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💪 Role of mathematics in digital currency
The situation is different with bitcoin. Since bitcoins are not stored globally or locally, no single entity serves as their custodian. They live as records on the block chain, a distributed ledger whose copies are exchanged by a volunteer network of linked computers. To “own” a bitcoin simply means to be able to transfer ownership of it to another person by recording the transfer in the block chain. What gives you this power? ECDSA private and public key pair access. What does this mean, and how does it affect bitcoin’s security?
Elliptic Curve Digital Signature Algorithm (ECDSA) is an acronym for Elliptic Curve Digital Signature Algorithm. It’s a method of “signing” data with an elliptic curve and a finite field in such a way that third parties can verify the signature’s validity while the signer maintains exclusive control over the signature’s development. The data that is signed in bitcoin is the transaction that transfers ownership.
The ECDSA has different signing and authentication procedures. Each procedure is a collection of arithmetic operations that make up an algorithm. The private key is used in the signing algorithm, and the public key is used in the authentication process. Later on, we’ll demonstrate an example of this.
😱 Bitcoin mathematics pdf
Maybe you think bitcoin is silly, maybe you think it’s risky and socially reckless, maybe you’re a bitcoin millionaire (in which case, hello), or maybe you just want to be a part of the blockchain revolution. Whatever your position on bitcoin and the blockchain, it seems that you can’t avoid them these days. And yet, every time I try to comprehend it, I’m met with some muddled, half-confused version of the facts.
But, in case you’re still puzzled, here’s a quick rundown of bitcoin’s math. A public ledger system known as the blockchain makes transactions in the bitcoin (or cryptocurrency) marketplace extremely safe. The elliptic curve digital signature algorithm is used to accomplish this. The algorithm is described in detail on the blog The Mathematical Investor, but for those with a basic understanding of elliptic curves, I’ll just say it’s related to modular arithmetic on the group structure of points on an elliptic curve. The blockchain’s aim is to serve as a decentralized digital public record of all bitcoin transactions, and it’s thought to be unhackable because it’s constantly modified and there’s no master version. In principle, this implies that stealing bitcoins, spending the same bitcoin twice, or inventing bitcoins out of thin air should be impossible.
📔 Blockchain explained
You may have a better understanding of cryptocurrencies and blockchain technology after reading our previous blog post. You may also be interested in learning the fundamentals of how they operate without being governed by a single central authority or server. More than just computer code underpins blockchain technology. It is, for the most part, highly reliant on mathematics. Jan Witte describes in his book “The Blockchain: A Gentle Four-Page Introduction” that blockchain technology is based on two mathematical concepts: hash tables and public key encryption (hence the “crypto” in “cryptocurrency”).
The “keys” in public key encryption are functions rather than numbers. It works like this: each party has a private key as well as a public key. To send a message to another party, one must encrypt the message using the public key of the other party. Using their private key, the other party may decrypt the message encoded with their public key. Essentially, the “public” and “private” functions are inverses of one another, because if the message is one function’s input and the output of that calculation is another function’s input, the result of that calculation is… The message that started it all. The Rivest–Shamir–Adleman (RSA) encryption is a well-known example of public key encryption, and it uses the modulus operator in its calculations, much like hash tables.
👐 Mathematical theory of blockchain
We discussed the evolving world of blockchain in a previous Math Investor article, stressing how it could affect financial services and investment. Blockchain is now being pursued by a number of companies, including many startups, to enable and streamline a variety of financial transactions.
It’s worth taking a quick look at how blockchain works mathematically. The following is based on an article written by Eric Rykwalder, one of the founders of Chain.com, a San Francisco-based blockchain tech company.
Participants enter data and certify their approval of the transaction using an elliptic curve digital signature algorithm in a publicly accessible ledger known as blockchain (ECDSA). An equation like y2 = x3 + an x + b is an elliptic curve. Since a = 0 and b = 7 in Bitcoin and most other implementations, this is simply y2 = x3 + 7. (see graph). Elliptic curves have a number of intriguing properties, including the fact that a nonvertical line intersecting two nontangent points often intersects a third point on the curve. On the curve, “addition” can be defined as finding the third point that corresponds to two given points. This is essentially what ECDSA does, with the exception that the operations are carried out modulo a large prime number M.