  
  [1X6 [33X[0;0YHash Functions[133X[101X
  
  
  [1X6.1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YA  hash function in [5Xdatastructures[105X is a function [22XH[122X which maps a value [22XX[122X to a
  small   integer   (where  a  small  integer  is  an  integer  in  the  range
  [10X[-2^28..2^28-1][110X on a 32-bit system, and [10X[-2^60..2^60-1][110X on a 64-bit system),
  under the requirement that if [22XX=Y[122X, then [22XH(X)=H(Y)[122X.[133X
  
  [33X[0;0YA  variety  of  hash functions is provided by [5Xdatastructures[105X, with different
  behaviours.  A  bad  choice of hash function can lead to serious performance
  problems.[133X
  
  [33X[0;0Y[5Xdatastructures[105X  does not guarantee consistency of hash values across release
  or [5XGAP[105X sessions.[133X
  
  
  [1X6.2 [33X[0;0YHash Functions for Basic Types[133X[101X
  
  [1X6.2-1 HashBasic[101X
  
  [33X[1;0Y[29X[2XHashBasic[102X( [3Xobj...[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya small integer[133X
  
  [33X[0;0YHashes any values built inductively from[133X
  
  [30X    [33X[0;6Ybuilt-in    types,    namely    integers,    booleans,   permutations,
        transformations, partial permutations, and[133X
  
  [30X    [33X[0;6Yconstructors for lists and records.[133X
  
  [33X[0;0YThis  function is variadic, treating more than one argument as equivalent to
  a    list   containing   the   arguments,   that   is   [10XHashBasic(x,y,z)   =
  HashBasic([x,y,z])[110X.[133X
  
  
  [1X6.3 [33X[0;0YHash Functions for Permutation Groups[133X[101X
  
  [33X[0;0Y[5Xdatastructures[105X   provides   two   hash  functions  for  permutation  groups;
  [2XHash_PermGroup_Fast[102X  ([14X6.3-1[114X)  is  the  faster one, with higher likelihood of
  collisions  and  [2XHash_PermGroup_Complete[102X  ([14X6.3-2[114X)  is  slower but provides a
  lower likelihood of collisions.[133X
  
  [1X6.3-1 Hash_PermGroup_Fast[101X
  
  [33X[1;0Y[29X[2XHash_PermGroup_Fast[102X( [3Xgroup[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya small integer[133X
  
  [33X[0;0Y[2XHash_PermGroup_Fast[102X is faster than [2XHash_PermGroup_Complete[102X ([14X6.3-2[114X), but will
  return  the  same  value for groups with the same size, orbits and degree of
  transitivity.[133X
  
  [1X6.3-2 Hash_PermGroup_Complete[101X
  
  [33X[1;0Y[29X[2XHash_PermGroup_Complete[102X( [3Xgroup[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya small integer[133X
  
  [33X[0;0Y[2XHash_PermGroup_Complete[102X  is  slower than [2XHash_PermGroup_Fast[102X ([14X6.3-1[114X), but is
  extremely unlikely to return the same hash for two different groups.[133X
  
