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 sign an elliptic curve point using an ECC signature scheme?

In Schnorr based ECC signature scheme, a message $M$ is signed with the private key $\mathit{sk}$ as $$s=\mathit{sk}\cdot h(M,R)+k$$ where $R=k\cdot P$ and $P$ is a base point. If $M$ is a point ...
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111 views

How does encryption work in elliptic curve cryptography?

So I think I understand a good amount of the theory behind elliptic curve cryptography, however I am slightly unclear on how exactly a message in encrypted and then how is it decrypted. So my ...
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591 views

How can I implement the elliptic curve MOV attack myself?

I understand and have implemented elliptic curve signatures in Python without the use of libraries like Sage, and would like to implement the MOV attack against certain weak types of elliptic curves. ...
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52 views

A standard extension of ECIES for multiple recipients (broadcast / multiparty)?

I have one sender, and a small number (~5) of recipients. The sender knows each recipient's public EC key. I want the sender to broadcast a single message in such a way that any one of the recipients ...
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RSA & DH at risk due to math advances, will this eventually affect elliptic curves too?

I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm ...
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3answers
145 views

Is there any alternative for extended euclidean algorithm to perform modulo division?

I'm implementing point addition and point doubling operations for elliptic curve cryptography. I have implemented extended euclidean algorithm to perform modulo division. It appears the that ...
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3answers
210 views

Do I understand (below) why Q = dP is easy while finding d is hard

As we all know for discussion of Dual_EC_DBRG, the point on an elliptic curve Q can be calculated from P and some (large) integer d $Q = dP$ And we know that knowledge of Q and P is not sufficient ...
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1answer
72 views

Schnorr Ring Signatures - wrong hash results

I don't know if I've placed the question right, It is half maths, half programming. I'm writing Schnorr Ring Signatures on Elliptic Curves in Java and I have a problem. I've found scheme on integers ...
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52 views

What happens if you multiply a point with its order on complete Edwards curves?

I was recently working with some ECC crypto and stumbled across the following phrase on the SafeCurves page: The rational points of a complete Edwards curve are the pairs (x,y) of elements of ...
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7k views

Is secp256r1 more secure than secp256k1?

Curves secp256r1 and secp256k1 are both examples of two elliptic curves used in various asymmetric cryptography. Googling for these shows most of the top results are Bitcoin related. I've heard the ...
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238 views

What is the most secure ECC Curve?

I have for a while used Koblitz curve (sect571k1), in ECDH and ECDSA. But I have started wonder if it is the most secure. I prefer security over efficiency. So the curve doesn't have to be the most ...
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88 views

All the affine points on the curve

I have calculated the affine points on the curve $x^2 + y^2 = 1 − 3x^2y^2$ over the field ${\mathbb{Z}}_{11}.$ Using $y^2 = \frac{1-x^2}{1+3x^2}$ I got the following points: ...
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242 views

How do I get the equivalent strength of an ECC key?

I know how to calculate the comparable symmetric strength of an RSA modulus: calculate the running time for a field sieve. This is how NIST gives approximate symmetric sizes for asymmetric algos in ...
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1answer
384 views

Koblitz encoding a message to a point, what is the “associated auxiliary base parameter”?

I am looking at the Koblitz method for encoding a message as an elliptic curve point. The first step given in the paper I'm reading is: "Choose an elliptic curve and its associated auxiliary base ...
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83 views

HD (Hierarchical Deterministic) Keys using Safe Curves?

Bitcoin's HD (Hierarchical Deterministic) Keys as described in BIP32 allow for a master key to be created (a private key and a chain code) such that a tree of both public and private keys can be ...
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6k views

ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
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2answers
189 views

Why are some x-coordinates unsuitable for an ECDSA generator point?

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ...
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983 views

Elliptic curve cryptography related key attacks

This question is an extension of Families of public/private keys in elliptic curve cryptography As described above, bitcoin "type 2" deterministic wallets use a root private/public key pair, where ...
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52 views

Translation of Schorr Ring Signature to ECSchnorr Ring Signature

I have to write an EC version of Schnorr Ring Signature Scheme. I've already wrote regular ECSchnorr Signature Scheme using this (page 128). I've found a scheme of Schnorr Ring Signature Scheme (page ...
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1answer
69 views

Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order

In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent ...
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35 views

Seeking an implementation of the Satoh algorithm for elliptical curve point counting

I would be very grateful if someone has an implementation of the Satoh algorithm (Fast Elliptic Curve Point Counting). Can someone point me to practical algorithm implementations or provide some ...
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66 views

How to compute projective cordinate Z in elliptic curve cryptography?

I was working on affine coordinates and struggling with the computation time taken for operations and then I was advised to use projective coordinates so that mul-inverse operation can be avoided Can ...
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32 views

Is it possible to re-generate ECPublic key from raw private key and curve params?

As per my understanding, in Elliptical curve cryptography, the EC PrivateKey is nothing but a bigInteger value. If you know the curve specs and you have this private BigInteger value (which I am ...
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1answer
75 views

How to perfrom modular division while numerator is lesser than the denominator?

I'm implementing point addition and point doubling of elliptic curve cryptography. The formula that I'm using for slope is Point addition: $S = \frac{(P_y-Q_y)}{(P_x-Q_x)}$ where $P$ and $Q$ are the ...
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Reasons for Chinese SM2 Digital Signature Algorithm

In the IETF RFC draft named "SM2 Digital Signature Algorithm" a signature algorithm is specified. The RFC does however not mention why this signature algorithm has been defined. Nor does it specify ...
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42 views

elliptic curve multiplication with negative factor

I'm learning the multiplication operation on EC. From most material I can found, the multiplication $nP$ is just: $$nP=P+P+\cdots +P+P$$ For negative factor, i.e. $(-n)P$, by above definition and the ...
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65 views

Cryptographic operations for NISTP256 can be implemented using montgomery method?

I need to implement cryptographic operations starting with the curve NISTP256. I have been told to implement it using Montgomery method. I read a pfd from ...
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32 views

Edwards / Montgomery ECC over binary extension fields

I recentely had a discussion about the redesign of our ECC code for the library I'm collaborating on and the person I was discussing with came up with Edwards and Montgomery curves over binary ...
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1answer
396 views

ECC keys vulnerable to brute force attack?

I have started learning about Elliptic curve cryptography. Since the key size required in ECC is relatively lesser than the key size in RSA to provide the same amount of strong encryptions, I wonder ...
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1answer
92 views

Is there any reason not to use EdDSA with Weierstrass curves?

I'm volunterely working for a crypto library and we're planning on adding Curve25519 support (finally). At the same point I had the idea of adding support for EdDSA in the same run. Our library is ...
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50 views

Elliptic Curve Blind Signature Implementation

I have seen this prior post: Elliptic Curve based blind signature implementation Currently I'm sizing up how difficult it would be to attain Elliptic Curve Blind signatures for an application I'm ...
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2answers
81 views

Is there a way to do single key-pair asymmetric encryption?

All of the information on asymmetric encryption I can find uses key-exchanging to get a mutual symmetric key. What I am trying to do is encrypt a message with a public key and have it only readable ...
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How to find the integer multiplicand from given two points?

I have two points on an elliptic curve $P(x_1,y_1)$ and $Q(x_2,y_2)$ and a scalar value $x$, where $P=x \cdot Q$. What is the best way that I could figure out the value of $x$? Given that I know all ...
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40 views

Is this ECC based messaging method secure?

I developing an encrypted messaging for ZeroNet (aP2P file-based network, website: zeronet.io) and would like to have some guidelines if any of this could work. My first idea: In ZeroNet every user ...
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17 views

Strength of a cryptography algorithm [duplicate]

I'm just read this article from Atmel corporation in comparing RSA with ECC cryptography algorithms. First of all please read these two paragraphs quoted from the article: P1: Strength of an ...
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1answer
79 views

Where can I get the correct and precise algorithms for elliptic curve cryptography?

I have been asked to implement cryptographic operations using elliptic curves. I would like to get precise algorithms for various processes like key generation, digital signature and verification. I ...
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1answer
99 views

About end to end algorithm SMS from Jo Mehmet Sollihagen Øztarman [closed]

This is a signcryption scheme from End-to-End Data Protection of SMS Messages by Jo Mehmet Sollihagen Øztarman (pdf, section 4.3.1): Public parameters C: an elliptic curve over GF(ph) with p ≥ ...
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65 views

EdDSA Verification vs. Cofactorless Verification

In the EdDSA for more curve paper the authors defines: Keys An EdDSA secret key is a $b$-bit string $k$. The hash $H(k) = (h_0, h_1, ... , h_{2b−1})$ determines an integer $s = ...
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1answer
52 views

Are private and public key sizes of Elliptic curve related?

I'm new to elliptic curve cryptography. I just want to know in the case where I take a random number (private key) and find its associated public key, does the size of the public key depends upon ...
3
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1answer
105 views

A (current or soon-to-be) NIST-recommended alternative to ECC?

So this comes from the professional rumor-mill, and I'm wondering if anyone might either debunk or shed light on this. My understanding is ECC is generally now preferred over RSA simply due to how ...
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39 views

ECDH security when no KDF is used

Let's suppose our device performs ECDH with a fixed, unknown, private key $prv$. It accepts as input any point $Q$ lying in the proper subgroup of the proper ellipict curve, then computes: $P = ...
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1answer
39 views

Is there a single-use signature scheme, where a second use of the private key discloses it to the world?

With ECDSA (and possibly DSA too) I'm aware that if the same value for $k$ is used with the same private key $D_A$ to sign two different messages, then anyone possessing the two messages $m_0$ and ...
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2answers
37 views

With EC secp256k1 is there a way of transforming a function of the private key to a function of the public key?

A key pair has a private key $D_A$ and a public key $Q_A$. $D_A$ is an integer less than the curve's $n$. Is there any (boolean) function of the private key $f(D_A)$ which can be transformed into a ...
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1answer
114 views

With ECDSA is there a way for the verifier to calculate any properties of $k$?

With ECDSA, given $(r,s)$ and $m$, is there a way for a verifier to calculate any (boolean) properties of $k$, without knowing $k$ or the private key $D_A$? (I understand that $k$ should be random, ...
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2answers
58 views

How does the size of the prime affect Elliptic Curve Bit Security?

I am using the MIRACL Library to implement an Elliptic Curve Diffie Hellman based Key Exchange according to ECDH-Scheme-Wikipedia. Referring to the Miracl Docs they suggest a few curves. Each curve ...
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39 views

Is it possible and safe to use SAKKE for signing/verification, rather than for encryption?

Is it safe to use the Sakai–Kasahara key encryption algorithm (SAKKE) for signing/verification, rather than for encryption? (Example at bitbucket.org) In particular, I want many Bobs to be able to ...
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2answers
396 views

Are all possible EC private keys valid?

I usually generate a key pair using openssl or Bouncy Castle. I'm using curve secp256k1. The 256bit private keys look fairly random. Do all values of "private ...
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1answer
321 views

Do Weak Elliptic Curves Exist?

I'm running into very contradictory opinions when try to understand if weak elliptic curves exist. I'm not interested in the case when a curve's weakness is attributed to properties of an EC's prime, ...
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1answer
67 views

Is it safe to generate ECDSA keys from the hash of the previous, over and over again?

I have memorized a very long and secure passphrase, that when hashed with sha256, I can use the result as an ECDSA private key, and use it as a brain-address for Bitcoin. Now I need more bitcoin ...
5
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1answer
115 views

Why not to use curve over field of $p^m$ with $p > 2$ for ECDSA?

I'm reading the ECDSA paper and they say you can only use ECDSA with odd-power fields $p$ or with binary fields $2^m$. Why not other power prime fields?