Elliptic Curve DSA

The Elliptic Curve Digital Signature Algorithm (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which uses Elliptic curve cryptography.

1 Key and signature size comparison to DSA
2 Signature generation algorithm
3 Signature verification algorithm
4 Security
5 See also
6 Notes
7 References
8 External links

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 $2^{80}$ 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.

Signature generation algorithm

Parameter
q	field size
FR	the basis used
a, b	field elements defining the point of the curve
DPS	DomainParameterSeed, optional
G	base point
n	order of G
h	cofactor

Suppose Alice wants to send a signed message to Bob. Initially, the curve parameters $(q, FR, a, b,[DomainParameterSeed,] G, n, h)$ must be agreed upon. $q$ is the field size; $FR$ is an indication of the basis used; $a$ and $b$ are two field elements that define the equation of the curve; $DomainParameterSeed$ is an optional bit string that is present if the elliptic curve was randomly generated in a verifiable fashion; $G$ is a base point of prime order on the curve (i.e., $G = (x_G, y_G)$ ); $n$ is the order of the point $G$ ; and $h$ is the cofactor (which is equal to the order of the curve divided by $n$ ).

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 $[1, n-1]$ ) and a public key $Q_A$ (where $Q_A = d_A G$ ). Let $L_n$ be the bit length of the group order $n$ .

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

Calculate $e = \textrm{HASH}(m)$ , where HASH is a cryptographic hash function, such as SHA-1, and let $z$ be the $L_n$ leftmost bits of $e$ .
Select a random integer $k$ from $[1, n-1]$ .
Calculate $r = x_1 \pmod{n}$ , where $(x_1, y_1) = k G$ . If $r = 0$ , go back to step 2.
Calculate $s = k^{-1}(z %2B r d_A) \pmod{n}$ . If $s = 0$ , go back to step 2.
The signature is the pair $(r, s)$ .

When computing $s$ , the string $z$ resulting from $\textrm{HASH}(m)$ shall be converted to an integer. Note that $z$ can be greater than $n$ but not longer.^[1]

It is crucial to select different $k$ for different signatures, otherwise the equation in step 4 can be solved for $d_A$ , the private key: Given two signatures $(r,s)$ and $(r,s')$ , employing the same unknown $k$ for different known messages $m$ and $m'$ , an attacker can calculate $z$ and $z'$ , and since $s-s' = k^{-1}(z-z')$ (all operations in this paragraph are done modulo $n$ ) the attacker can find $k = \frac{z-z'}{s-s'}$ . Since $s = k^{-1}(z %2B r d_A)$ , the attacker can now calculate the private key $d_A = \frac{s k - z}{r}$ . This cryptographic failure was used, for example, to extract the signing key used in the PlayStation 3 gaming console.^[2]

Signature verification algorithm

For Bob to authenticate Alice's signature, he must have a copy of her public key $Q_A$ . If he does not trust the source of $Q_A$ , he needs to validate the key ( $O$ here indicates the identity element):

Check that $Q_A$ is not equal to $O$ and its coordinates are otherwise valid
Check that $Q_A$ lies on the curve
Check that $nQ_A = O$

After that, Bob follows these steps:

Verify that $r$ and $s$ are integers in $[1, n-1]$ . If not, the signature is invalid.
Calculate $e = \textrm{HASH}(m)$ , where HASH is the same function used in the signature generation. Let $z$ be the $L_n$ leftmost bits of $e$ .
Calculate $w = s^{-1} \pmod{n}$ .
Calculate $u_1 = zw \pmod{n}$ and $u_2 = rw \pmod{n}$ .
Calculate $(x_1, y_1) = u_1 G %2B u_2 Q_A$ .
The signature is valid if $r = x_1 \pmod{n}$ , invalid otherwise.

Note that using Straus's algorithm (also known as Shamir's trick) a sum of two scalar multiplications $u_1 G %2B u_2 Q_A$ can be calculated faster than with two scalar multiplications.^[3]

Security

On March 29th, 2011, two researchers published a IACR paper^[4] demonstrating that it is possible to retrieve a TLS private key of a server using OpenSSL that authenticates with Elliptic Curves DSA and binary curves via a timing attack^[5].

Notes

^ FIPS 186-3, pp. 19 and 26
^ http://events.ccc.de/congress/2010/Fahrplan/attachments/1780_27c3_console_hacking_2010.pdf, page 123–128
^ http://www.lirmm.fr/~imbert/talks/laurent_Asilomar_08.pdf The Double-Base Number System in Elliptic Curve Cryptography
^ http://eprint.iacr.org/2011/232
^ https://www.kb.cert.org/vuls/id/536044

References

Accredited Standards Committee 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, Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography, Version 2.0, May 21, 2009.
López, J. and Dahab, R. An Overview of Elliptic Curve Cryptography, Technical Report IC-00-10, State University of Campinas, 2000.
Daniel J. Bernstein, Pippenger's exponentiation algorithm, 2002.
Daniel R. L. Brown, Generic Groups, Collision Resistance, and ECDSA, Designs, Codes and Cryptography, 35, 119-152, 2005. 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

Digital Signature Standard; includes info on ECDSA

Public-key cryptography

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