# Elliptic Curve DSA

Template:Use mdy dates In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.

## Key and signature-size comparison to DSA

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 the equivalent of about $2^{80}$ operations 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: $4t$ bits, where $t$ is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.

## Signature generation algorithm

Suppose Alice wants to send a signed message to Bob. Initially, they must agree on the curve parameters $({\textrm {CURVE}},G,n)$ . In addition to the field and equation of the curve, we need $G$ , a base point of prime order on the curve; $n$ is the multiplicative order of the point $G$ .

Parameter
CURVE the elliptic curve field and equation used
G elliptic curve base point, a generator of the elliptic curve with large prime order n
n integer order of G, means that $n\times G=O$ For Alice to sign a message $m$ , she follows these steps:

## Signature verification algorithm

For Bob to authenticate Alice's signature, he must have a copy of her public-key curve point $Q_{A}$ . Bob can verify $Q_{A}$ is a valid curve point as follows:

After that, Bob follows these steps:

Note that using Straus's algorithm (also known as Shamir's trick) a sum of two scalar multiplications $u_{1}\times G+u_{2}\times Q_{A}$ can be calculated faster than with two scalar multiplications.

## Correctness of the algorithm

It is not immediately obvious why verification even functions correctly. To see why, denote as $C$ the curve point computed in step 6 of verification,

Because elliptic curve scalar multiplication distributes over addition,

Expanding the definition of $s$ from signature step 6,

Since the inverse of an inverse is the original element, and the product of an element's inverse and the element is the identity, we are left with

From the definition of $r$ , this is verification step 6.

This shows only that a correctly signed message will verify correctly; many other properties are required for a secure signature algorithm.

## Security

In December 2010, a group calling itself fail0verflow announced recovery of the ECDSA private key used by Sony to sign software for the PlayStation 3 game console. However, this attack only worked because Sony did not properly implement the algorithm, because $k$ was static instead of random. As pointed out in the Signature generation algorithm Section above, this makes $d_{A}$ solvable and the entire algorithm useless.

On March 29, 2011, two researchers published an IACR paper demonstrating that it is possible to retrieve a TLS private key of a server using OpenSSL that authenticates with Elliptic Curves DSA over a binary field via a timing attack. The vulnerability was fixed in OpenSSL 1.0.0e.

In August 2013, it was revealed that bugs in some implementations of the Java class SecureRandom sometimes generated collisions in the k value. As discussed above, this allowed solution of the private key, in turn allowing stealing bitcoins from the containing wallet on Android app implementations, which use Java and rely on ECDSA to authenticate transactions.

This issue can be prevented by deterministic generation of k, as described by RFC 6979.