## Bitcoin’s Mathematical Elliptic Curve Digital Signature Algorithm

24 December 2019

Before we discuss what an Elliptic Curve Digital Signature Algorithm does it is important to first understand an essential aspect of cryptocurrency – security. With the crypto existence being mostly online, there is a considerable risk attached to in terms of hacking. For every transaction, a unique address generates. This generated address essentially is the sent location for each and every cryptocurrency. If this were a stand-alone implementation for transactions, it would be easy for individuals to transfer funds from any account. This is the very reason why private keys are in play.

When an address is created for cryptocurrency, a private key is the first thing that is generated. From the private key, a corresponding public key is generated and then by hashing that public key an address is obtained. An address can’t be chosen first, with a private key being derived from that as this would enable individuals to derive one’s private key from their address.

Public Key Cryptography

The essential components of public-key cryptography are Public keys, Private Keys and digital signatures. Irrespective of the mathematical basis used to implement such a cryptographic system, it must satisfy the following for this purpose:

• It must be infeasible to derive the private key corresponding to a given public key.
• It must be possible for users to prove that they know a private key that corresponds to a public key without revealing the key itself.

One way to do this is with the application of elliptic curves. This is the method used by most cryptocurrencies, despite the existence of alternatives such as RSA, which uses prime numbers. This is because a 256-bit elliptic curve private key is just as secure as a 3072 bit RSA (Rivest-Shamir-Adleman) private key.

Elliptic-curve cryptography is a type of public-key cryptography. It is based on the algebraic structure of elliptic curves over finite fields. It is used for key agreement, digital signatures, pseudorandom generators, and other tasks.

The Benefits of an Elliptic Curve Digital Signature Algorithm (ECDSA)

Elliptic Curve Digital Signature Algorithm is a Digital Signature Algorithm that uses keys that are derived from elliptic curve cryptography. Though functionally, it provides the same outcome as other digital signing algorithms, Elliptic Curve Digital Signature Algorithm is based on elliptic curve cryptography, which is more efficient. As a result, Elliptic Curve Digital Signature Algorithm requires smaller keys to provide equivalent security and is, therefore, more efficient.

Finite Fields

In the context of Elliptic Curve Digital Signature Algorithm, finite fields are a predefined range of numbers (positive numbers) within which every calculation must fall. Anything beyond this range is rounded off to fit the range.

Bitcoin’s protocol adopts an Elliptic Curve Digital Signature Algorithm and in the process selects a set of numbers for the elliptic curve and its finite field representation. These which are fixed for all users of the protocol. The parameters include the equation used, the field’s prime modulo, and a base point that falls on the curve. The order of the base point is not independently selected but is a function of the other parameters. Another way of looking at the ESDSA is graphically. Consider the multiple times the point can be added to itself until its slope becomes a vertical line (or infinite). The base point is selected in a manner such that the order is a large prime number.

Bitcoin uses huge numbers for its prime modulo, base point, and order. In fact, all practical applications of Elliptic Curve Digital Signature Algorithm use enormous values. The security of the algorithm is dependent on these values being large, and therefore impractical to brute force or reverse engineer.

Part of what makes a private key random is the generation of an unpredictably chosen number. The number can be from 1 through to the order. From this the public key is then derived from the private key by scalar multiplication of the base point many times equal to the value of the private key.

Signing The Data Securely

Once we have both private and public keys, we can sign data. For this, data can be of any length. Usually, data is first hashed to generate a number carrying the same number of bits as the order of the curve.

Now that we have some data and a signature for that data, a third party who has one’s public key can receive our data and signature, and verify that we are the senders.

In essence, Elliptic curve cryptography is a powerful technology that can enable faster and more secure cryptography across the Internet. Its application in cryptocurrency is brilliant and has made it far more secure. With cryptocurrency being a domain where the risk of being hacked and stolen from is prominent, security measures such as an Elliptic Curve Digital Signature Algorithm is vital. With several layers of security, be it as a public key, private key or digital signatures, the Elliptic Curve Digital Signature Algorithm makes cryptocurrency a whole lot safer. The outcome from the algorithm ensures that the only person spending your cryptocurrency is you.

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