It only takes a minute to sign up. I hear that there should be a point at infinity on secpk1. I wounder how to calculate it and what does it even mean. Are there more than one point at infinity? Geometrically thinking on an elliptic curve, adding two points that are symmetric against x-axis, should not be feasible because any vertical line on x-y plane will only intersect the curve at TWO points not THREE.

Let the EC be given in Weierstrass equation. Well, we cannot represent it! But, magically, it can be encoded in implementations. For geometrical meaning please see the below image from Wikipedia Elliptic Curves The group law. Arithmetically can define the addition rules in affine coordinates. These are derived from the line intersection and tangent equations. The formulas are. For Jacobian Coordinate operations see this link.

One may define a conventional representation of the point at infinity in the Cartesian coordinate system by picking any coordinate not on the curse, e. And the addition law is extended as:. As noted in commentthe point at infinity has a more natural representation in projective coordinates. For the secpk1 curve, there is a point of infinity which is N the total points for this curve.

To calculate use scalar multiplication of N and the base generator point for the curve. Another way is to go 2coin. Then subtract by pressing -1 you will be at infinity, press one more time, you are at N Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. What is the point at infinity on secpk1 and how to calculate it? Ask Question.

Asked 1 year, 3 months ago. Active 1 month ago. Viewed times. PouJa PouJa 8 8 bronze badges. Active Oldest Votes. The Group Law on Affine Coordinates Arithmetically can define the addition rules in affine coordinates. By the way what is the random looking output value is that my computer program in python is giving as output? For the programming part, unfortunately, off-topic here.

Sign up. Branch: master. Find file Copy path. No definitions found in this file. Cannot retrieve contributors at this time. Raw Blame History. EventQueue ; import javax. JFrame ; import javax. JTextField ; import javax. JLabel ; import javax. JPanel ; import java. Font ; import java. Insets ; import javax. JButton ; import javax. SwingConstants ; import java. ActionListener ; import java. BigInteger ; import java. Random ; import java.If you need to generate x or ed keys then see the genpkey subcommand. These are text files containing base encoded data. A typical traditional format private key file in PEM format will look something like the following, in a file with a ".

PKCS8 private key files, like the above, are capable of holding many different types of private key - not just EC keys. You can convert between these formats if you like. All of the conversion commands can read either the encrypted or unencrypted forms of the files however you must specify whether you want the output to be encrypted or not.

You can replace the first argument "aescbc" with any other valid openssl cipher name see Manual:enc 1 for a list of valid cipher names. Note that by default in the above traditional format EC Private Key files are not encrypted you have to explicitly state that the file should be encrypted, and what cipher to usewhilst for PKCS8 files the opposite is true. The default is to encrypt - you have to explicitly state that you do not want encryption applied if appropriate using the "-nocrypt" option.

This is a binary format and so is not directly human readable - unlike a PEM file. Often it is more convenient to work with PEM files for this reason. Note that you cannot encrypt a traditional format EC Private Key in DER format and in fact if you attempt to do so the argument is silently ignored! It is possible to create a public key file from a private key file although obviously not the other way around!

OpenSSL contains a large set of pre-defined curves that can be used. The full list of built-in curves can be obtained through the following command:. Keys can be generated from the ecparam command, either through a pre-existing parameters file or directly by selecting the name of the curve.

Information on the parameters that have been used to generate the key are embedded in the key file itself.

### ECDSA Sign/Verify

By default, when creating a parameters file, or generating a key, openssl will only store the name of the curve in the generated parameters or key file, not the full set of explicit parameters associated with that name.

For example:. This will simply confirm the name of the curve in the parameters file by printing out the following:. If you wish to examine the specific details of the parameters associated with a particular named curve then this can be achieved as follows:. The above command shows the details for a built-in named curve from a file, but this can also be done directly using the "-name" argument instead of "-in". The output will look similar to the following:. The meaning of each of these parameters is discussed further on this page. Parameters and key files can be generated to include the full explicit parameters instead of just the name of the curve if desired. This might be important if, for example, not all the target systems know the details of the named curve.

In OpenSSL version 1. Attempting to use a parameters file or key file in versions of OpenSSL less than 1. This problem can be avoided if explicit parameters are used instead.

So under OpenSSL 1. The full parameters are included rather than just the name. This can now be processed by versions of OpenSSL less than 1. So under 1. This will correctly display the parameters, even though this version of OpenSSL does not know about this curve. This key file can now be processed by versions of openssl that do not know about the brainpool curve.

It should be noted however that once the parameters have been converted from the curve name format into explicit parameters it is not possible to change them back again, i.

Jump to: navigationsearch. Personal tools Not logged in Talk Contributions Log in. Navigation Main page Recent changes Random page Help.ECDSA relies on the math of the cyclic groups of elliptic curves over finite fields and on the difficulty of the ECDLP problem elliptic-curve discrete logarithm problem.

Elliptic curves, used in cryptography, define:. Generator point Gused for scalar multiplication on the curve multiply integer by EC point. Order n of the subgroup of EC points, generated by Gwhich defines the length of the private keys e.

For example, the bit elliptic curve secpk1 has:. The private key is generated as a random integer in the range [ Generate securely a random number k in the range [ The proof s is by idea verifiable using the corresponding pubKey.

ECDSA signatures are 2 times longer than the signer's private key for the curve used during the signing process. For example, for bit elliptic curves like secpk1 the ECDSA signature is bits 64 bytes and for bit curves like secpr1 the signature is bits. The output is boolean value: valid or invalid signature.

The general idea of the signature verification is to recover the point R' using the public key and check whether it is same point Rgenerated randomly during the signing process. The signing signing encodes a random point R represented by its x-coordinate only through elliptic-curve transformations using the private key privKey and the message hash h into a number swhich is the proof that the message signer knows the private key privKey.

The signature verification decodes the proof number s from the signature back to its original point Rusing the public key pubKey and the message hash h and compares the x-coordinate of the recovered R with the r value from the signature. Read this section only if you like math.

Most developer may skip it. It is not obvious, but let's play a bit with the equations. Now, replace s1 in the point R'. The final step is to compare the point R' decoded by the pubKey with the point R encoded by the privKey. The algorithm in fact compares only the x-coordinates of R' and R : the integers r' and r.

## Point at Infinity

It is important to know that the ECDSA signature scheme allows the public key to be recovered from the signed message together with the signature. The recovery process is based on some mathematical computations described in the SECG: SEC 1 standard and returns 0, 1 or 2 possible EC points that are valid public keyscorresponding to the signature.

The public key recovery from the ECDSA signature is very useful in bandwidth constrained or storage constrained environments such as blockchain systemswhen transmission or storage of the public keys cannot be afforded. Practical Cryptography for Developers. Cryptography - Overview. Hash Functions. MAC and Key Derivation. Secure Random Generators.See on GitHub. In the previous post I gave you an overview of public-key cryptography and its relation with the blockchain.

In the hope of making the topic as clear as possible, the article might seem a little verbose. By the way, your patience will be rewarded as the course will be downhill from here.

In this overwhelming context, our only input is the private key. The public key is uniquely derived from the private key, be it uncompressed or compressed. OS X For your information, Bitcoin Core developers are slowly moving away from OpenSSL towards their own implementation of secpk1 crypto.

A private key is a byte number chosen at random, and you know that 32 bytes make for a very big number, as big as. The output file ec-priv. The file can be quickly decoded to text so that you can see the raw hexes:.

The key reflects our identity, so we want to keep it secret and safe. Watch out! By default, a public key is made of two byte numbers, the so-called uncompressed form. The numbers represent the coordinates of a point on the secpk1 elliptic curve, which has the following formula:. After all, this is what makes EC cryptography secure.

## Questions tagged [secp256k1]

Due to its dependent nature, can be implied from and the curve formula. In fact, the compressed form saves space by omitting the value. The first byte becomes 02 for even values of and 03 for odd values. My ends with 8cso the new prefix is 02 :. The keypair generation task is cumbersome, yet not difficult with the aid of the OpenSSL library.

I wrote a helper function in ec. The first two struct s belong to the arbitrary-precision arithmetic area of OpenSSL, because we need to deal with very big numbers. All the other types relate to EC crypto. Loading the private key is easy, but requires an intermediate step. The public key derivation needs a deeper understanding of EC math, which is not the aim of this series.

Basically, we locate a fixed point on the curve the generatorgroup in the code and multiply it by the scalar private keya virtually irreversible operation in modular arithmetic. The resulting is a second point, the public key pub.

Eventually, the public key is loaded into the keypair:. The third step is the most complicated. The conversion form is set first, which in turn affects the length of the public key 33 or Try running the program and compare it with the output of the openssl command line tool.

Home Questions Tags Users Unanswered. Questions tagged [secpk1].

Ephemeral Diffie-Hellman with RSA (DHE-RSA)

Ask Question. Learn more… Top users Synonyms. Filter by. Sorted by. Tagged with. Apply filter. How to calculate v for signature? I have to create raw transaction and sign on it manually too.

How to calculate v in signature? According to this How to generate public key from private key in Rust?

### The Math Behind Bitcoin

How to generate Ethereum public key from private key in Rust? I found rust-secpk1 and it seems to be what I need, but there is no documentation at all, which makes this crate for me, Rust newbie, Roman Frolov 2, 2 2 gold badges 6 6 silver badges 24 24 bronze badges. I would like to change it to secpr1 curve for my private network. In normal circumstances, all I need to change is the Consy 3 3 silver badges 13 13 bronze badges.

Can Ethereum Contracts Store Bitcoin? I know that Ethereum and Bitcoin both you the same underlying elliptic curve, so I'm wondering, is it possible to store bitcoin in an Ethereum contract? Loourr 1 1 bronze badge. Secpk1 bindings are not compiled. Pure JS implementation will be used I am trying to create a meteor dapp that can sign a piece of data and then recover the address it was signed with.ECC implements all major capabilities of the asymmetric cryptosystems: encryptionsignatures and key exchange.

The ECC cryptography is considered a natural modern successor of the RSA cryptosystem, because ECC uses smaller keys and signatures than RSA for the same level of security and provides very fast key generationfast key agreement and fast signatures. The private keys in the ECC are integers in the range of the curve's field size, typically bit integers. Example of bit ECC private key hex encoded, 32 bytes, 64 hex digits is: 0xb64e85c3fbbaeaaa9dae8ea6a8b The key generation in the ECC cryptography is as simple as securely generating a random integer in certain range, so it is extremely fast.

Any number within the range is valid ECC private key. Thus the compressed public keycorresponding to a bit ECC private key, is a bit integer. Example of ECC public key corresponding to the above private key, encoded in the Ethereum format, as hex with prefix 02 or 03 is: 0x02f54ba86dc1ccb5bedd23f01ed87e4ac47fcda13d41de1a.

In this format the public key actually takes 33 bytes 66 hex digitswhich can be optimized to exactly bits. ECC crypto algorithms can use different underlying elliptic curves. Different curves provide different level of security cryptographic strengthdifferent performance speed and different key lengthand also may involve different algorithms.

ECC curvesadopted in the popular cryptographic libraries and security standards, have name named curves, e. ECC keys have lengthwhich directly depends on the underlying curve. In most applications like OpenSSL, OpenSSH and Bitcoin the default key length for the ECC private keys is bitsbut depending on the curve many different ECC key sizes are possible: bit curve secpr1bit curve sectk1bit curve secpk1bit curves secpk1 and Curvebit curve sectk1bit curves p and secpr1bit curve sectr1bit curve Curvebit curve CurveGoldilocksbit curve Mbit curve Pbit curve sectk1 and many others.

Elliptic-curve cryptography ECC provides several groups of algorithms, based on the math of the elliptic curves over finite fields:. All these algorithms use a curve behind like secpk1curve or p for the calculations and rely of the difficulty of the ECDLP elliptic curve discrete logarithm problem. Let's get into details about the elliptic curves over finite fields. Cryptography uses elliptic curves in a simplified form Weierstras formwhich is defined as:.

This is a visualization of the above elliptic curve:. This means that the field is a square matrix of size p x p and the points on the curve are limited to integer coordinates within the field only.

All algebraic operations within the field like point addition and multiplication result in another point within the field.