For example, af2f0fb8fbb0d2ed1c1cd2a1ec0fb85daa is the hash of hello world , and 30ede9ea08ff1adb8aa6be05fdf84aeacabb5 is the hash of hello worle. This behaviour makes it very difficult to predict what input gives a particular output. For example, what input gives aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa as a hash? It's effectively impossible to work it out. People will often build lookup tables that map these inputs to outputs in order to perform quick reversals later.
These tables are called rainbow tables and rely on the input already having been hashed. The second step is to get the idea of a proof of work. It might be impossible to find a hash specifically with a string consisting of nothing but the letter "a" but what if we asked for a hash with a single zero at the front?
Altering the last letter of hello world took 26 attempts to finally get hello worlC which equates to 0d7eae0fab3abc2cccc0bb4aabb24ffaf8c. Why is this useful? Because it creates a puzzle whose difficulty is measurable and which it's impossible to perform better than blind guessing. That second property is important because it's the only way to create a fair "mining" system. Miners solve such puzzles as above but which are far more difficult.
For example, find a hash that looks like this: xxxx Each hash is can be considered to be just a number. For example, the hash ab3abc2cccc0bb4aabb24ffaf8c has a numeric value of So in mining, the miners have to achieve a hash with a numeric value lower than a specified number. This number is called the target. If your hash attempt gives you a number less than the target, which is the same thing as having a bunch of zeros at the front of the hash, then you win and you get to "mine the block".
To find such a small hash takes millions of attempts, or more accurately, the whole mining network, with everyone trying at the same time, needs millions of billions of tries to get it right. The part of the content that they are hashing and are allowed to change, a single number, in order to try and get a hash beginning with zeros, is called the nonce.
The current block reward of 25 Bitcoins is given to the miner who successfully "mines the block" finds the appropriate hash. It's not really that mining "generates" the Bitcoin in any sense, it's just that it's written into Bitcoin code that a transaction block starts with a unique transaction called a "coinbase" transaction, which is the only type of transaction with no inputs. It only has an output, consisting of the reward plus the transaction fees. To make any sense of Bitcoin's solution to this problem, you need to understand also what is meant by "distributed timestamp server" and how proof of work hashes can be used to construct this.
It is, very briefly, explained in Sections 3 and 4 of the bitcoin whitepaper. What grants this ability? What does that mean and how does that secure bitcoin? With bitcoin, the data that is signed is the transaction that transfers ownership.
ECDSA has separate procedures for signing and verification. Each procedure is an algorithm composed of a few arithmetic operations. The signing algorithm makes use of the private key, and the verification process makes use of the public key. We will show an example of this later. Elliptic curves have useful properties. For example, a non-vertical line intersecting two non-tangent points on the curve will always intersect a third point on the curve.
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 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. 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. The parameters we will use are:. The calculation looks like this:. Here we have to pause for a bit of sleight-of-hand: how do we perform division in the context of a finite field, where the result must always be an integer?
We have to multiply by the inverse, which space does not permit us to define here we refer you to here and here if interested. In the case at hand, you will have to trust us for the moment that:. As with the private key, the public key is normally represented by a hexadecimal string.
But wait, how do we get from a point on a plane, described by two numbers, to a single number? From this partial information we can recover both coordinates. The data can be of any length. The usual first step is to hash the data to generate a number containing the same number of bits as the order of the curve.
The recipe for signing is as follows:. As a reminder, in step 4, if the numbers result in a fraction which in real life they almost always will , the numerator should be multiplied by the inverse of the denominator. In step 1, it is important that k not be repeated in different signatures and that it not be guessable by a third party.
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