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15

I don't believe that there's any way to generate the vanity hashes without iterating. In base 58, there's $\log_2(58) \approx 5.858$ bits per letter, so fixing 8 letters would need in average $58^8/2 = 2^{\log_2(58)·8}/2 \approx 2^{46}$ iterations. Note that Bitcoin addresses always start with a 1 by convention (this comes from the version field), and the ...


14

There is a draft RFC which describes a way to implement deterministic (EC)DSA (with test vectors). In this draft, both $h(m)$ (the hash of the message) and $x$ are used as input to a deterministic PRNG which uses HMAC (that's HMAC-DRBG as specified by NIST); the PRNG output is used to yield $k$. I am not sure your simple multiplication with $x$ would be ...


12

First of all, I'm no expert in this area. Generally $n$ bit ECC seems to have a security level of about $n/2$, but I found some claims that it's lower for certain types of curves. RFC4492 - Elliptic Curve Cryptography (ECC) Cipher Suites contains the following table: for Transport Layer Security (TLS) Symmetric | ECC ...


11

Here are five test vectors for secp256k1, which I just generated with my own code. My code is a generic implementation of elliptic curves; it has been tested for many curves for which test vectors were available (in particular the NIST curves) so I tend to believe that it is correct. Each test vector is a value $m$ (chosen randomly modulo the curve order ...


10

On a general basis, you want to keep encryption and signature keys disjoint, because they tend to have distinct life cycles. In broad terms, an encryption key should be escrowed, because loss of the private key implies loss of the data which is encrypted relatively to the public key. However, a signature key must not be escrowed, since the proof value of a ...


8

Rabin signatures have a very fast verification algorithm: a simple squaring modulo some integer. RSA signature verification (with a public exponent equal to 3) is also very fast. These signature algorithms are simple to implement and will beat ECDSA for verification speed, even if batch verification is used for ECDSA. The Niederreiter digital signature ...


8

Disclaimer: I don't know Javascript and I do not practice BouncyCastle. However, I do know Java, and ASN.1. ASN.1 is a notation for structured data, and DER is a set of rules for transforming a data structure (described in ASN.1) into a sequence of bytes, and back. This is ASN.1, namely the description of the structure which an ECDSA signature exhibits: ...


7

ECDSA is actually a kind-of computational zero-knowledge protocol, played by the signer, with a "reduction function" as impartial verifier. For that matter, ECDSA is not very different from plain DSA. Things basically go this way. There is a known public group $\mathbb{G}$ which I will denote additively, with $G$ as generator, and of size $q$ (a known prime ...


7

Yes. Modern cryptosystems are designed and analysed under the assumption that the key is never used for anything else. If you use your encryption keys for digital signatures, you are violating that assumption, and it is very easy to construct schemes where this violation will compromise security. It is possible to construct schemes that can use the same ...


6

I'm considering switching to ECDSA, would this require less space with the same level of encryption? The answer to that question is yes, both ECDSA signatures and public keys are much smaller than RSA signatures and public keys of similar security levels. If you compare a 192-bit ECDSA curve compared to a 1k RSA key (which are roughly the same security ...


6

There isn't a simple answer, as speed of batching depends on a number of parameters. First, the speed of the signature and the speed of the batching is largely independent. If you have two signature algorithms S1 and S2 that both permit batching technique B1, then generally they will both permit batching technique B2. If S1 is faster than S2 for individual ...


6

Well, you understand that Elliptic Curves define an operation on points we denote as +; that is, if $A$ and $B$ are two (not necessarily distinct) points, then $A+B$ is a third point (which will be distinct unless either $A$ or $B$ are the 'point-at-infinity'). If $A$ and $B$ are the same, the operation is usually called doubling instead of addition. Now, ...


6

In their 1998 SAC paper, M'Raihi et al showed how to use hash functions to turn Schnorr signatures (quite similar to (EC)DSA) deterministic, and proved that if the original signature scheme (with randomness) is secure, so is the deterministic one. Bernstein et al's recent EdDSA signature scheme uses the same technique to avoid randomness.


6

No, signing the hash of the public key cannot introduce a weakness on a secure signature scheme. When we have a signature scheme, we assume that it is secure in an chosen text model, where the attacker has access to the public key, and can ask any text of his choosing to be signed. We can see that any such scheme (such as ECCDSA, or so we believe) cannot ...


5

I recommend you use Rabin signatures. Rabin signatures without batch verification are likely to be faster than most other signatures with batch verification. Moreover, read Dan Bernstein's work. He has shown how to make Rabin signatures even faster. For standard Rabin signatures, verification requires approximately one modular multiplication modulo n ...


5

I'm surprised that Daniel J. Bernstein's EdDSA has not been mentioned. High-speed high-security signatures Even faster batch verification. The software performs a batch of 64 separate signature verifications (verifying 64 signatures of 64 messages under 64 public keys) in only 8.55 million cycles, i.e., under 134000 cycles per signature. The ...


5

I think your question can be reduced to the question whether there is a significant subset of weak public/private key pairs in any of the EC groups you mention. I am not aware of any such weakness, but if it exists, it would put a large dent in the security of Elliptic Curve Cryptography as a whole. If there is no significant risk you will get a key pair ...


5

The write up on Wikipedia is pretty good. I won't go into all the detail that they do there, but your private key is a randomly selected integer $d_A$ selected from $[1,n-1]$ where $n$ is the order of the group. The public key is $Q_A=d_AG$ where $G$ is the base point on the curve defined in the publicly agreed upon parameters.


5

You got tripped up by the fact that there are two different group operations in play here, and they don't play nice with each other. This is implicit in the notation, and it's easy to get tripped up, because the notation expresses both operations in the same way -- but they are not the same. This is arguably a pitfall in the notation: the assumption is ...


5

The paper "On the Joint Security of Encryption and Signature in EMV" shows that ECIES and EC-Schnorr signatures can be used together without compromising security: In the random oracle model ECIES-KEM and EC-Schnorr are jointly secure if the gap-DLP problem and gap-DH problem are both hard Ed25519 is extremely similar to EC-Schnorr, and both ECIES ...


4

It looks like your main question is determining why $k{_{E}}^{-1} = 2$, correct? As mentioned in the comments to the question, this is because it is the modular multiplicative inverse. The multiplicative inverse is a number, $x^{-1}$, such that $x·x^{-1}=1$. However, since we are in modulo 19, we want to find $x^{-1}$ such that $x·x^{-1}\equiv1 \bmod 19$. ...


4

Well, the normal rules apply, i.e. $(aG + bG) = (a + b)G$, so as long as you add $a$ and $b$ correctly, everything should work fine. Note that you don't actually have to reduce the result of the addition for the point multiplication to give the same result, however your implementation may require the number to be smaller than the order. Also make sure that ...


4

One rationale for avoiding randomized schemes in general, and in MACs in particular, is that the random in such schemes tends to increases the size of cryptograms or reduce the size of the payload. An example is scheme 2 in ISO/IEC 9796-2 RSA signature with message recovery, where the size of the random/salt field is directly antagonist with the amount of ...


4

The most important property of cryptographic PRNGs is that it's indistinguishable from true random numbers, unless you know the seed, or you have huge computational resources. Two important consequences of this requirement are: You can't find the seed from observing the output You can't predict more outputs from observing some of them. An attacker who ...


4

Well, lets go through the issues: It seems to be possible to retrieve the (public) key used for creating an ECDSA signature just from the signature alone Nope, not quite. You also need the message being signed. And, with that, it doesn't give you the unique public key; it does allow you to narrow it down to two possibilities (assuming you're using a ...


4

You are using the wrong value as the modulus; you ought to be using the value $r$ (which is also listed in the document). $p$ is the characteristic of the field that the elliptic curve you're using is defined on. In this case, we're not interested in that; instead what we're interested in is the order of the curve, that is, that value $r$ such that $rP = ...


4

First of all I do not know your implementation, but it seems that you have some basic misunderstandings. Signature: ECDSA(sha256(Data) ) ECDSA is typically implemented in a way that you do not explicitly hash the data prior to passing it to the signing algorithm (but as this might be your own implementation and signing may still work correctly). ...


3

I'm just reading the book Advances in Elliptic Curve Cryptography, and Chapter II (by Dan Brown) is about provable security of ECDSA. It lists some necessary conditions for the ECDSA components (group, conversion function, RNG, hash function), each with an associated forgery. For example, the group has to be resistant against discrete logarithms, as well as ...


3

Well, the most common representation of 'point at infinity' would be a value that consists solely of zeros; that is, if normal points are encoded as a series of 64 bytes, then the point at infinity would be encoded as 64 00 bytes. On the other hand, it wouldn't appear to apply to ECDSA; ECDSA signatures consist of two integers between 1 and the curve order, ...


3

All points on an elliptic curve verify, by definition, the curve equation, usually written as $Y^2 = X^3 + aX + b$, with two given $a$ and $b$ parameters (these two parameters actually define the curve). So, if you know $X$, you can use the curve equation to recompute $Y^2$. A square root extraction will yield $Y$ or $-Y$. The compressed point format ...



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