# Tagged Questions

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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### What is more efficient, pairing based cryptography or non pairing based cryptography? [closed]

As far as I know non pairing pairing based cryptography is less time consuming than pairing based because, pairing based uses complex operations. Are there any advantages of pairing based cryptography ...
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### Why develop Edward curve formulas that deviate from unification?

Edward curves were considered initially because they provide a unified formula for both doubling and addition, thus having inherent side-channel resistance. But a lot of work has been done recently ...
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### Can I use an ECDH Shared Secret from the same Private / Public Key Pair?

I'd like to know if using the ECDH shared secret of a static EC Private Key with it's own corresponding static EC Public Key causes a problem / weakness. (edit) not asking if it's ok to re-use the ...
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### How to generate own secure elliptic curves?

I know that the algorithm used to generate the Brainpool curves and the NIST curves is published. The algorithm should be this one (RFC5639 Appendix A). From what it looks like it's rather slow to ...
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### EC: Why does $h>200$ need to hold?

From BSI-TR-03111 (PDF), on page 15: The class number of the principal order belonging to the endomorphism ring of E SHOULD be at least 200. This value commonly is referred to as $h$ in that ...
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### What are the differences between the elliptic curve equations?

I think we're all aware of the "classical" Weierstrass (short?) elliptic curve equation: $y^2\equiv x^3 + ax +b \pmod p$. Well known examples of these curves include the NIST's and Brainpool ones. ...
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### Doubling a point on an elliptic curve

I'm working with the elliptic curve $\mathcal{E} : y^2 = x^3 + 11x^2 + 17x + 25$ over $(\mathbb{Z}_{31},+,\cdot)$ and am trying to double $P=[2,7]$. Following the instructions here, I'm doing the ...
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### Supersingular Isogeny Key Exchange Software [closed]

Are there any optimised implementations of the Supersingular Isogeny Key Exchange by Defeo, Jao, and Plut. It seems like this would be a great drop-in replacement for Diffie-Hellman in both OpenSSL ...
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### Is the inverse of a point on an elliptic curve over $\mathbb{Z}_p$ always in the group?

I'm working on a zero knowledge proof system that uses ECC over $\mathbb{Z}_p$ (currently using NIST P-256 since mbed TLS doesn't support group operations on Curve25519, but the problem should be ...
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### Converting ECC Code from python to Java. Extended Euclidean Algorithm not working. [closed]

I'm in the process of converting a Python program I found for calculating ECDSA public keypairs from a given private key. In this particular case it's on the Bitcoin curve. https://raw....
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### Is it possible to choose which point will have the public key of a given Elliptic Curve?

I am wondering if there is a feasible way that, given a specific elliptic curve (such as secp256r1), I could create a keypair where the public key has a given $x$-coordinate. If it is not possible, is ...
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### PQ Key Exchange based on Elliptic Curve Isogenies

I was reading the Wikipedia article on Post Quantum Cryptography and was interested in opinions as to whether the Supersingular Elliptic Curve Isogeny Diffie-Hellman listed there would be a good Post ...
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### Key exchange using ECDH vs ECIES

I'm a beginner to ECC crypto programming. Does any one explain to me the difference with using ECDH for shared key exchange and use of ECIES by encrypting shared key with the public key of the ...
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### Why does anyone use elliptic curves for a CSPRNG?

I saw Martijn Grooten's talk on elliptic curves at BSides London this year, and it helped me understand how elliptic curve crypto works, especially in the case of Diffie-Hellman (ECDH). He also ...
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### Is it right about an example of bilinear map $e$?

The equation for the elliptic curve is $y^{2} = x^{3} + x$ and is defined over the field $F_{q}$ for some prime $q\equiv 3 \pmod 4$, and set $q=307$. Choose random generator $g=[182, 240]$. My ...
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### Trying to understand the use of ECC in TLS certificates

I was looking the certificate of this website: https://www.cloudflare.com that has an ECC based certification. I'm just curious to know if is possible to understand which elliptic curve is used and ...
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### 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 ...
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### Would key stretching help mitigate concerns with “verifiably random”?

Daniel J. Bernstein (and others) have expressed concern over how "verifiably random" curve parameters are generated. He points out that hashing a public seed doesn't prevent, say, the US government (...
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I have this curve in E(F131) : $$Y^2 = X^3 + X + 2$$ I want calculate the sum P + Q considering that $$P= (5,1)$$ and $$Q = (60, 49)$$ For calculate the result i use these formulas: $$xr = ((... 1answer 143 views ### Ring signatures in ECC Are ring signatures possible with elliptic curves? If so how. The original paper by Rivest, Shamir, Tauman seems to require an invertible trapdoor function. But I've only seen algorithms for secret ... 1answer 259 views ### Logjam on Elliptic Curves? I think we're all aware of the Logjam attack. From now on we know that re-using primes for DH is a bad idea. But we also say that elliptic curves are safe from the attack (relying on the NFS), ... 2answers 131 views ### Severity of Cooking NIST P Curve Constants Bruce Schneier and Gregory Maxwell have both stated that they believe the constants chosen for NIST's P curves (i.e. P-256r) are cooked. DJB has put together a detailed list of red flags but, outside ... 2answers 401 views ### Elliptic Curve Cryptography Encryption and text representation implementation I'm writing a coursework and right now I've implemented the ECDSA algorithm, but I also need to encrypt and decrypt small text files (.txt) using elliptic curve cryptography. The problem is that I do ... 0answers 106 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 ... 1answer 132 views ### Inversion Free Direct Conversion between Twisted Edwards (X,Y,Z) and Montgomery (X,Z) The Wikipedia page for Montgomery curves shows how to convert points on a twisted Edwards curve to and from points on an equivalent Montgomery curve. However, their description and the original ... 2answers 383 views ### Elliptic curve brute forcing I have elliptic curve of equation y^2 \equiv x^3 -x . And the coordinate of points Q and P. I want to solve Q=[k]P (where k is the unknown) by testing all possible k. Is this the right ... 1answer 98 views ### what is the public information in Elliptic curve cryptosystems [closed] Currently my knowledge about Elliptic curve is quite limited to the textbook and I don't know how a practical Elliptic curve cryptosystem works. I read an example about key exchange using Elliptic ... 1answer 97 views ### Second generator for secp256k1 curve I want to get a group element h on the elliptic curve secp256k1. The important thing is that no one should know the discrete logarithm of h with respect to g. That is, h should be created from ... 1answer 803 views ### ECC vs RSA: how to compare key sizes? I know and I have understood the details of RSA, elliptic curve cryptography, (EC)DH and (EC)DSA. I keep reading everywhere that (if we don't consider non-deterministic computers) "ECC can achieve ... 1answer 147 views ### What does signed fixed window method mean in ECC? I am studying (sliding) window method in Elliptic Curve Cryptography (ECC) but I am confused by the term, signed fixed window method. By the way term is used in a research paper and not in the book ... 1answer 339 views ### Non adjacent form of an integer is unique I have tried to look up the proof for NAF (Non-adjacent form) being unique for every integer, but as far as I have seen, textbooks only mention it as a property of NAF, but no proof is given. Also I ... 0answers 50 views ### Generating interactive, secure multiple ECC key pairs deterministically In Elliptic Curve Cryptography (ECC) assuming user A has a private public key pair of S, P accordingly with generator point G which as we know is:$$P=S*G Assuming that user A wants to ...
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I'm having a hard time understanding why people use constant-time techniques to counter time-attacks, when blinding seems as good and cheaper to implement. Why do people avoid blinding in ECC?
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### Authenticated EC key exchange without a signing/signature scheme?

From my little understanding of EC-based authenticated key exchange protocols, I believe that it is not possible for authenticated key exchange without a signing/signature scheme. Is this correct? ...
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### Elliptic curve cryptography G*G

I understand how ECC works for the point multiplications and stuff. All we normally do is to multiply a scalar number (lets say d as a private key) with the base point generator ...
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### Is there any patent free EC point compression available? If not, should I GPL/LGPL my code?

I'm currently using the BouncyCastle implementation in uncompressed mode, and considering releasing an optimized version of EC code that uses point compression. It appears the only thing limiting me ...
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### Is it ever unsafe to compress an EC point?

I am working with a library that outputs EC points in uncompressed form. To save space, I'm considering modifying said library to use compressed EC points. Assuming that I keep track of the sign bit ...
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### EC based password authenticated key exchange protocol

I would like to know: What is the currently best (efficiency and security wise) known EC based password authenticated key exchange protocol? (i.e. is there any EC based protocol may be similar to DH ...
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### Average/approximate difference in value between valid consecutive $x$ coordinates in ECC?

From my basic understanding not all values of $x$ coordinates can satisfy a given elliptic curve equation, i.e. some $x$ coordinate values are not valid points on the curve because $x^3+ax+b$ is not a ...
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### Is only one shared secret generated by ECDH per key pair?

I'm confused about ECDH. Using their public keys and private keys, two entities can arrive on a shared secret. But from the equations I've seen, it looks like ONLY the numbers present in their key ...
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### Is there any pattern in points on EC?

I read some where in crypto.stackexchange answer (related to EC-SRP protocol) that there is pattern in points on elliptic curves, i.e. if given some points containing mix of correct and wrong points ...
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### How does ECDHE_RSA key exchange mechanism work?

Using Wireshark, I found these data exchanged with google.com over TLS: Client Hello possible cipher suites and possible curve types (eg. secp256r1) sent Server Hello cipher suite selected ...
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### How can ECDSA signatures be shortened (to be used as a product key)?

So I made my own serial key generation software, using ECDSA, for use in my own applications and it works great so far! To keep the serial key short enough I use a 128 bit EC curve. My final signature ...
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### What does variable-base point/scalar multiplication mean in ECC?

I am a bit confuse about the term, variable-base point/scalar multiplication, in Elliptic Curve Cryptography. What I have understood so far. It means that the base or point on EC is variable/unknown. ...
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### 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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### What aspect of elliptic curve encryption paradigms makes them especially susceptible to quantum based attack algorithms?

This was a statement made during a talk at today's DFN-CERT conference but unfortunately it wasn't explained further. Can anyone shed light on why elliptic curves are susceptible to quantum based ...
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### In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
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### What are the computational benefits of primes close to the power of 2?

Recently I was reading some article about the Bernstein's Curve25519. This is a particular Montgomery curve over $\mathbb{F}_q$ where $q = {2^{255}-19}$. What I missed or was unable to understand is ...