User:Vbuterin/K of N redundant offline private key proposal
There is an important tradeoff between security and recoverability when storing an offline wallet. The simplest setup, storing the entire private key in a single place in a lockbox, provides medium security but is vulnerable to accidental loss or destruction through accidents such as fire. Storing a simple backup of the key mitigates this risk, but at the same time decreases security, and the opposite strategy, splitting the key into multiple pieces, all of which must somehow be combined to retrieve the original key, increases security but makes the accidental loss problem even worse. The optimal solution is a setup which combines the two strategies, allowing the key to easily be split up into n pieces, any k of which can be used to recover the original key. The simplest such setup is storing k parts of the key and then an XOR of all k parts, but that approach is limiting as it only allows for n = k+1. What this document proposes is a setup which allows any n and k to be used. The steps are the following:
1. Kpiece key splitting
This proposal provides a scheme where a key can be broken into up to 8 parts, and each part is serialized as a 59character Base58Checkencoded message that starts with the characters "6s".
If generating a new random key, simply generate K random 256bit values. If N and K are both 8, then the 8th value needs to have its most significant bit set to zero to prevent an overflow condition described later.
If splitting a specific key, then simply generate K1 random 256bit values, taking care that the value to be used in position #8 has its most significant bit set to zero, then XOR them all together along with the specific key. The result of the XOR operation should be used in a position other than the one with a constraint on the highest bit.
Nearly all 256bit values are valid Bitcoin private keys  the set of 256bit values constituting invalid private keys all start with 127 or more consecutive 0 or 1 bits and are extremely unlikely to be encountered by accident with an appropriate source of random numbers.
Pseudocode implementation (doesn't apply anymore)
def generate(P,k):
v = array(k)
for i in range(0,k2):
while hex(sha256(v[i]))[0:2] != '00':
v[i] = random(1,2^256)
remainder = P
for i in range(0,k2):
remainder = remainder XOR v[i]
while hex(sha256(v[k2]))[0:2] != '00' or hex(sha256(v[k1]))[0:2] != '00':
v[k2] = random(1,2^256)
v[k1] = remainder XOR v[k2]
So far, what we have are k
pieces that can be trivially recombined (through XOR) into the original private key, and all of which have a basic checksum to provide some protection against mistakes.
2. N piece generation
The next step will transform these k
pieces, once again referred to as v(1)
to v(k)
, into n
pieces, any k
of which can be used to find the original values v(1)
to v(k)
.
Piece i
will have the following format:
 This message contains a 39byte payload.
 Each message representing a single part will be 59 base58 characters starting with '6s'.
 Byte 0:
0x4f
(unique "format" identifier to make keyparts in base 58 start with '6s' to be recognizable visually, when used properly in combination with byte 1)  Byte 1: 0x93+k where k = 1...8. This will result in the third character being a digit 1 thru 8 so the user can see the value of k. (e.g. 6s1, 6s2, 6s3, 6s4, 6s5, 6s6, 6s7, 6s8 prefixes). Highest allowable value: 0x9c
 Bytes 2,3 = Bitmapped:
 bit 15...13 (MSB): The fixed value 011. This is necessary for the uservisible prefix to work as intended.
 bit 12: compression flag (0=none, 1=compressed)
 bits 11...9: which key part this is, minus 1
 bits 8...0: SHA256(resulting bitcoin address)[0...1] & 0x01FF, for the purpose matching parts together and verification they produce the correct address
 Bytes 4 thru 38: The number
v(1) + v(2)*2^i + v(3)*3^i + v(4)*4^i + ...
There are 280 bits allocated to the number (35 bytes). This will fit all of the 256bit numbers generated in the scheme, multiplied by their respective factors. However, there is one exception: if a user generates an 8of8 code, then the factors used to create the eighth code will overflow and require 281 bits whenever v(8) exceeds a certain threshold. In order to prevent this, the value for v(8) needs to be chosen with its eight most significant bits set to zero.
Example
For example, if you want your private key to require three out of five pieces to reconstitute, the final pieces will be (byte lengths will depend on exactly what v(1)
, v(2)
and v(3)
are):

0x86 1 33 v(1) + v(2)*2 + v(3)*3

0x86 2 33 v(1) + v(2)*4 + v(3)*9

0x86 3 33 v(1) + v(2)*8 + v(3)*27

0x86 4 33 v(1) + v(2)*16 + v(3)*81

0x86 5 34 v(1) + v(2)*32 + v(3)*243
When you're done, the pieces can be kept as hex or base58  either format works fine, and stored wherever you want. You might, for example, store one in a safe at home, another at a safety deposit box at the local bank, a third on your computer, a fourth with a friend and memorize the fifth.
Key Reconstitution
To reconstitute the private key, simply solve the linear system from any three pieces, inferring from the second byte of the pieces what all the coefficients are, to get the original values v(1)
to v(k)
. Once you have these values, XOR all of them to get your final result.
For example imagine that you have pieces 1, 3 and 4 from above, letting pc(i)
be the value from piece i
not including the three byte prefix. You thus have:
v(1) + v(2)*2 + v(3)*3 = pc(1)
v(1) + v(2)*8 + v(3)*27 = pc(2)
v(1) + v(2)*16 + v(3)*81 = pc(3)
The simplest method to solve linear systems is elimination, which would work as follows:
v(1) + v(2)*8 + v(3)*27 = pc(2)
 v(1)  v(2)*2  v(3)*3 = pc(1)
.
v(2)*6 + v(3)*24 = pc(2)  pc(1)
v(1) + v(2)*16 + v(3)*81 = pc(3)
 v(1)  v(2)*2  v(3)*3 = pc(1)
.
v(2)*14 + v(3)*78 = pc(3)  pc(1)
v(2)*14 + v(3)*78 = pc(3)  pc(1)
v(2)*6  v(3)*24 = pc(2)  pc(1) <how did you get this? shouldn't it be + v(3)*24 not v(3)*24? Casascius (talk) 06:12, 6 August 2012 (GMT)

v
v(2)*84 + v(3)*468 = pc(3)*6  pc(1)*6
v(2)*84  v(3)*336 = pc(2)*14  pc(1)*14
.
v(3)*132 = pc(3)*6  pc(2)*14 + pc(1)*8
v(3) = (pc(3)*6  pc(2)*14 + pc(1)*8) / 132
v(2) = v(3)*78/14 + pc(3)  pc(1)
v(1) = pc(2)  v(2)*8  v(3)*27
The general pattern is to take k
equations, combine all adjacent pairs to get k1
equations without any coefficient for v(1)
and repeat the process until you're down to a single equation of the form v(k)*a = b
. From there, just work your way back up the chain of intermediate equations to get all the other values.
You actually don't have to know what n
is because you can simply assume that n
is the number of pieces that you have, and if you have too many pieces solving the linear system will simply lead you to discover that the n+1
st, n+2
nd, etc pieces are all equal to zero.
The scheme can easily be adapted to make the private key the EC product of v(1)
, v(2)
, etc rather than an XOR, and even the linear systems can be changed to an multiplicative/exponential equivalent if desired, so it's pretty adaptable.