Questions tagged [elliptic-curves]

Elliptic curves are algebraic-geometric structures with applications in cryptography. Such a curve consists of the set of solutions to a cubic equation over a finite field equipped with a group operation. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider more specific tags such as discrete-logarithm and ecdsa.

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11
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1answer
2k views

What is the relationship between p (prime), n (order) and h (cofactor) of an elliptic curve?

I am reading up on ECC and having trouble understanding how these are related. In a finite field, all point operations are taken modulo $p$. $n$ is the order of the generator $G$ — which apparently ...
146
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4answers
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Should we trust the NIST-recommended ECC parameters?

Recent articles in the media, based upon Snowden documents, have suggested that the NSA has actively tried to enable surveillance by embedding weaknesses in commercially-deployed technology -- ...
20
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1answer
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ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
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1answer
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After ECDH with Curve25519, is it pointless to use anything stronger than AES128?

Is the following reasoning correct: After ECDH with Curve25519, the resulting shared secret will be an EC public key with a bit strength of 128 bits. This public key would then be hashed (let's say ...
150
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4answers
101k views

Why is elliptic curve cryptography not widely used, compared to RSA?

I recently ran across elliptic curve crypto-systems: An Introduction to the Theory of Elliptic Curves (Brown University) Elliptic Curve Cryptography (Wikipedia) Performance analysis of identity ...
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5answers
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Who uses Dual_EC_DRBG?

Recent news articles have suggested that the NSA may be involved in trying to influence the cryptography in public standards or commercially deployed software, to enable the NSA to decrypt the ...
20
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2answers
4k views

Using same keypair for Diffie-Hellman and signing

Are there any security risks using a single key-pair for both key-exchange and signing? I'm mainly interested in using Curve25519 for key-exchange and Ed25519 for signing. But similar combinations, ...
17
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2answers
4k views

Is there a feasible method by which NIST ECC curves over prime fields could be intentionally rigged?

The NIST elliptic curves P-192, P-224, P-256, P-384, and P-521, prescribed in FIPS 186-4 appendix D.1.2, are generated according to a well defined process, but using an arbitrary random-looking seed ...
9
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1answer
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Why can ECC key sizes be smaller than RSA keys for similar security?

I understand how ECC is based on the discrete log problem and RSA on integer factorization. I've read several references that show how a solution to either of these problems can typically be adapted ...
33
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2answers
12k views

How does recovering the public key from an ECDSA signature work?

It is possible to recover the public key from an ECDSA signature values $(r,s)$? Please explain how this works.
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2answers
939 views

Summarize the mathematical problem at the heart of breaking a Curve25519 public key

It's pretty easy to generate a Curve25519 private key: generate 32 random bytes of data and then do: e[0] &= 248 e[31] &= 127 e[31] |= 64 You can then ...
3
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1answer
544 views

ECDH security when no KDF is used

Let's suppose our device performs ECDH with a fixed, unknown, private key $\text{prv}$. It accepts as input any point $Q$ lying in the proper subgroup of the proper elliptic curve, then computes: $P =...
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What is so special about elliptic curves?

There seems to be sources like this, this also, and some introductions that discuss elliptic curves in general and how they're used. But what I'd like to know is why these particular curves are so ...
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2answers
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When using Curve25519, why does the private key always have a fixed bit at 2^254?

When using Curve25519, the private key always seems to have a fixed bit set at position 2^254. Why is that? Is there any good reason to use a fixed positioned most-significant-bit in the private key?
26
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1answer
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ECDSA Compressed public key point back to uncompressed public key point

From the ECDH demo here, if I generate a private key for Alice I can get _ P = 1175846487558108474218546536054752289210804601041 Which gives the following public ...
19
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1answer
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Mapping points between elliptic curves and the integers

My primary question is: Is there an easy way to create a bijective mapping from points on an elliptic curve E (over a finite field) to the integers (desirably to $\mathbb{Z}^*_q$ where $q$ is the ...
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4answers
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How does one calculate the scalar multiplication on elliptic curves?

I found this example online: In the elliptic curve group defined by $$y^2 = x^3 + 9x + 17 \quad \text{over } \mathbb{F}_{23},$$ what is the discrete logarithm $k$ of $Q = (4,5)$ to the base $...
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1answer
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Why are elliptic curve variants of RSA “chiefly of academic interest”?

Yesterday I was thinking about elliptic curve variants of popular protocols/algorithms (ECDH, ECES[1], etc) and the thought occured that I had never seen an elliptic curve variant of RSA. My ...
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2answers
1k views

Why are NaCl secret keys 64 bytes for signing, but 32 bytes for box?

Ed25519 secret and public keys can both be represented in 32 bytes. Why does NaCl use 64 byte signing keys?
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3answers
2k views

Elliptic Curves of different forms

Looking at http://safecurves.cr.yp.to/ to find a safe curve, I find that most curves described here are of a different form from that generally used. In Bouncy Castle, for example, ...
3
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1answer
219 views

Key equivalence across different elliptic curves

Is it possible to prove key-equivalence across elliptic curves of different order? Specifically: Suppose I have a key $x$ valid for both curves listed below On curve $g$ (for example, Curve25519) it ...
22
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7answers
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Current mathematics theory used in cryptography/coding theory

What are the mainstream techniques borrowed from algebraic geometry (or some other branch of mathematics) which are currently used in cryptography/coding theory? I've only heard about a small subset ...
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2answers
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Schnorr signatures: multisignature support

Schnorr signature is mentioned as a promising upgrade to bitcoin to improve scalability. It support multisignature, several signatures can be aggregated into a single, new signature. But I fail to ...
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2answers
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How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
9
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1answer
4k views

Mapping of message onto elliptic curve and reverse it?

I would like to perform a variant of Elliptic Curve ElGamal in java using the BouncyCastle libraries. I currently face the difficulty of mapping a message $m$ onto the elliptic curve $E_p$. I have so ...
6
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1answer
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What does the $\|$ operation mean in cryptographic notation?

I am studying elliptic curves problems, which also includes study of related protocols such as ECIES. The problem is that I don't understand the notation $\|$. What does this operation mean? Some ...
39
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4answers
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Basic explanation of Elliptic Curve Cryptography?

I have been studying Elliptic Curve Cryptography as part of a course based on the book Cryptography and Network Security. The text for provides an excellent theoretical definition of the algorithm but ...
40
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4answers
33k views

Is secp256r1 more secure than secp256k1?

Curves secp256r1 and secp256k1 are both examples of two elliptic curves used in various asymmetric cryptography. Googling for these shows most of the top results are Bitcoin related. I've heard the ...
37
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1answer
29k views

Explaining weakness of Dual EC DRBG to wider audience?

I have an audience of senior (non-technical) executives and senior technical people who are taking the backdoor in Dual_EC_DRBG and considering it as a weakness of Elliptic curves in general. I can ...
24
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3answers
6k views

Can ECDSA signatures be safely made “deterministic”?

Using the terminology of the ECDSA wikipedia page, ECDSA (and DSA) signatures require a random k value for each signature which ensures that the signature is different each time even if the message ...
14
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1answer
2k views

Is pairing based cryptography ready for productive use?

I'm currently testing one among those many interesting cryptographic protocols based on bilinear maps. It's quite hard to understand the underlying fundamentals, especially since there are several ...
21
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2answers
4k views

How does the MOV attack work?

What exactly is the MOV attack, how does it actually work, and what is it used for? It's explained briefly here and I'd like to know what it is more / what is it fully used for.
14
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4answers
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EC Schnorr signature: multiple standard?

I'm working on some EC-Schnorr signature code. Reading various papers on that, it seems EC-Schnorr is not standardized as well as ECDSA. For example, I found two main differences in two main actors ...
18
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5answers
4k views

Is there a situation where RSA cannot be replaced with ECC + symmetric algorithms? If no, why do we still use it?

RSA is both an asymmetric encryption algorithm and a digital signature algorithm. However, in recent years, many cryptographic protocols (TLS, for example) have moved away from the use of RSA to ...
16
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1answer
2k views

Edwards / Montgomery ECC with Weierstrass Implementation?

So let's assume I want to perform Ed448 or Ed25519 digital signatures or want to perform a DH key-exchange. Assume further that those curves (Curve448 or Curve25519) are required. But the problem is, ...
13
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1answer
422 views

Logjam on Elliptic Curves?

I think we're all aware of the Logjam attack. From now on we know that re-using primes for DH is a bad idea. But we also say that elliptic curves are safe from the attack (relying on the NFS), ...
11
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1answer
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Trying to better understand the failure of the Index Calculus for ECDLP

So I'm going to give you guys my understanding and then if you would be so kind as to tell me where I'm off the mark (hopefully I'm not completely wrong). So basically the index calculus for the ...
5
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2answers
735 views

Is ECC over real numbers possible?

Many elliptic-curve cryptosystems today use GF(p) or GF(2^m). What if, say, we use big floating numbers with the classical point addition formulas - is a cryptosystem possible to build on that?
27
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3answers
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Why would anyone use an elliptic curve with a cofactor > 1?

In cryptography, an elliptic curve is a group based on a finite field $GF(p^k)$; this group has $n$ elements on it, and we work on a prime-sized subgroup of size $q$. We denote the value $h = n/q$ as ...
15
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1answer
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Does Curve25519 only provide 112 bit security?

In a recent mail on the IETF CFRG mailing list it was claimed that: The (currently missing) security considerations (or somewhere) should describe why Curve25519 is ok when used in contexts where ...
7
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2answers
798 views

Besides key and ciphertext sizes what are other advantages of elliptic curve versions of various protocols?

There are elliptic curve variants of Diffie-Hellman, ElGamal, DSA and possibly other protocols/algorithms. I know that these elliptic curve variants have smaller key and ciphertext sizes which will ...
4
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1answer
458 views

How should ECDSA handle the null hash?

In ECDSA, assume your public key is $Q=(x,y)$, then when performing the verification of any message leading to a null hash value (that is $H(M)=0$), the signature $(r,s)=(x,x)$ would always lead to a ...
4
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1answer
823 views

Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack

The curve $E(\mathbb{F}_{47}):y^2=x^3+x+38$ has order $61$ and $61|47^3-1$ so the embedding degree of $E$ is $3$ and therefore the MOV attack, presumably using some sort of distortion map and a ...
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2answers
276 views

Using two elliptic curves to do a range proof

Suppose Alice holds a secret value $a$ to which she has publicly committed to using two elliptic curves of distinct order. The curves are $g$ and $g'$ of orders $q$ and $q'$ (with $q < q'$) and ...
17
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1answer
3k views

How to calculate elliptic curve parameters?

I'm having a rough time understanding the math behind elliptic curves. I want to implement ECDH where user can define a, b, and p parameters of elliptic curve. How can I calculate generator base ...
9
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2answers
950 views

Is it safe to reuse ECDH asymmetric keys for authentication?

Alice, Bob, and Carol each generate ECDH keypairs. Alice and Bob establish a communication channel and negotiate an AliceBob secret. The question is: Is it safe for Alice and/or Bob to reuse their ...
4
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3answers
667 views

Why is it impossible to find the private key from the public key?

From what I understand, based on the article A Beginner’s Guide: Private and Public Key Cryptography Deciphered, the public key is generated by performing $n$ tangent plus mirroring operations from ...
0
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4answers
1k views

Optimized modular multiplicative inverse for Bitcoin (secp256k1)

I have written an application to brute-force attack Bitcoin addresses for OpenCL. It implements a simple exhaustive search starting from public key G (private key 1) and point increment (addition of G)...
11
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1answer
1k views

Is every point on an elliptic curve of a prime order group a generator?

If the order of elliptic group is prime then every point is a generator of that group. I tested the above statement on some elliptic curves and found it true. Does that really work on all curves? Is ...
26
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3answers
38k views

How can I use SSL/TLS with Perfect Forward Secrecy?

I'm new to the field of cryptography, but I want to make the web a better web by setting up the sites that I host with Perfect Forward Secrecy. I have a list of questions regarding the setup of ...