Editing Secret sharing (section)

==Trivial secret sharing==
Note: ''n'' is the total number of 'players', among whom the shares are distributed, and ''t'' is the minimum number of players required to reveal the secret.

=== ''t'' = 1 ===
''t'' = 1 secret sharing is trivial. The secret can simply be distributed to all ''n'' participants.

=== ''t'' = ''n'' ===
There are several {{nowrap|(''t'', ''n'')}} secret-sharing schemes for {{nowrap|1=''t'' = ''n''}}, when all shares are necessary to recover the secret:

# Encode the secret as a [[Binary numeral system|binary]] number ''s'' of any length. For each player ''i'', where ''i'' is one fewer than the total number of players, give a random binary number ''p<sub>i</sub>'' of the same length as ''s''. To the player without a share, give the share calculated as {{nowrap|1=''p''<sub>''n''</sub> = ''s'' ⊕ ''p''<sub>1</sub> ⊕ ''p''<sub>2</sub> ⊕ ... ⊕ ''p''<sub>''n''−1</sub>}}, where ⊕ denotes [[bitwise operation#XOR|bitwise exclusive or]]. The secret is the bitwise exclusive-or of all the players' numbers (''p''<sub>''i''</sub>, for 1 ≤ ''i'' ≤ ''n'').
# Instead, (1) can be performed using the binary operation in any [[Group (mathematics)|group]].  For example, take the cyclic group of integers with addition modulo 2<sup>32</sup>, which corresponds to 32-bit integers with addition defined with the binary overflow being discarded.  The secret ''s'' can be partitioned into a vector of ''M'' 32-bit integers, which we call ''v''<sub>secret</sub>.  Then {{nowrap|(''n'' − 1)}} of the players are each given a vector of ''M'' 32-bit integers that is drawn independently from a uniform probability distribution, with player ''i'' receiving ''v<sub>i</sub>''.  The remaining player is given ''v<sub>n</sub>'' = ''v''<sub>secret</sub> − ''v''<sub>1</sub> − ''v''<sub>2</sub> − ... − ''v''<sub>''n''−1</sub>.  The secret vector can then be recovered by summing across all the players' vectors.

=== 1 < ''t'' < ''n'' ===
The difficulty{{clarify|is this a major "difficulty" that's still unresolved? If not, then why to mention it at all?|date=April 2023}} lies in creating schemes that are still secure, but do not require all ''n'' shares.

When space efficiency is not a concern, trivial {{nowrap|1=''t'' = ''n''}} schemes can be used to reveal a secret to any desired subsets of the players simply by applying the scheme for each subset. For example, to reveal a secret ''s'' to any two of the three players Alice, Bob and Carol, create three (<math>\binom{3}{2}</math>) different {{nowrap|1=''t'' = ''n'' = 2}} secret shares for ''s'', giving the three sets of two shares to Alice and Bob, Alice and Carol, and Bob and Carol.

=== ''t'' belonging to any desired subset of {1, 2, ..., ''n''} ===
For example, imagine that the board of directors of a company would like to protect their secret formula. The president of the company should be able to access the formula when needed, but in an emergency any 3 of the 12 board members would be able to unlock the secret formula together. One of the ways this can be accomplished is by a secret-sharing scheme with {{nowrap|1=''t'' = 3}} and {{nowrap|1=''n'' = 15}}, where 3 shares are given to the president, and one share is given to each board member.