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II.5. FURTHER DISCUSSION using .net vs 2010 toget qr-codes for web,windows application Java programming language is very simple to analyze. A QR Code ISO/IEC18004 for .NET more complex design might be more secure, although it could also be more complex to analyze.

With added complexity, one can never discount the possible appearance of an attack. For DSA, it s possible that somebody could attack the conversion function. For example, DSA could be insecure because the conversion function used is not almost-bijective or for some other reason.

One could assume that the DSA conversion function is almost-bijective and try to nd a provable security result, but nobody has done this yet. The intuition that a one-way conversion function imparts some kind of security attribute is not entirely ungrounded. Almost-invertibility means that the public key can be recovered from the message and signature (with reasonable probability).

A one-way conversion function seems to prevent this. This di erence does not have an impact on GMR security. It could have other impacts such as anonymity (hiding the signer s identity) or e ciency (omitting the public key).

Hiding the public key is not a stated objective of ECDSA. Non-Pseuodrandom k : No result has shown that k needs to be indistinguishable from a uniform random integer in [1, q 1]. Indeed, since ECDSA is not meant to provide con dentiality, the need for indistinguishability is not clear.

Intuitively, a weaker condition than pseudo-randomness ought to be su cient for ECDSA. Certainly, the private keys must be unguessable and arithmetically unbiased, because of known attacks, but these are weaker conditions than pseudo-randomness. To see why pseudo-randomness might not be necessary for k, consider the following.

Choose truly random private keys k subject to the condition that their hashes display a given pattern. Such k fail to be pseudo-random because they can be distinguished by applying the hash function, yet they do not seem to be weak. They are unlikely to have an attackable arithmetic bias.

They may have enough entropy to be unguessable. Also, some of the results do not involve a signing oracle and therefore do not require the ephemeral private keys k to be generated pseudo-randomly. Deterministic k : In some of the proofs, the signing oracle value has the property that the same message query always gives the same signature response.

Technically, this means the proof is only applicable to the deterministic mode of ECDSA signing, where k is chosen as a secret deterministic function of the message m being signed. An intuitive explanation that the deterministic mode is more secure is that it reveals less signatures and therefere less information about the private key. A very cautious implementation of ECDSA could use the deterministic mode so that these provable security results apply.


Attack-Like Attribute qr bidimensional barcode for .NET s of ECDSA. Despite the proofs of GMR security of ECDSA, it might be argued that GMR security itself is not the right de nition.

Logically speaking, of course, a de nition, by de nition, cannot be right or wrong. Nonetheless, cryptology is a practical science, not a purely mathematical one, and therefore de nitions ought to be tailored to pertinent concerns, not purely arbitrary ones. With this perspective, some alternative de nitions of security for signatures in which ECDSA can be deemed insecure are explored and assessed for their pertinence.

Many of the attributes that we explore hinge on the particular conversion function f used in ECDSA. Altering f to avoid these attributes could potentially do more harm than good, diminishing the reputational security of ECDSA and the provable security of ECDSA. Accordingly, addressing these attributes is best handled through other means.

Signature Non-Anomyity : Given a valid ECDSA signature (r, s) on message m, the associated public key Y can be recovered, as follows. (Note that this does not violate the GMR de nition of signature security.) Solve for the public key as Y = [r 1 ]([s]R [H(m)]G), where R is selected from f 1 (r), the set of points in the preimage of r.

Self-Signed Signatures : A signature of a message is self-signed if the message contains the signature. A self-signed ECDSA signature can be generated as follows. Choose random k and s.

Compute r = f ([k]G). Form the message m containing the signature (r, s). Compute e = H(m).

Now solve for a private key x that makes this signature valid, which can be found as x = ( sk e)/r (mod q). This attribute does not violate GMR security. Indeed, it may be a useful attribute in the sense that it can be used to ensure that the private key was not stolen.

It may also be useful for server-assisted key generation, where a server adds entropy to the message m so the signer s private key x has enough entropy. Additional modi cations to the self-signed siganture veri cation are necessary, however, if the server cannot be trusted and the signer s entropy for k is weak. Unknown Private Key : A valid ECDSA signature can be generated without knowing the private key and yet not violate the GMR de nition of signature security, as follows.

This can be done for any elliptic curve domain parameters and any message m, by rst generating a random value of the signature (r, s) and then solving for the public key as Y = [r 1 ]([s]R [H(m)]G), where R f 1 (r), the set of points in the preimage of r. If f 1 (r) = {}, then just try another value of r..

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