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Elliptic Curve Digital Signature Algorithm

  (Redirected from ECDSA)

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

Contents

Key and signature-size comparison to DSAEdit

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. For example, at a security level of 80 bits (meaning an attacker requires a maximum of about   operations to find the private key) the size of an ECDSA public key would be 160 bits, whereas the size of a DSA public key is at least 1024 bits. On the other hand, the signature size is the same for both DSA and ECDSA: apx.   bits, where   is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.

Signature generation algorithmEdit

Suppose Alice wants to send a signed message to Bob. Initially, they must agree on the curve parameters  . In addition to the field and equation of the curve, we need  , a base point of prime order on the curve;   is the multiplicative order of the point  .

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  

Alice creates a key pair, consisting of a private key integer  , randomly selected in the interval  ; and a public key curve point  . We use   to denote elliptic curve point multiplication by a scalar.

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

  1. Calculate  , where HASH is a cryptographic hash function, such as SHA-2.
  2. Let   be the   leftmost bits of  , where   is the bit length of the group order  .
  3. Select a cryptographically secure random integer   from  .
  4. Calculate the curve point  .
  5. Calculate  . If  , go back to step 3.
  6. Calculate  . If  , go back to step 3.
  7. The signature is the pair  .

When computing  , the string   resulting from   shall be converted to an integer. Note that   can be greater than   but not longer.[1]

As the standard notes, it is crucial to select different   for different signatures, otherwise the equation in step 6 can be solved for  , the private key: Given two signatures   and  , employing the same unknown   for different known messages   and  , an attacker can calculate   and  , and since   (all operations in this paragraph are done modulo  ) the attacker can find  . Since  , the attacker can now calculate the private key  . This implementation failure was used, for example, to extract the signing key used for the PlayStation 3 gaming-console.[2] Another way ECDSA signature may leak private keys is when   is generated by a faulty random number generator. Such a failure in random number generation caused users of Android Bitcoin Wallet to lose their funds in August 2013.[3] To ensure that   is unique for each message one may bypass random number generation completely and generate deterministic signatures by deriving   from both the message and the private key.[4]

Signature verification algorithmEdit

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

  1. Check that   is not equal to the identity element  , and its coordinates are otherwise valid
  2. Check that   lies on the curve
  3. Check that  

After that, Bob follows these steps:

  1. Verify that   and   are integers in  . If not, the signature is invalid.
  2. Calculate  , where HASH is the same function used in the signature generation.
  3. Let   be the   leftmost bits of  .
  4. Calculate  .
  5. Calculate   and  .
  6. Calculate the curve point  . If   then the signature is invalid.
  7. The signature is valid if  , invalid otherwise.

Note that using Shamir's trick, a sum of two scalar multiplications   can be calculated faster than two scalar multiplications done independently.[5]

Correctness of the algorithmEdit

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

 

From the definition of the public key as  ,

 

Because elliptic curve scalar multiplication distributes over addition,

 

Expanding the definition of   and   from verification step 5,

 

Collecting the common term  ,

 

Expanding the definition of   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  , 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.

SecurityEdit

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   was static instead of random. As pointed out in the Signature generation algorithm section above, this makes   solvable and the entire algorithm useless.[6]

On March 29, 2011, two researchers published an IACR paper[7] 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.[8] The vulnerability was fixed in OpenSSL 1.0.0e.[9]

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.[10]

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

See alsoEdit

NotesEdit

ReferencesEdit

External linksEdit