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ch4 fixed equations

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Andreas M. Antonopoulos 2014-03-01 23:40:15 -08:00
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@ -24,7 +24,7 @@ In the most simple form, the +private key+ is a number. The private key be used
===== Generating a private key from a random number
A private key is a number, between +1+ and +n - 1+ where +latexmath:[\(\n ~= 1.158 * 10^77\)]+ is the order of the elliptic curve used in bitcoin (See <<secp256k1>>). To create such a key, we just pick a 256-bit random number, and check that it is less than +n - 1+, or pick another random number and check again, until we get one less than +n - 1+. The constant +n+ is defined in any elliptic curve cryptography library. In programming terms, this is usually achieved by feeding a larger string of random bits, collected from a cryptographically-secure source of randomness, into the SHA-256 hash algorithm which will conveniently produce a 256-bit number.
A private key is a number, between +1+ and +n - 1+ where latexmath:[\(n ~= 1.158 * 10^77\)] is the order of the elliptic curve used in bitcoin (See <<secp256k1>>). To create such a key, we just pick a 256-bit random number, and check that it is less than +n - 1+, or pick another random number and check again, until we get one less than +n - 1+. The constant +n+ is defined in any elliptic curve cryptography library. In programming terms, this is usually achieved by feeding a larger string of random bits, collected from a cryptographically-secure source of randomness, into the SHA-256 hash algorithm which will conveniently produce a 256-bit number.
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@ -102,7 +102,7 @@ KwSSD6LKk8nUQSkS2cDqBZ2AqGdGs2BMer2yMn9byxJydor5GWJX
==== From Private Key to Public Key
The public key is calculated from the private key using elliptic curve multiplication, which is irreversible: +latexmath:[\(\K = k * G\)]+ where +k+ is the private key, +G+ is a constant point called the _Generator Point_ and +K+ is the resulting public key. The reverse (division), or calculating +k+ if you know +K+ is as difficult as trying all possible values of +k+, ie a brute-force search.
The public key is calculated from the private key using elliptic curve multiplication, which is irreversible: latexmath:[\(K = k * G\)]+ where +k+ is the private key, +G+ is a constant point called the _Generator Point_ and +K+ is the resulting public key. The reverse (division), or calculating +k+ if you know +K+ is as difficult as trying all possible values of +k+, ie a brute-force search.
The public key is a point on the elliptic curve, and consists of a pair of coordinates +(x,y)+, normally represented by a 512-bit number with the added prefix +04+.
@ -267,7 +267,7 @@ or
\end{equation}
++++
where +latexmath:[\(\p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1\)]+, a very large prime number.
where +latexmath:[\(p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1\)]+, a very large prime number.
The +mod p+ indicates that this curve is over a finite field of prime order +p+, also written as latexmath:[\(\mathbb{F}_p\)]. The curve looks like a pattern of dots scattered in two dimensions, which makes it difficult to visualize. However, the math is identical as that of an elliptic curve over the real numbers shown above.