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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How to use the non adjacent form (NAF) algorithm to implement scalar multiplication? [on hold]

I'm confused by the following NAF-algorithm. How could I possibly use it to implement scalar multiplication for elliptic curves? ...
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how scalar multiplication using periodic sequence in ECC be created?

if it is prepare in java then what steps would be taken? how NIST recommended binary curve be used in the scalar multiple algorithm
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1answer
62 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 ...
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35 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 ...
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1answer
66 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 ...
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102 views

If we should not reuse primes in DH, shouldn't we not reuse ECDH elliptic curve properties?

An article How is NSA breaking so much crypto? describes NSA's methods for breaking encryption. If a client and server are speaking Diffie-Hellman, they first need to agree on a large prime number ...
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102 views

What happens if no final subtraction is done in Montgomery multiplication?

I'm doing Montgomery arithmetic modulo $N = 2^{255}-19$ for the Curve25519, picking $R = 2^{256}$ for Montgomery. After multiplying two numbers $0 \leq A,B < N$ in the Montgomery representation ...
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1answer
94 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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What is th purpose of m and q in elliptic curve cryptography protocols?

In crypto protocols that contains calculus on elliptic curves I can often see $\dfrac{m}{q}$$Q$ where $m$ stands for order of EC points group and $q$ is the order of corresponding subgroup of $m$. $Q ...
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1answer
227 views

Simple digital signature example that one could compute without a computer?

I am working on a document to explain Bitcoin to students. But I am having a hard time translating the principle described in §2 of the Bitcoin whitepaper in layman's terms. There is a great question ...
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24 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 ...
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49 views

Could someone explain the given algorithm?

The following snapshot of the algorithm is taken from the book Guide to Elliptic Curve Cryptography published by Springer 2004. I don't understand that why at the statement 9.1 if both $T_1$ and ...
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2answers
878 views

Understanding elliptic curve encryption [closed]

I'm having a hard time understanding the elliptic curve encryption. One thing thing I don't understand is listing all the points on the curve mod p. Suppose I have the following elliptic curve: $y^2 = ...
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0answers
55 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 ...
3
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0answers
30 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 ...
5
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1answer
176 views

Finding the subgroup in isogeny-based cryptography

Isogeny-based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is theorem: Elliptic curves ...
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1answer
37 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 ...
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24 views

Pairing on BN curves, GMP code

Is it possible to write a small implementation of tate pairing using BN curves using GMP. It does not need to be efficient. I just want to understand the steps. Thank you.
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94 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 ...
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1answer
28 views

Is it possible for the Rho method against an Elliptic Curve to take more than the sqrt of the total state space?

Is it possible for the Rho method against an Elliptic Curve to take more than the sqrt of the total state space? It the reason why this is not generally done because of a meet-in-the-middle attack?
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2answers
50 views

Stripping off message authentication or signature

If attackers can strip off RSA / EC / -DSA digital signature and conduct CCA on AES-CTR or CBC payload, why can't they do the same for AES-GCM?
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2answers
75 views

What is necessary for generating an elliptic curve?

Let's say I want to generate my own elliptic curve with an order whose bit length is $n$ (specifically 2048, 4096, and/or 8192)? How would I do this? What needs to be done? What software can do this? ...
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1answer
49 views

Does OpenSSL apply ASN1 encoding to the hash before signing using ECDSA?

I read on stack overflow that OpenSSL performs ASN1 encoding to the hash before signing it for, for ECDSA. In other words, OpenSSL performs the following steps when for an Elliptic curve key ...
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53 views

Software timing attack using Kocher method

What's the minimum number of random sample points needed in Kocher's timing attack, so that we can determine enough valid measurements of $A_{i,r}$ and $D_{i,r}$? I'm working from this paper: Volker ...
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2answers
559 views

Creating serial key generator using ECDSA, how to get signature short enough?

I've written some questions in this stackoverflow and got great responses but now I'm trying to wrap it all together. I have for the last couple of weeks been building a serial key generator project ...
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34 views

How common are non-RSA digital certificates?

Is there a statistic available that shows just how common are DSA or ECC certificates amongst webservers? I know that RSA-based certificates are the most common, however I'd like to know, if there is ...
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Is an instruction similar to PCLMULQDQ valuable in post quantum key exchange?

This paper describes a Quantum Key Exchange based on the Ring Learning With Errors problem. When used with ECC, there is only a slight performance impact. Assuming this is the popularized approach ...
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1answer
55 views

Comparison Affine Coordinates and Projective Coordinates Addition in Excel

Kurve : EC : $y^2=x^3 + x + 1$ Generator:$(1,7)$ $p=23$ Result in Affine use Excel: $P=(1,7)$, $Q=(7,11) \implies P+Q=(18,20)$ Result in Projective use Excel: $P=(1:7:1), Q=(7:11:1) \implies ...
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2answers
74 views

Is the Discrete logarithm problem suitable for this pairing scheme?

Let $Ans$ be the product of two pairings : $e(g,h)^{k} \times e(g,h)^{r}=Ans$ If everybody knows only $[g,h,e(g,h)^{k}]$ but $[r,Ans]$ is not known. In the discrete logarithm problem, the user knows ...
3
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1answer
21 views

Reuse of TLS client key/certificate in challenge-response protocol

The situation: We have a custom PKI with clients communicating with the server over standard SSL/TLS encrypted channel. PKI uses ECC, server certificate supports ECDHE_ECDSA key exchange mechanism and ...
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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 ...
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1answer
6k views

ECDSA vs ECIES vs ECDH

Recently I started studying Elliptic Curve Cryptography and I just loved it. I want to transfer some big data (like 3KB), What is the best method, ECDSA, ECIES, or ECDH (and why)? I am confused, how ...
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2answers
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elliptic curve point doubling in Jacobian coordinates

I am writing an application that uses Elliptic curve Diffie–Hellman for authentication. I found two formulas for point doubling in Jacobian coordinates. 1st) \begin{equation} X_1 = (3x^2 + aZ^4)^2 ...
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1answer
50 views

Proper forward secrecy [closed]

Currently I have a protocol using a simple RSA to AES handshake. I have been reading more and more and would like to implement proper forward secrecy, but at the same time I'd like to improve the ...
2
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0answers
104 views

RSA_DH vs ECDH implementation

In ECDH protocol is possible, naturally, to use the same algorithm for calculate a secret key for both communication parties (Alice and Bob for example). It is possible to design also a same algorithm ...
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1answer
118 views

Which attacks are possible against raw/textbook elliptic curve?

A quick question, we know that raw RSA is a no go. To solve this we have different PKCS standards forcing structure on the input messages. For EC the story is something else. For signatures we have ...
2
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1answer
86 views

Is there an asymmetric algorithm that can perform double encryption?

I'm looking for any asymmetric algorithm can perform the serial encryption, by serial I mean double or more encryption with different key(same key size). RSA can not do it since the cipher of the ...
4
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2answers
309 views

Using Lattice-based cryptography for TLS\SSL

Given the general benefits of Lattice-based cryptography, such as: Post quantum Security Security from worst case scenario Efficiency What could the outlook of shifting from RSA \ ECC-based ...
3
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1answer
62 views

Why inversion and multiplication operations are costly in elliptic curves?

There are several algorithms for efficient scalar multiplication of an arbitrary point P(x,y) by some positive integer k in elliptic curves defined over $F_{p}$ or $F_{2^{m}}$. The scalar ...
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3answers
111 views

ECDH-ECDSA Combination

I am doing research on cryptography primitives at a basic level and I faced a question on encryption methods. I understood that ECDH is an approach to for secure key exchange between two parties ...
6
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1answer
95 views

Encoding scalar values to points on Ed25519

I'm interested exploring key derivation and threshold signature protocol that require point arithmetic (addition) on the private scalar values and $S$ values of the signatures in ed25519. ...
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1answer
39 views

Non-Degeneracy of the Weil pairing

In this YouTube video, Dan Boneh mentions that if both points are defined on the base field then the pairing is degenerate. Why is that? And specifically is this true if I use the Weil Pairing?
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1answer
159 views

great discovery in the field of elliptic curves cryptography?

Prof. Adi Shamir says in The Cryptographers' Panel 2016: i think that NSA has made a great discovery in the field of elliptic curves cryptography and NSA wants to avoid the increased use and ...
3
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3answers
177 views

Supersingular Isogeny Key Exchange broken?

Found this report detailing a quantum algorithm for computing isogenies between supersingular elliptic curves. http://cacr.uwaterloo.ca/techreports/2014/cacr2014-24.pdf with the quote ...
2
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1answer
94 views

Elliptic curve ElGamal with homomorphic mapping

I am interested in ElGamal due to the fact that you can achieve some degree of homomorphic properties. I became interested in applying ElGamal to elliptic curves, and found this other question with an ...
3
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2answers
264 views

Is there any reason to use RSA or DSA when we have ECC?

I am having trouble coming up with a use case for RSA or DSA. It appears that ECC is better in every way. Is this true? I am looking for cases where RSA/DSA is superior to ECC, not where it is used ...
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Named Elliptical Curve parameters

Are named curve parameters always the same? I know this may be a stupid question however I think this is the case. For example the secp256r1 is defined in this documet ...
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1answer
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Why do the subexponential algoriths for the DLP not work for the ECDLP?

Elliptic curve cryptography is much more secure for the same parameters because attacks that work on the DLP do not work on the ECDLP. Why do the attacks fail in the latter case?
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How do we calculate DHKey using A's public key and B's private key?

I have 2 set of public/private keys. I would like to know how I can calculate DHKey. e.g: ...