# 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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### 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 ...
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### 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 ...
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 ...
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?
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### 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?
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### 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 ...
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### 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 ...
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 ...
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 ...
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 ...
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### “Probability” of an ECDSA signature

The article Elliptic Curve Digital Signature Algorithm in Bitcoin's wiki talks about signatures having probabilities (as in ...
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### 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'...
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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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### 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 $Q = aG$...
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### 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 ...
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### 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 ...
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 ...
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 ...
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?
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### 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, ...
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 ...
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 ...
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 ...
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 ...
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### 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) ...
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### 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 ...
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 $B$?
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### 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 ...
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?