What is an example of a bitcoin math problem

Here is an extremely simplified sketch of the problem, but it should give a pretty good idea of what the problem is. The data: This is the hash of the lastest block.
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As mentioned, blockchain is the underlying technology of bitcoin. Blockchain is a public distributed ledger in which transactions are recorded in chronological order. Any record or transaction added to the blockchain cannot be modified or altered, meaning transactions are safe from hacking. A block is the smallest unit of a blockchain, and it is a container that holds all the transaction details.

A block has four fields, or primary attributes:. SHA is a cryptographic hash algorithm that produces a unique bit alphanumeric hash value for any given input, and that is the unique feature of this cryptographic algorithm: Whatever input you give, it will always produce a bit hash. Bitcoin mining is the process of verifying bitcoin transactions and recording them in the public blockchain ledger. In blockchain, the transactions are verified by bitcoin users, so basically the transactions have to be verified by the participants of the network. Those who have the required hardware and computing power are called miners.

We will talk more about them later, but the important concept to understand here is that there is nothing like a centralized body—a regulatory body, a governing body, a bank—to make bitcoin transactions go through. Any user with mining hardware and Internet access can be a participant and contribute to the mining community. The process is solved based on a difficult mathematical puzzle called proof of work.

The proof of work is needed to validate the transaction and for the miner to earn a reward. All the miners are completing amongst themselves to mine a particular transaction; the miner who first solves the puzzle gets the reward. Miners are the network participants who have the necessary hardware and computing power to validate the transactions.

To understand bitcoin mining, you have to first understand the three major concepts of blockchain. In the bitcoin network, as mentioned, users called miners are trying to solve a mathematical puzzle. The puzzle is solved by varying a nonce that produces a hash value lower than a predefined condition, which is called a target.

As of today, Bitcoin miners who solve a puzzle get a reward of Once a block is added to the blockchain, the bitcoins associated with the transactions can be spent and the transfer from one account to the other can be made. To generate the hash, Bitcoin miners use the SHA hashing algorithm and define the hash value. If it is less than the defined condition the target , then the puzzle is deemed to be solved.

If not, then they keep modifying the nonce value and repeat the SHA hashing function to generate the hash value again, and they keep doing this process until they get the hash value that is less than the target. To do that, what would the steps be? First, transaction data is shared with bitcoin users from the memory pool.

The transaction sits in an unmined pool of memory transactions. In a memory pool, unconfirmed transactions wait until they are verified and included in a new block. Bitcoin miners compete to validate the transaction using proof of work. The miner who solves the puzzle first shares the result across the other nodes. Once the block has been verified, the nonce has been generated, then the nodes will start granting their approval.

If maximum nodes grant their approval, the block becomes valid and is added to the blockchain. The miner who has solved the puzzle will also receive a reward of The 10 bitcoins for which the transaction was initiated now will be transferred from Beyonce to Jennifer. In proof of work, a predefined condition the target is adjusted for every 2, blocks, which is approximately every 14 days.

The average time to mine a block is 10 minutes, and to keep the time frame for block generation within 10 minutes, the target keeps adjusting itself. The difficulty of the puzzle changes depending on the time it takes to mine a block. This is how the difficulty of a block is generated: It is the hash target of the first block divided by the hash target of the current block.

mining pools - What are bitcoin miners really solving? - Bitcoin Stack Exchange

This is the difficulty being changed after every 2, blocks, so basically it is very hard to generate the proof of work—but it is very easy for the miners to verify once someone have solved the puzzle. And once the majority of the miners reach a consensus, the block gets validated and added to the blockchain. What if someone tries to hack the data? Each block has solved a puzzle and generated a hash value of its own, which is its identifier.

Now suppose a person tries to tamper with block B and change the data.

Introduction

The data is aggregated in the block, so if the data of the block changes, then the hash value that is the digital signature of the block will also change. It will therefore corrupt the chain after it—the blocks ahead of block B will all get delinked, because the previous hash value of block C will not remain valid. For a hacker to make the entire blockchain valid for the block B that has been changed, he or she would have to change the hash value of all the blocks ahead of block B. This would require a huge amount of computing power and is next to impossible.

With this method, blockchain is non-hackable and prevents data modification. In the early days of bitcoin, miners used to solve the mathematical puzzles using regular processors—controlling processor units CPUs.

Bitcoin Mining

It used to take a lot of time for mining Bitcoins and other cryptocurrencies, even though the difficulty levels were easier than today. As mentioned above, the difficulty level keeps changing and growing, so the miners also had to increase their processing power. They discovered that graphical processing units GPUs proved to be more efficient than regular CPUs, but this also had the drawback of consuming more electricity. A miner has to calculate the return on investment based on the hardware and the cost of electricity and other resources needed to do the mining.

Today miners use hardware called ASIC application-specific integrated circuit , which was specifically introduced for mining Bitcoin and other cryptocurrencies. A further property is that a non-vertical line tangent to the curve at one point will intersect precisely one other point on the curve. For example:. The process of scalar multiplication is normally simplified by using a combination of point addition and point doubling operations.

Here, 7P has been broken down into two point doubling steps and two point addition steps. A finite field, in the context of ECDSA, can be thought of as a predefined range of positive numbers within which every calculation must fall. The simplest way to think about this is calculating remainders, as represented by the modulus mod operator. Here our finite field is modulo 7, and all mod operations over this field yield a result falling within a range from 0 to 6. ECDSA uses elliptic curves in the context of a finite field, which greatly changes their appearance but not their underlying equations or special properties.

The same equation plotted above, in a finite field of modulo 67, looks like this:.

Point addition and doubling are now slightly different visually. Lines drawn on this graph will wrap around the horizontal and vertical directions, just like in a game of Asteroids, maintaining the same slope. So adding points 2, 22 and 6, 25 looks like this:. A protocol such as bitcoin selects a set of parameters for the elliptic curve and its finite field representation that is fixed for all users of the protocol. The parameters include the equation used, the prime modulo of the field, and a base point that falls on the curve.

The order of the base point, which is not independently selected but is a function of the other parameters, can be thought of graphically as the number of times the point can be added to itself until its slope is infinite, or a vertical line. The base point is selected such that the order is a large prime number. Bitcoin uses very large numbers for its base point, prime modulo, and order. The security of the algorithm relies on these values being large, and therefore impractical to brute force or reverse engineer.


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Who chose these numbers, and why? A great deal of research , and a fair amount of intrigue , surrounds the selection of appropriate parameters. After all, a large, seemingly random number could hide a backdoor method of reconstructing the private key. In brief, this particular realization goes by the name of secpk1 and is part of a family of elliptic curve solutions over finite fields proposed for use in cryptography.

With these formalities out of the way, we are now in a position to understand private and public keys and how they are related. The public key is derived from the private key by scalar multiplication of the base point a number of times equal to the value of the private key. Expressed as an equation:. This shows that the maximum possible number of private keys and thus bitcoin addresses is equal to the order. In a continuous field we could plot the tangent line and pinpoint the public key on the graph, but there are some equations that accomplish the same thing in the context of finite fields.

In practice, computation of the public key is broken down into a number of point doubling and point addition operations starting from the base point.