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 ...

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Bitcoin multi-sign and the amount of storage it required [migrated]

I read this paper (Threshold-optimal DSA/ECDSA signatures and an application to Bitcoin wallet security), the authors said that in the optimal threshold signature that they propose players requires ...
0
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1answer
22 views

Message Expansion / Encryption Blowup Factor / Ciphertext Expansion of ECC

In order to complete the following table with asymptotic times and message expansions, $\quad \quad \quad \quad \quad \quad \quad \quad \quad$ RSA $\quad$ McEliece $\quad$ ECC Encryption Speed $\...
2
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1answer
32 views

Security model scalar multiplication in NaCl

I'm having trouble understanding what the “security model” for “scalar multiplication” in NaCl is. Security model crypto_scalarmult is designed to be ...
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How can I find all points on an elliptic curve?

I have an elliptic curve $$y^2=x^3 + 8 \times x + 6 \mod 13$$ How do I find all the points on this EC? Also, could you give me more examples?
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1answer
129 views

Do I need to prove this?

I am using ABE scheme that has already proven under BDHE assumption. Here is the scheme https://eprint.iacr.org/2008/290.pdf In the key generation algorithm, I want to tie the user secret key ...
1
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1answer
66 views

Given $ g^s, g^y , g^r, g^t, g^{st-rs}, g^{(yr+d)/t}$ , is it hard to distinguish $e(g,g)^{syr}$ from a random value?

Where $g$ is a group element in bilinear group $G$, $e(g,g)∈GT$ and $s, y, r, t, d$ are randomly chosen. I understand it is very similar to the conventional DBDH problem, but $g^t, g^{st-rs}, g^{(yr+...
2
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1answer
41 views

Given $g^a, Y$, is it hard to distinguish $e(g,g)^{ab}$ from a random value?

where $g$ is a group element in bilinear group $G$ $Y = M.e(g,g)^{ab}$ $M$ is a message Does anyone know the answer or suggest some material for reference? Many Thanks
5
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64 views

Using a single Ed25519 key for encryption and signature

The libsodium documentation contains a function crypto_sign_ed25519_pk_to_curve25519 that converts an Ed25519 key into a Curve25519 one, so it can be used for both ...
1
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1answer
23 views

Time efficiency of Bitcoin Multi-signature Vs. threshold signature

I read this paper (Securing Bitcoin wallets via a new DSA/ECDSA threshold signature scheme) that illustrated that threshold signature is the best solution to avoid single point of failure but I think ...
0
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2answers
65 views

security of using digital signature as key derivation material

I want all keys in my iOS application to rely on the Secure Enclave for security as the iOS keychain becomes insecure on jailbroken devices. Currently, the Secure Enclave currently only supports ...
0
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1answer
46 views

Is this ECDHE approach secure?

First of all sorry if this question could be trivial, but I did some researches and it seems to me that this is a standard approach, but I want to be really sure. I'm developing a signed/encrypted ...
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0answers
53 views

Why do openssl elliptic curve digital signatures differ by one byte [duplicate]

For some curves defined in Openssl 1.0.2h if I generate a key and generate a digital signature the size of the signature block changes by one byte. Why does this happen? My test file is, y.sh ...
6
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1answer
87 views

What does Shor's algorithm tell us about the complexity class of RSA and the DLP?

If quantum computers operate in BQP and (using Shor's algorithm) they are able to factor large integers and break the discrete log problem, what does that tell us about the complexity class of these ...
1
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1answer
69 views

Is brainpoolP512r1 safe?

I'm using brainpoolP512r1 elliptic curve for DSA and DH in my application. Recently I've found out on this website that brainpoolP256t1 and brainpoolP384t1 aren't secure. Unfortunately my curve wasn't ...
1
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0answers
71 views

Feasible attacks on ECRSA cryptosystem

In ECRSA cryptosystem, I want to know the feasible attacks. For illustrations we have two prime $p$ and $q$ such that $p \equiv 2 \pmod 3$, $q \equiv 2 \pmod 3$ and generate key pair as follows: $$n = ...
0
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0answers
26 views

elliptic curve and embedding degree

I am new in ECC. I am confused what the embedding degree in elliptic curve represents and what is the impact of its values on the curve and security ( small values or large values? What does the ...
0
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0answers
44 views

Found weil pairing. Index Calculus method on the results of weil pairing

Consider Elliptic curve with p = 59, A = 1, B= 0, P = (25,30) and Q=(35,31). So I tried to solve this using MOV attack. The torsion point for them E[5] is R(-25,30i) where is sqroot -1 Chosen two ...
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0answers
55 views

Why is RSA still being used? [duplicate]

In hybrid encrytion, I still see that some site's use RSA in their https connection, so now I wonder, why do they not use ECC instead of RSA, ECC requires less computational power and encrypt's and ...
2
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1answer
136 views

How does DH work when combined with ECC or RSA?

I know that Diffie-Hellman is used to create keys in a secure way over an insecure channel. But there is one thing which I cannot understand: I see that a lot of sites use ECC or RSA alongside DH. ...
3
votes
1answer
80 views

tripartite diffie hellman with Weil pairing

I try to understand how the tripartite Diffie-Hellman key exchange works. I read Joux's paper for this: https://www.semanticscholar.org/paper/A-One-Round-Protocol-for-Tripartite-Diffie-Hellman-Joux/...
3
votes
1answer
53 views

Is a hash function used to expand the key after ECC shared secret is complete?

I am trying not to solicit opinions, but I have not been able to find a reference to the question at hand. I have designed an ECC engine in silicon that handles any curve in the form of $y^2 = [ax^3 +...
4
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2answers
104 views

Recover elliptic curve order from ECDSA signatures

I need the elliptic curve order to recover the private key from two signed messages with ECDSA. What I have: two signatures signed by the private-key I want to get the messages that have been ...
2
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0answers
30 views

Boneh/Franklin Identity based encryption with Tate pairing

Boneh/Franklin developed an identity based encryption scheme based on the Weil Pairing. This algorithm has also been standardised in IEEE P1363.3 . I know that this algorithm can also be implemented ...
4
votes
1answer
52 views

Key derivation with Curve25519 for data encryption

I'm new when it comes to cryptography so this question might be a little bit stupid: Can I derive a key with Curve25519 from a (let's say) root key and encrypt the file with a secret key algorithm? ...
0
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1answer
35 views

meaning of Reduce f modulo the order of the base point G

I am trying to perform some math operations related to Elliptic curve cryptography, and came across this sentence: Reduce f modulo the order of the base point G. What does it mean?
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1answer
235 views

ECDSA key recovery - floating point values

I'm currently attempting to recover an ECDSA key. I have $m$, $m'$ and signatures $(r, s)$, $(r', s')$, and I know that $k$ is constant, the curve is NIST P-192 and the hash function of the. As such,...
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0answers
34 views

hash function for elliptic curve co-ordinates

Is there any hash function which takes the co-ordinates of an elliptic curve $E_p(a,b)$ as input and gives an integer value i.e. $h(.) : \{(x,y) \in E_p(a,b)\} \rightarrow \mathbb{Z}$
0
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1answer
19 views

How many field operations are needed when you compute kG in elliptic curves with a multiple additions or the double-ans-add-algorithm?

For an assignment, we have to calculate how many field computations are needed to calculate kG in an elliptic curve. They want us to show this for two different ways of calculating kG. The first way: ...
2
votes
1answer
71 views

If I encrypt data with two different ECC Private keys, how secure is the result?

I'm wondering whether encrypting data with two different 256-bit ECC keys will result in a more secure encryption (minimum 384 bit equivalent) or will result in data that's effectively encrypted with ...
1
vote
1answer
71 views

Scalar Multiplication for Elliptic Curve

Let $\mathbb{E}$ be the elliptic curve $y^2 = x^3 + 6x \text{ mod } 11$ and consider the point $P = (2, 3)$ on it. How do I compute $3P$? I have been able to figure out what $2P$ is, $2P = (5,10)$. ...
11
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1answer
1k views

Why can an elliptic curve private key be 1?

I often see in papers (e.g. this one) that for an elliptic curve with generator point $G$ and order $n$ the private key $d$ can take on any integer value in the range $[1, n)$. When $d = 1$ the ...
3
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2answers
308 views

Which elliptic curves are quantum resistant? [closed]

If I want to learn about quantum resistant crytography what are the best resources? Which type of elliptic curves should I be studying?
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1answer
94 views

Creating ECDH using OpenSSL

For academic reasons, I'm playing around with OpennSSL 1.0.2g. I tested RSA encryption/decryption. I created key exchange with DHKE. But I'm struggling to find a way, to create ECDH, using only ...
3
votes
1answer
130 views

Quantum vs. regular computing time to break ECC?

How long exactly would it take for a regular computer to crack an elliptic curve public/private key via bruteforce, vs. a quantum computer using Shor's algorithm with a couple thousand qubits? Can ...
1
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1answer
26 views

Single-scalar multiplication with sign bit

I want to know if there's an "easy" timing-free way to compute the sign of $y$-coordinate of $r G$, for a secret scalar $r$ and $G$ the generator of an elliptic curve. We have several choices for ...
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3answers
85 views

What is proper size of window Using RSA Window method?

I want to Use Window method for RSA modular Exponentiation. because of SCA. But I don`t know what is proper window size.
0
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1answer
41 views

Is a KDF needed when using X25519 for ECDH and XChaCha20-Poly1305 for AEAD?

Is a KDF needed when using X25519 for ECDH and XChaCha20-Poly1305 or XSalsa20-Poly1305 for authenticated encryption? My hypothesis is "no" because the key from X25519 is less than 256 bits long; ...
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0answers
40 views

Has anybody implemented BGN cryptosystem to multiply two plain texts?

I have tried to use this code as follows, http://stackoverflow.com/questions/33581962/bgn-implementation-in-java but the issue is, when it generate the pairing, the output is in GTfiniteelement (for ...
2
votes
1answer
69 views

Multiplying integers modulo $2^{255}-19$ using the Curve25519 polynomial reduction algorithm

I'm reading the Curve25519 paper and I see that in page 11, under the title "Multiplying integers modulo $2^{255}-19$", it says: The coefficients of $x^{10}$; $x^{11}$; ...; $x^{18}$ in $uv$ are ...
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1answer
32 views

How to find HEXA value of String [closed]

This may be very easy thing, but i am just unable to understand this. I have a string "Hi", and in HEXA its written like this ...
7
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1answer
236 views

How many qubits are required to break RSA 2048 or 4096 with a universal quantum computer?

So in the news this week, IBM have created a universal quantum computer with 5 fully functional qubits. Logic and Moore's law dictates they will be able to scale this up to a lot more qubits within a ...
3
votes
1answer
73 views

ECFP harder than ECDLP ?

Given two points $P$ and $Q = \sum_{i=1}^{n} x_i.P$ over $E_p(a, b)$ for $x_1,x_2,...,x_n \in \mathbb F_p$. The Elliptic Curve Factorization Problem (ECFP) is to find the points $x_1.P,x_2.P,...,x_n.P$...
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0answers
56 views

Weil Pairing - Miller's Algorithm

I'm trying to implement Weil Pairing using Miller's algorithm. I have got couple of questions. How to select $m$? As stated in this link page 13, I interated from $1$ to order of point $P$ such that ...
6
votes
1answer
103 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 ...
5
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1answer
110 views

Backdoor in NIST elliptic curves

Let $E$ be an elliptic curve defined over a finite field $F_q$ with prime order $n$ and $P,Q \in E$ and $k$ be private key such that $kP=Q$. Since $n$ is prime, $E$ is isomorphic to $Z_n$. Suppose $\...
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0answers
59 views

Smart Card choice for PKI implementation

I'm seeking to implement a national digital signature standard on a smart card. I feel like this is a good place to ask if anybody is acquainted with a hardware supplier offering smart cards that ...
4
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0answers
52 views

EC Schnorr signature: multiple standard?

I 'm working on some EC-Schnorr signature code. Reading various papers on that, it's seems EC-Schnorr is not standardized as well as ECDSA. For example, I found two main differences in two main ...
1
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1answer
43 views

Attacks on schemes based on elliptic curves when the transmitted points are not on the curve

Some elliptic curve schemes require to send a curve point during the normal execution of the protocol. For example, ElGamal encryption and ElGamal signature require this. On the other hand, ECDSA does ...
2
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0answers
34 views

Some confusions about Repeated Doubling Algorithm?

The following repeated point doubling algorithm is taken from the book Guide to Elliptic Curve Cryptography by D. Hankerson, A. Menezes, and S. Vanstone on page#93. Clearly, this algorithm is ...
6
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2answers
108 views

Is there a theorem to determine the elliptic curve parameters based on the group order?

By Hasse's theorem we know that range of the group order of the elliptic curve. And similarly, there exist a theorem on the admissible order of elliptic curves. Suppose by the theorem on the ...