliptic Curve DSA

Elliptic Curve DSA (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which operates on elliptic curve groups. As with elliptic curve cryptography in general, the bit size of the public key believed to be needed for ECDSA is about twice the size of the security level, in bits. By comparison, at a security level of 80 bits, meaning an attacker requires about the equivalent of about $2^\left\{80\right\}$ signature generations to find the private key, the size of a DSA public key is at least 1024 bits, whereas the size of an ECDSA public key would be 160 bits. On the other hand, the signature size is the same for both DSA and ECDSA: $4 t$ bits, where $t$ is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.

ignature generation algorithm

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

For Alice to sign a message $m$, she follows these steps:

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

ignature verification algorithm

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

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

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

ee also

* Elliptic curve cryptography

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.

* [http://csrc.nist.gov/cryptval/dss.htm Digital Signature Standard; includes info on ECDSA]

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