# 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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### Mapping of message onto elliptic curve and reverse it

I would like to perform a variant of Elliptic Curve ElGamal in java using the BouncyCastle libraries. I currently face the difficulty of mapping a message $m$ onto the elliptic curve $E_p$. I have so ...
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### Trying to better understand the failure of the Index Calculus for ECDLP

So I'm going to give you guys my understanding and then if you would be so kind as to tell me where I'm off the mark (hopefully I'm not completely wrong). So basically the index calculus for the ...
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### What crypto system allows for 3 parties: Party 1 who makes an assertion, Party 2 mutates the assertion, Party 3 validates it

I'm looking for the cryptographic equivalent of a Drivers license where the issuer can be verified, the issuer doesn't need to know who you showed the drivers license to, but also allows per-...
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### Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack

The curve $E(\mathbb{F}_{47}):y^2=x^3+x+38$ has order $61$ and $61|47^3-1$ so the embedding degree of $E$ is $3$ and therefore the MOV attack, presumably using some sort of distortion map and a ...
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### ECC algorithm pollard's $\rho$ complexity

One of the methods to break a ECDLP is Pollard's rho algorithm. When ECDLP is defined over a finite field $F_p$, and given a relation $S=w.T$, where S and T are a member of $F_p$. Then ECDLP is to ...
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### 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? ...
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### 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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### 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 ...
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### Are EC public/private keys significantly weakened by having a known byte?

I would like to use different EC keys for different purposes in an app, and I would like to easily see the purpose for which particular key (pair) was generated. With ~65K attempts, I can generate a ...
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### Safe curves in Weierstrass form?

I would like to implement a protocol using elliptic curves. I'm thinking of using MIRACL so using curves in their Weierstrass form is preferable as it they are supported by this framework. I don't ...
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### ECDH and ECDSA in PGP with known public key

How are used ECDH and ECDSA in combination with public key ? Usually these methods are used for establish a secret themselves, ...
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### Calculating $\mathbb F_{p^2}$-rational points of an elliptic curve defined over $\mathbb F_p$

How can I calculate points on an elliptic curve defined over $\mathbb F_p$, for example $y^2 \equiv x^3 + 1 \pmod p$, with coordinates in $\mathbb F_{p^2}$? (points might have complex number format in ...
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### 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 ...
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### Implementing CD serial key system

I am trying to create a system where to unlock the application one needs to enter a serial code. I have read many articles on the theme but there are two problems bugging me. One is, If I have a ...
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### Are there any elliptic curve asymmetric encryption algorithms?

RSA offers the functionality of encrypting (short messages, or symmetric keys) with a public key, and decrypting with a private key. However, RSA key generation is extremely expensive, especially for ...
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### Blockwise Montgomery multiplication

I have to implement a 256*256 bit Montgomery multiplier for pairing computations. The straightforward approach is to use a bit-serial version, but I would like to utilize the built-in 64*64 bits ...
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### Proof of elliptic curve difficulty

Are there any proofs that cryptographic functions on an elliptic curve are any more difficult than the analogues over modulo arithmetic? While at present, ECC appears to be more difficult, as it is ...
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### How Were secp*k1 elliptic curve generators chosen?

The Koblitz elliptic curves specified in the SEC2 document https://docs.google.com/viewer?url=http%3A%2F%2Fwww.secg.org%2Fcollateral%2Fsec2_final.pdf all have the nice feature that the parameters are ...
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### Why a key pair is used in OpenSSL ECDSA_sign or ECDSA_do_sign instead of only private key ?

If the digital signature in ECC (ECDSA) is made using a private key, why is the OpenSSL library using ECKEY pair as an input parameter instead of the private key only ?
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### OpenPGP Public-Key algorithm

What means when pgpdump shows this kind of algorithm in the Public-Key Encrypted Session Key Packet? ...
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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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### What is the difference between “secp…” and “sect…”?

The National Institute of Standards and Technology (NIST) recommended elliptic curve domain parameters to have names such as “secp…” and “sect…”. For example: “secp224k1” and “sect571k1”. What is ...
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### Elliptic Curve Crypto, is a distributed signing method possible using Shamir's Secret Sharing?

Note: A distributed signature scheme exists for RSA: Practical Threshold Signatures, Victor Shoup. Is it possible to adapt such scheme for ECC? A centralized signing machine is vulnerable to ...
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### How can I tell which curve a given ECDSA implementation uses? (P-521 or something else)

I'd like to test and see if certain software uses P-521 ECC curves, or if it uses another variant. Without having access to the sourcecode, or the specification, is there any way for me to test which ...
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### Using single EC cert/keying material to derive symmetric encryption key (for storage)?

The situation involves a single party (single certificate) who would want to AES encrypt a file that they can later decrypt. Assume the EC certificate + EC keys have a purpose i.e. "File encryption" (...
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### How to calculate y value from ((y*y) mod prime) efficiently

i am working ECC-224 bit. can any one tell me, how to calculate y value from ((y*y) mod prime) efficiently for large bit numbers.
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### Optimal same-base exponentiation?

I've (finally) implemented the answer to this question in our library, which stated how to transform montgomery curves (and points) to weierstrass curves (and points). Now, for scalar multiplication, ...
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### 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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### Which multiplicatively homomorphic encryption scheme supports encryption of 0?

I want a multiplicatively homomorphic encryption scheme that supports encryption of 0 (e.g. Elgamal doesn't support). I also want the multiplication to be operated on the ciphertext of 0, i.e., if ...
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### Are RSA or ECC vulnerable to an attack where the same (unknown) plaintext is encrypted with multiple public keys?

I'm not sure what this attack model is called - it's not known-plaintext and also not quite cipher-text-only. It is similar to this question except the general case (not just two keys) and using keys ...
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### 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 ...
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### Is there a flaw in this ECC blind signature scheme?

Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ...
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### Using the same private key for two ECC key pairs

Let $(d_1,Q_1)$ and $(d_2,Q_2)$ be ECC key pairs over two different elliptic curves (say NIST P-224 and NIST P-256). According to the Elliptic Curve Discrete Logarithm Problem (ECDLP), if the private ...
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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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### 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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### 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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### 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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### Finite fields and ECC

I understand modular arithmetic(or at least I think I do!) and I've tried to read and learn about how the Math in RSA works(and I think it went pretty well). I've been reading up on ECC and it looks ...
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### 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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### What curve and key length to use in ECDSA with BouncyCastle

I'm developing a client/server system in Java which is not interacting with third party software, so I don't have to worry about compatibility. At a certain point, I need the client and server to ...
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### Elliptic Curve Cryptography Encryption Results

I'm playing around with a package that does Elliptic Curve Cryptography from http://jecc.sourceforge.net/ Every time I encrypt a value it produced a different result (same private key). However I'm ...
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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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### Difference between Pseudo Mersenne primes and Generalized Mersenne primes

The field prime numbers $p$ proposed by the NIST standards are referred to as Generalized Mersenne prime numbers [1] and as Pseudo Mersenne prime numbers [2]. Is there a difference between Pseudo ...
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### 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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### 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 ...
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### counting points on elliptic curve

Given an elliptic curve with equation $y^2=x^3+ax+b$, and i want to find the number of points $(a,b)\in E(\mathbb{F}_p)$ where the polynomial has repeated roots, how do i do it? I have an intuition it ...
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### Counting points on elliptic curve over binary field

How to count number of rational points on elliptic curve over binary field?
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### 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 ...
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. ...