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To this point, we ve described a digital signature as the private-key encryption of a digest. Now we come to DSA, which does not encrypt data. Although DSA uses the digest of the data, it does not encrypt the digest. Your first thought is likely to be, If it can t encrypt data, how can it produce a digital signature Remember that DH cannot be used to encrypt data but can be used to solve the key distribution problem. Similarly, even though DSA cannot be used to encrypt data, it can be used to create a digital signature. A digital signature is a chunk of data that comes from the message and the private key. Only that particular message coupled with that particular private key will produce that particular signature. If you accomplish that by encrypting the digest, great. If you accomplish that in some other way, that s fine, too. With DSA, the signer digests the message with SHA-1 and treats that digest as a number (it s a big number: 160 bits long). Another number sent to the algorithm is a random or pseudo-random value, usually called k. The last input is the private key. The algorithm then performs some mathematical operations, one of which is modular exponentiation, the same function at the heart of DH and RSA. The output is two numbers, usually called r and s. These two numbers are the signature. The verifier computes the SHA-1 digest of the message. Is it the same digest that the signer produced The verifier does not have that digest available but does have r and s. Using the digest as a number, along with the public key and the s, the verifier performs some mathematical operations. The result of the computations is a number called v. If v is the same as r, the signature is verified (see Figure 5-12). At its most basic, DSA computes the same number in two different ways. In Diffie-Hellman, two parties can generate the same secret value even though each one is using different input. The same thing is happening here with DSA. Two parties produce the same number using different input. The two sets of input are related. Well, they should be related; if something breaks down, the final answers will differ. Each side has three inputs. The signer has the digest, k, and the private key. The verifier has the digest, s, and the public key. The digests are related; they should be the same thing. If that relationship breaks downsay, the signed data is not the same as the data being verified and the two parties produce different digests the final answer from each individual will differ. The k and s are related (they re not the same number, but they re related). If the signature is wrong, the s will be wrong and the two
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Figure 5-12
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The Digital Signature
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players will produce different final answers. The private key and the public key are also related; they are partners related mathematically. If that relationship is not there if the public key used to verify is not the partner to the private key used to sign the two agents will produce different final answers. The security of DSA lies in the discrete log problem, the same problem that gives DH its security. So the size of DSA keys will be the same as that of DH keys. As always, you can find more detailed information in the RSA Labs FAQ on the accompanying CD.
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This algorithm looks a lot like DSA. The signer has three inputs: the digest, k, and the private key. The output is r and s. The verifier has the digest, s, and the public key. The output is v. If v and r are the same, the signature is verified; if they re not the same, something went wrong. What went wrong Was it the wrong digest The wrong public key Was the signature mangled in transmission You probably can t know exactly what happened, but you do know that something went wrong. The math underlying ECDSA is elliptic curves, so key size is the same as with ECDH.
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