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.

Filter by
Sorted by
Tagged with
0
votes
0answers
22 views

ECC public key encryption without symmetric cipher

Imagine the following scenario. A process is running in background and permanently encrypting some data. An adversary has full control of the process, e.g. it can dump the process memory any time and ...
1
vote
1answer
85 views

Public Key generation for Ed25519 vs X25519

It is my understanding that EdDSA uses a slight variant of Curve25519 (typically used for ECDH), called Ed25519. Given the same private key, are the differences between the two algorithms enough to ...
3
votes
2answers
67 views

Size of $E$ over $\mathbb{F}_p$ contains $p+1$ points

I am struggling to prove this claim: I proved that the map $x\mapsto x^3+1$ is a bijection from $\mathbb{F}_p$ to itself if we have that $p\equiv 2\bmod{3}$. We have to use this fact to prove that ...
3
votes
3answers
214 views

Is RSA in decline across the board?

From what I gather from the internet (source), the recommended practice for 2019 and beyond is to avoid RSA and use ECDH and ECDSA. Is this the general case?
2
votes
2answers
228 views

Operation on elliptic curves

Let $Y = xG$ be a point on an elliptic curve, $G$ the generator point and $x$ a scalar. Without knowing $x$, is it possible to calculate $x^nG$, being $n$ a natural number?
0
votes
0answers
24 views

Compute shared public keys

I want to compute, in a distributed way, the following shared public keys on an elliptic curve: $(xG, x^2G,...,x^nG)$, being $x$ a secret scalar that no single party knows, $G$ the public ...
1
vote
2answers
48 views

Understanding Montgomery's parameterization of elliptic curves

I'm having a bit of trouble understanding the translation of affine coordinates to projective coordinates in Montgomery curve ECM. Would be very thankful if someone could explain it by expanding the ...
25
votes
3answers
7k 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 ...
0
votes
1answer
67 views

Can we retrieve his private key using his public key in ECC?

A paper wallet is the name given to an obsolete and unsafe method of storing bitcoin which was popular between 2011 and 2016. It works by having a single private key and bitcoin address, being printed ...
4
votes
0answers
60 views

Regarding the need to hash the shared secret in X25519 with the public keys

I was looking at the LibSodium documentation where it says [...] and to mitigate subtle attacks due to the fact many $(p, n)$ [public key - secret scalar] pairs produce the same result, using the ...
0
votes
1answer
138 views

“Probability” of an ECDSA signature

The article Elliptic Curve Digital Signature Algorithm in Bitcoin's wiki talks about signatures having probabilities (as in ...
1
vote
0answers
37 views

Need help understanding SPAKE2 setup values

I am trying to write a simulation of the SPAKE2 protocol in python (just so I can get a better understanding of the protocol altogether). I am reading through the ietf draft here: Datatracker. There'...
11
votes
3answers
725 views

What does the special form of the base point of secp256k1 allow?

The popular ECC parameters secp256k1 are documented in SEC2 as using curve $y^2\equiv x^3+a\cdot x+b\pmod p$ with $a=0$, $b=7$, $p=2^{256}-2^{32}-\mathtt{3d1_h}$, base point $G$ with the apparently ...
0
votes
1answer
62 views

Key clamping in curve25519 not evident in generated key's binary representation

I understand with curve25519 that the private key for secret.box is clamped... I understand that this clamping process to clear ...
0
votes
1answer
58 views

Is this the right way to implement ElGamal scheme over Elliptic Curves over prime field? [duplicate]

I'm fairly new to Cryptography, especially elliptic curves in general. I learned to do Point Multiplication, Scalar Multiplication and also programmatically implemented them. But I was trying to do ...
3
votes
1answer
573 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 =...
2
votes
2answers
2k views

Security of elliptic curves

How can we say an elliptic curve is secure and can be used for cryptographic applications?
0
votes
1answer
76 views

Order of subgroups formed by Elliptic Curves with a Cofactor

In this question: Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? The answer indicates that the order of all points on the curve over the finite field ...
22
votes
2answers
3k views

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 ...
0
votes
1answer
77 views

Choice of finite fields for use in elliptic curves

this is maybe a basic question but I'm trying to better understand elliptic curve cryptography at a fundamental level. I understand that a finite field is required in order to define a boundary for ...
1
vote
1answer
577 views

Using Montgomery ladder to calculate the coordinates

In one of my assignments I need to solve the problem below: For a Montgomery curve $3y^2 = x^3+x^2+x$ over ${\mathbb{F}}_{11}$ and point $P = (9,8)$, compute the $x$ coordinate of $3P$ using the ...
0
votes
1answer
68 views

Does the nonce really have to be hashed as part of the challenge in a Schnorr signature?

From this article: https://tlu.tarilabs.com/cryptography/digital_signatures/introduction_schnorr_signatures.html#why-do-we-need-the-nonce The article states that the challenge ...
0
votes
1answer
94 views

Why does curve25519 use a cofactor of 8?

This cofactor (as I understand it) effectively discards valid points that satisfy the curve equation over the finite field. Why would one wish to reduce the number of possible private keys, it seems ...
0
votes
2answers
73 views

Elliptic curve commitments mod p

As far as I understand secp256k1 is defined over the group p with p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F I don't really ...
1
vote
1answer
177 views

Schnorr NIZK over Ed25519

I am trying to implement the following Schnorr non-interactive zero-knowledge protocol: https://tools.ietf.org/html/rfc8235#page-7 I'm using the libsodium 1.0.16 and GNU MP libraries. I just can't ...
0
votes
0answers
39 views

Signing with ECDSA [duplicate]

I am new to ECC. I was reading this post https://andrea.corbellini.name/2015/05/30/elliptic-curve-cryptography-ecdh-and-ecdsa/ I want to know if there is any intuition behind the following formula: $...
0
votes
2answers
118 views

The use of Elliptic Curves as part of a blockchain transaction

As I understand it, Elliptic-Curve Cryptography is used in the verification step of a transaction (i.e. when creating a digital signature), but not in the creation and security of a 'block' (when ...
1
vote
1answer
2k views

What is the recommended minimum key length for ECDSA signature

I want to identify the proportion of certificates that use unrecommend ECDSA key length for TLS certificates based on some data I collected. By looking at a standard like NIST for example, I find ...
1
vote
0answers
38 views

short signature for EC

i'm building a low-power wireless sensor network in which each slave node has a public/private ECC key pair -- generated by the node itself during manufacturing.... the slave node is also provisioned ...
0
votes
2answers
95 views

How does this formula work $(aG + bG) = (a + b) G$ in ECDSA?

Please explain how does this formula $(aG + bG) = (a + b) G$ work in ECDSA? According to the source: $a$ and $b$ are different private keys Suppose $a = 3$ $b = 4$ then the public key is $Q = aG$...
0
votes
1answer
32 views

Is it possible to execute elliptic curve encryption on small sensor-tags CC2650?

I am working on a project wherein ECC needs to be implemented on small devices, namely, CC2650 sensor-tags for authentication. The ECC implementation should be on Contiki OS. I have read some articles ...
0
votes
1answer
219 views

Is there a feasible way to generate an RSA key manually the same way as it is for an ECC one?

In elliptic curves, a private key is just a random number, and one relatively small compared to other crypto systems (256 bits for ECC vs 4096 bits for RSA for example). Suppose I don't trust ...
0
votes
1answer
58 views

No way to do ECDH with OpenSSL from the command line?

I've scoured this website and the OpenSSL wiki pages, and done numerous internet searches, and I've come to the seemingly incredible conclusion that one cannot generate an ECDH shared secret key using ...
1
vote
1answer
54 views

Elliptic curve one time signatures

This is kind of an academic question, but I wonder if it's possible to build an intentionally one-time signature scheme with elliptic curves? I assume you could do it by supplying ECDSA with ...
3
votes
1answer
416 views

ECDSA public key recovery is discovered by whom?

I'm looking for the history of the method (ECDSA public key recovery from signature). Where did this implementation first appear in (is it bitcoin?) and who discovered this method?
1
vote
0answers
41 views

Using ECC CDH test vectors with ECDH when h >1

I am writing formal tests for a system with a number of crypto requirements including support for ECDSA, ECDH and HMAC. The system is required to support the following EC's: NIST curves P-224, 384, ...
5
votes
1answer
630 views

Why doesn't this replay attack work on ECDSA?

I've just started working with elliptic curves and ECSDA in particular, so my understanding of the underlying math isn't great. The thing I'm currently stuck on is trying to understand why replay ...
3
votes
0answers
66 views

Using (EC)DH to generate a signature

Say I have access to a system A that is limited to performing (EC)DH, followed by key derivation to produce a secret key. This secret key is later used to provide integrity protection. There is a ...
4
votes
1answer
368 views

Could a C25519/ED25519 cryptographic module be FIPS certified?

NIST algorithms include ECDH and ECDSA. NIST also specifies curves. Is the use of NIST curves required for FIPS certification or could other curves theoretically be certified if someone were willing ...
1
vote
1answer
169 views

Using ECDH for encryption and decryption [duplicate]

I'm playing with cryptography and its use with typescript on one side and PHP on the other side. Now I'm looking for routines that can encrypt and decrypt with ecdh's private and shared keys. Any ...
2
votes
1answer
54 views

Variants of Bilinear Diffie-Hellman Assumption

Could someone point me to the paper/reference where the following variant of q-strong Bilinear Diffie-Hellman assumption was used? Given $s \in \mathbb{Z}_p^*$ and $g, g^{\frac{1}{s}}, g^{s}, g^{s^2},...
2
votes
2answers
71 views

Is there a key exchange protocol that requires only one message?

Say I want to exchange a secret with someone, but I only get to send one message to the other person, and then we encrypt with that secret. Diffie-Hellman and ECDH require multiple messages to be sent ...
1
vote
0answers
62 views

How to find kernel of isogeny from the dual isogeny

Let $E$ be a supersingular elliptic curve over $\mathbb{F}_{p^2}$, where $p = \ell_A^{e_A} \ell_B^{e_B} f \pm 1$ for some primes $\ell_A, \ell_B$. Let $R \in E[\ell_A^{e_A}]$ be a point of order $\...
4
votes
1answer
224 views

ECIES/ ECDHE/ EC-ElGamal encryption comparison

I need to choose an encryption system, so I am trying to understand the differences between the existing options. I always find that people compare ECIES (Elliptic Curve Integrated Encryption Scheme) ...
1
vote
1answer
43 views

Ensure Data Integrity In An ECDH Key Excange

Been playing around with the inner workings of onion routing and I have a problem. If I wanted to send the 2nd node of a relay network an ephemeral ECC public key, it has to go through node 1, so that ...
0
votes
1answer
35 views

For discrete elliptic curves, can you find G, if you are given b and B?

I know you cannot find $b$ if you are given $B$ and $G$, where $B = [b]G$, but can you find $G$ given $b$ and $B$?
0
votes
1answer
58 views

Post-Quantum Public Key Cryptography with EC math properties

Is there any quantum resistant public key cryptography with similar properties of elliptic curves? Assuming lowercase for scalars and uppercase for points. The properties I'm interested are: Reusing ...
1
vote
0answers
53 views

Elliptic curve discrete logarithm problem

I'd like to know what is the maximum bits of the finite field that we can solve the ECDLP in a "regular" computer in trivial time. Is there any recent data about this?
2
votes
1answer
52 views

How does the order of Q affect the time it takes to solve ECDLP?

I use Sagemath's built-in function discrete_log() to solve ECDLP and according to the documentation it uses Pohling-Hellman algorithm to solve an ECDLP. This is ...
0
votes
1answer
112 views

Question About ECC Primitives (GnuPG)

In version 2.1, GnuPG featured primitives that use elliptic curves. For encryption, one option was ECDH, which confused me. I thought ECDH was for key agreement, not public key encryption. How does ...