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9

Clearing the lower 3 bits of the secret key ensures that is it a multiple of 8, which in turn ensures that no information, small as it may be, about the secret key is leaked in the case of an active small-subgroup attack. The typical simple Diffie-Hellman key exchange works like this: $$ \text{Alice} \xrightarrow{\hspace{3cm} a G \hspace{3cm}} \text{Bob} ...


6

You could try the 112-bit secp112r1 cited in [1]. Before you do this, the problem with that paper is they actually show how to break the discrete log problem on this curve! And this was back in 2012. So any export-strength implementation of ECC is definitely breakable by governments, research groups and sufficiently determined/resourceful commercial ...


5

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 ...


5

If you can store the private key with some pre-computed work, then you can pick almost any public key you want. So in a way, it depends on the implementation. Here's a diagram of how Ed25519 works, note how keys are generated: (Image source.) A more detailed description (that is simpler than the actual paper) of the process is in these slides (slides 9 ...


5

Trevor Perrin wrote a library doing exactly that. Explanation can be found on in the curves mailing list archives. To convert a Curve25519 public key $x_C$ into an Ed25519 public key $y_E$, with a Ed25519 sign bit of $0$: $$y_E = \frac{x_C - 1}{x_C + 1} \mod 2^{255}-19$$ The Ed25519 private key may need to be adjusted to match the sign bit of $0$: if ...


4

All of these are answered by the SafeCurves project: $x^2 + y^2 \equiv 1 + dx^2y^2 \pmod p$ Edwards curves can be converted to Montgomery form.Montgomery curves can be converted to Weierstrass form.Some, but not all, Weierstrass curves can be converted to Montgomery form. The Montgomery ladder (applicable only to Edwards and Montgomery curves) is faster ...


4

In the usual definition of security of Elliptic Curves, curve25519 security is in fact 126 bits. If look at safecurves's rho page you can see the rho complexity for curve25519 is $2^{125.8}$ in accordance to what you say. Curve25519 author basically doesn't accept that definition of security. In the Curve25519 paper he states in section 1: Every known ...


4

Assuming this is the paper you're talking about, your modification completely eliminates resilience to collisions in the underlying hash function $H$. The EdDSA scheme (and in the Schnorr scheme on which it is based) is highly resilient against collisions in $H$. Specifically, in the generic group model, the Schnorr scheme has been proven to be secure even ...


3

Threshold (robust) m-of-n variant of Schnorr signature scheme is known: Douglas R. Stinson, Reto Strobl - Provably Secure Distributed Schnorr Signatures and a (t, n) Threshold Scheme for Implicit Certificates Major hints on intended usage are from Ripple page mentioned. Points 4 and 3 are explicit: produce a signature, in a theshold m-of-n way. This could ...


3

The core of the problem is finding a near first pre-image on the function $A = aB$ on an elliptic curve, where $A$ is the public key, and $a$ the private key¹. For a normal hash function you $ 2^m $ operations to fix $m$ specific bits.² In particular a full pre-image takes $ 2^n $ hash function calls. A full pre-image on $A = aB$ is equivalent to solving ...


3

ge_scalarmult_base returns GroupElementP3 which doesn't have (x, y) as members. It has X, Y, Z from which you can compute x = X / Z and y = Y / Z. So you have two choices: Compress the point with ge_tobytes: byte[32] Abytes; fe y; ge_scalarmult_base(&A,sk); ge_tobytes(Abytes, &A); fe_frombytes(&y, Abytes); // your code here using `y` ...


3

To perform an Ed25519 signature operation, you need to know three values, denoted by $\sf RH$, $a$ and $A$ in the diagram. Now, as it happens, these values are not independent: $A$ can be derived from $a$, and both $\sf RH$ and $a$ can be derived from the seed $k$. Thus, all you really need to store is the seed $k$; everything else can be derived from ...


3

ECDH is the same as what Curve25519 uses mathematically. The issue is that converting Curve25519 into Weierstrauß form is a bad idea, because it introduces issues relating to the potential failure of the addition law, which are difficult to address well. Keeping the curve in Montgomery or twisted Edwards form finesses these difficulties. ECDSA has issues ...


3

No, this does not weaken ed25519 in any way. Known plaintext will not have any effect on a signature algorithm, if it did it would make that algorithm completely useless.


3

AFAIK, no. However, Ed25519 keys can be converted to Curve25519 keys. My Ed25519 library supports this (or well, it supports DH with Ed25519 keys). Whether it is secure to use the same key for both signing and Diffie-Hellman, I don't exactly know. This answer suggests that it is very likely, but it still needs more study.


3

Edwards curves have unified addition so adding a point to itself returns the correct result. This differs from Weierstrass curves, where adding a point to itself gives the wrong result and you must use doubling. So your expectation that addition and doubling should return the same point is correct. High performance implementations of ECC use some form of ...


2

PLEASE NOTE: The code I link to below has not yet been reviewed by anyone with professional cryptography experience. I expect that it contains bugs, and it is definitely not production-ready. I am still learning about the JCA; there are parts of the code I have not finished, and there are parts that I will most likely go back and redo. That said, the tests ...


1

Unless I am missing something, I believe that you can simply compute $R = s*B - h*A$ using point subtraction.


1

The results should be the same as you have noted. Although I would think that most people would have a function more akin to ScalarMult(int, point) instead of dbl. Perhaps you could look at dbl and check if it is really a doubling function. Another check you could perform is to look at the $x$-coordinates of either outputs. Are they the same? If only the ...


1

Ed25519 or more general the EdDSA (Edwards-curve Digital Signature Algorithm) approach can be considered as a variant of ElGamal signatures (such as Schnorr or DSA). They all are signatures following the hash-then-sign approach. This simply means that you can sign arbitrary length messages by hashing them to a constant size string using a secure ...


1

Ed25519 is a specific implementation of EdDSA using the Twisted Edwards curve: x^2 + y^2 = 1 + (121665/121666) * (x^2)(y^2) It's known as high speed high security signature algorithm. For using the code you pointed out, you need to feed sk, pk, m, sm. So first you need to call publickey function with sk, then call signature function with m, sk and pk. PK ...


1

Related to Curve25519 Curve25519 seems to be secure so far. Yet, you have to remind yourself that Dr. Bernstein specified Curve25519 for key-exchange. Meaning: key-generation, transaction signing, and verification are somewhat different beasts – you might want to cross-check on that before jumping toward Curve25519. Sure, Curve25519-java supports signing… ...


1

Support for this has now (2014-08-05) been added to Sodium. Implement ed25519 -> curve25519 keys conversion Look at test/default/ed25519_convert.c for some example code.


1

Looking at the Ed25519 paper, it seems that a key pair consists of a private key $k$ (just a $k$-bit string) and a public key $A$, where $A = a · B$ and $a$ is derived from the first half of the hash $H(k)$ ($B$ is the base point of an elliptic curve (or actually a twisted Edwards curve equivalent to an elliptic curve) and $·$ is the scalar multiplication in ...



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