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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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1answer
67 views

Point doubling with only one coordinate

In many source codes that implement ECDH, there is a function that multiplies the base point of that curve with a constant. This function usually takes as arguments the constant and just one ...
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294 views

Difference between Pure EdDSA (ed25519) and HashEdDSA (ed25519ph)

My question refers to EdDSA as specified in RFC 8032. I get from the RFC that ed25519 and ed25519ph are two different instances of EdDSA mainly differing in the fact that that in the case of ...
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1answer
116 views

Why do Edwards curves protect against side-channel attacks?

From Wikipedia: One of the attractive feature of the Edwards Addition law is that it is strongly unified i.e. it can also be used to double a point, simplifying protection against side-channel ...
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212 views

Representations of secret keys on Curve25519

https://tools.ietf.org/html/draft-josefsson-tls-curve25519-06#appendix-A.2 gives the following as a secret key / public key combo: ...
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1answer
105 views

Can the RSA accumulator scheme be converted to Elliptic Curve math?

Is it possible to translate the RSA accumulator scheme directly to EC without requiring bilinear pairings? In RSA we have: $A_{n+1} = A_n^c$ st. $\{c \: \textrm{prime} \: | \: c \in [\...
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1answer
67 views

Secure Communication

Focus: I have to design a secure keep alive communication protocol and was wondering if it was necessary to sign the ciphertext after the session key has been generated as an attacker will not know ...
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1answer
64 views

Computational Complexity: ECC multiplication vs Modular multiplication

How does performing scalar multiplication on an elliptic curve compare to exponentiation in a multiplicative group modulo a prime? I.e. on a given elliptic curve of size $|t|$, what's the complexity ...
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55 views

Difficulty of Reversing Elliptic Curve

In ECC, it is apparently easy to verify the final point given the starting point and the number of hops. But it is difficult to compute the number of hops given just the starting point and the final ...
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1answer
43 views

formulas for adding points on curve25519

Curve25519 is a Montgomery curve. https://hyperelliptic.org/EFD/g1p/auto-montgom-xz.html#diffadd-dadd-1987-m-3 gives a set of formulas for adding two points (well, more specifically, the X coordinate ...
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1k views

Why are co-factors 4 and 8 so popular when co-factor is more than one?

For elliptic curve cryptography, I seem to keep coming across curves with either co-factors of 4 or 8 whenever it is a non-prime order group. Is this a co-incidence? Have we studied ECC for curves ...
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1answer
59 views

Why does Ed25519 scalar multiplication allow values larger than the subgroup order?

The GeScalarMultBase function is documented like so. From the way it is documented we see that it expects a little-endian value and has a precondition that constrains the range it accepts. ...
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678 views

Is it safe to reuse a ECDSA nonce for two signatures if the public keys are different?

We denote the s value of an ECDSA signature $(r, s)$ on a message $m$ as: $s=\frac{H(m)+xr}{k}$ Assume two ECDSA signatures sharing the same nonce $(r, s_1) , (r, s_2)$ on two messages $m_1, m_2$, ...
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77 views

Check validity of generated parameters for SIDH

In section 4.1 of the paper Towards Quantum-Resistant Cryptosystems From Supersingular Elliptic Curve Isogenies by Feo, Jao and Plût it is described how you generate valid parameters for the SIDH ...
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55 views

Get points of an Elliptic Curve defined over a Finite Field on Twisted Edwards Extended Coordinates

I'm working on a crypto library, and I need to perform some tests for the implementation of: Point Addition. Point Subtraction. Point Doubling. Scalar Mul Point. The operations are performed on ...
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1answer
345 views

curve25519 by openSSL

How can i generate ec curve25519 keys using openSSL? When I run openssl ecparam -name curve25519 -genkey -noout -out private.ec.key I have this message ...
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1answer
40 views

Does any problem arise when the order of an elliptic curve is equal to its prime field modulus? [duplicate]

Regarding cryptographic schemes in elliptic curve cryptography, is there a problem with having the order of an elliptic curve being equal to its prime field modulus? That is, an elliptic curve where $...
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52 views

Modifying Elliptic Curve Parameters

For context, I was watching this bit of the video: which goes over this source code. The piece is about elliptic curve cryptography and how it works. I want to use some of this knowledge to make my ...
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0answers
28 views

What is the cryptography involved in the initial setup of a cryptocurrency?

I keep hearing that when a cryptocurrency is created it goes through an initial setup phase wherein cryptographic parameters are created that are used by the cryptocurrency network throughout its ...
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41 views

The ECC private key is generated with 0x00 at the beginning.(prefix)

I created a private key using the prime256v1 curve. My purpose is to get a 32 byte private key. However, the private key is preceded by 0x00, resulting in 33 bytes. Why is this happening? The only ...
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30 views

How to get a random point of a specific EC group with cofactor Not-Equal 1?

We got a EC group generated with point G, and the cofactor of E(G) is with the similar size of the Order. Now we need a random point of E(G) and not revealing the "logarithm" of the random point, so ...
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1answer
82 views

Are there any security risks in using Elliptic Curves defined over fields $\mathbf{F}_{p^n}$ where $n>1$

I've recently been studying elliptic curves, and I've found that most of the current implementations use fields $\mathbf{Z_p}$ or in some cases $\mathbf{F}_{2^n}$. All the reasons I've seen for not ...
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1answer
52 views

Replacing elliptic curve diffie-hellman primitive with elliptic curve cofactor diffie-hellman for specifc curves?

From what I've read about elliptic curve Diffie-hellman with and without cofactor (I am pretty new to the whole thing so I am not able to understand everything) is that when the cofactor of the curve $...
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1answer
63 views

Why does Hasse's theorem sometimes seem to be invalid?

In order to generate secure elliptic curves, this answer recommends to Calculate the cardinal $|E(\mathbb{F}_p)|$ Check this cardinal is in the hasse interval (with $p$ prime) and to ...
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1answer
121 views

Elliptic Curve Cryptography insecure when input does not lie on the curve?

I am new to Elliptic Curve Cryptography and I was reading up on it online when I came across this link. It stated the following. Unfortunately, there is a gap between ECDLP difficulty and ECC ...
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1answer
88 views

Reasoning about WebCrypto ECDSA choices: P-256/384/521, SHA-1/256/384/512?

When implementing EC signing/verification in Javascript, the only options available via the WebCrypto API are: Curves: P-256, P-384, or P-521 Hashes: SHA-1, SHA-256, SHA-384, or SHA-512 If I was ...
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1answer
61 views

Valid private keys on curve25519

Given that valid private keys on curve25519 must be less than the order of the curve which is (as I understand) already smaller than 2^256, AND a valid key must be clamped to be divisible by 8 and ...
3
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1answer
55 views

Is inversion always cheap with Twisted Edwards curves?

I'm reading on Jubjub, which is planned for the next upgrade of Zcash. It is based on a Twisted Edwards curve with parameters $a = -1$ and $d = −(10240/10241)$. The reading says Jubjub does not need ...
3
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1answer
111 views

What makes lattice-based cryptography quantum-resistant?

As opposed to RSA or elliptic curve cryptography?
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2answers
64 views

Complexity of number field sieve theorem does not match with security of elliptic curves

Number field sieve algorithm can is used to break discrete logarithm on field $F_{p^n}$. The algorithm has time complexity $\exp((c+o(1))\cdot(\log p^n)^{1/3}\cdot(\log \log p^n)^{2/3}$. Originally ...
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2answers
71 views

Diffie-Hellman Primitives in SP800-56A

I wonder if someone can give an explain about the different between two Diffie-Hellman Primitives defined in SP800-56A, CH5.7.1 5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive 5.7....
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1answer
65 views

Question regarding of the ECC test vector format

I am trying to find some ECC test vector for using. I just find some post (like this) and github resource (like this ) They are good reference to my C test code but I'd like to get some more advice... ...
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2answers
237 views

Invalid curve attack: finding low order points

Background Here's a description of page 182 of "Guide to Elliptic Curve Cryptography" by Hankerson, Menezes and Vanstone. Here's a quote from that page: The main observation in invalid-curve ...
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1answer
39 views

Given a point $c$ in a field $Z_p$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$?

If we have a point in a field $c$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$ ?
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27 views

Key Value Based Key Derivation

For a system that is using public key cryptography to authenticate users and their actions I'm trying to solve (ease) UI/UX problem so that users will be able to use login/passwords they're accustomed ...
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2answers
75 views

SIDH cryptosystem question

I'm trying to understand the SIDH cryptosystem and got confused at this point: Alice fixes base $\{P_A,Q_A\}$ so that it generates $E_0[l_A^{e_A}]$. Then she chooses secret parameters $m_A,n_A$ and ...
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1answer
84 views

NIST elliptic curves behaving anamolous in OPENSSL benchmark

I tried to collect some benchmarks on NIST elliptic curves using charm library. The charm library is just a wrapper over OPENSSL. I experimented with prime192v1 (P-192), secp224r1 (P-224), prime256v1 (...
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1answer
115 views

Naming convention for NIST elliptic curves in OPENSSL

NIST standardized 5 elliptic curves (P-192, P-224, P-256, P-384, P-521) for prime fields. When I looked into openssl, these curves are named as prime192v1, secp224r1, prime256v1, secp384r1, secp521r1. ...
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1answer
182 views

Is Curve P-384 equal to secp384r1?

I am a bit confused with different notations of elliptic curves. Specifically, I am comparing the NIST specification with the SECG specification. More specifically I want to know if the NIST curve $...
2
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0answers
88 views

Security of an Elliptic Curve Public Key with a “Small” x-coordinate

Consider an elliptic curve over a finite field $F_p$ with $p$ prime and order $n$. Let $Q$ be a generator for the field. Given a public key point $P = aQ$, suppose we have an algorithm that finds an ...
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1answer
130 views

Do Weil, Tate, and Ate pairings exist on all elliptic curves?

I don't know much about the math behind elliptic curves. Do Weil, Tate and Ate pairings exist on all elliptic curves? If the answer is negative, then what pairings do MNT, BN and SS curves have? ...
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1answer
136 views

What does the number 256 in pairing curve BN256 indicate?

There are many pairing based elliptic curves like MNT curves, BN curves, SS curves etc., When we say BN256 curve, what does the number 256 indicate? Is it some group order or number of bits required ...
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1answer
78 views

Is there a concept of embedding degree for non-pairing based elliptic curves?

From this post, I learned the concept of embedding degree. Intuitively, if embedding degree of an elliptic curve $E(F_p)$ is $k$, it means there is a way to transform points in $E(F_p)$ to $F_{p^k}$. ...
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2answers
177 views

EC Key Compression

Using the secp256k1 curve, will the below yield the same result? Generate private key -> compress private key -> generate public key Generate private key -> generate public key -> compress public key ...
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1answer
54 views

Elliptic curve over prime field with high order roots of unity

Suppose I have an elliptic curve defined over a prime field $\operatorname{GF}(p)$ where $p$ is a large prime (e.g. 256-bit). Suppose also that $p = kn +1$, where $n$ is a relatively large power of $2$...
3
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1answer
158 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) ...
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1answer
42 views

Is there a lower limit on message length for signature?

I was working on a tool that signs small messages (~20 bytes) when a question occurred about message size: What would be the risk of using extremely small/restricted input (say, 5 bytes of hexadecimal ...
0
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1answer
51 views

Question about using Montgomery form for elliptic curve operations on bls12-381

Since the prime for bls12-381 is not of a form to allow easy modular reduction , is the best approach to use the Montgomery multiplication + reduction algorithm? ...
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2answers
276 views

In Elliptic Curve, what does the point at infinity look like?

We know that for each point $P$ in curve $E$ there exists a minimum scalar $k$ such that $k*P$ equals the point at infinity. And the book Cryptography Theory and Practice by Douglas R. Stinson only ...
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1answer
39 views

Is there a way to project one elliptic-curve element to a subgroup with certain size?

For discrete logarithm we can pick a random number $n$ and project it to a subgroup. E.g. given a prime $p$ with $p-1 = 2\cdot a \cdot b +1$ we can compute $n^{((p-1)/a)} \equiv n_a \mod p$ after this ...
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0answers
82 views

ECC with 512bit compatible curves

I understand that given solutions for solving a discrete logarithm problem are on the order of 𝑂(2𝑛/2), ergo, 256bit private keys based on 25519 or secp256k1 have an effective bit strength of ...