MinWise Independent Linear Permutations A set of permutations cal F subseteq S_n is  minwise independent  if for any set X subseteq n and any x in X, when  pi is chosen at random in cal F we have bf P left(minpi(X)  pi(x)right)  1over X. This notion was introduced by Broder, Charikar, Frieze and Mitzenmacher and is motivated by an algorithm for filtering nearduplicate web documents.  Linear permutations  are an important class of permutations. Let p be a (large) prime and let cal F_pp_a,b:;1leq aleq p1,,0leq bleq p1 where for xin p0,1,ldots,p1, p_a,b(x)axbpmod p. For Xsubseteq p we let F(X)max_xin Xleftbf P_a,b(minp(X)p(x))right where bf P_a,b is over p chosen uniformly at random from cal F_p. We show that as k,p toinfty, bf E_XF(X)1over kOleft((log k)3over k32right) confirming that a simply chosen random linear permutation will suffice for an average set from the point of view of approximate minwise independence.

A link between two classes of permutation polynomials Abstract   The study of permutation polynomials over finite fields has attracted many scholarsu0027 attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations.  In this paper, we find an equivalent relation between permutation polynomials of the form    f  (  x  )    x  h   (    x      2    n    xe2x88x92  1    )     over      F      2    2  n        and permutations of the form    F  (  x  )        H    1    (  xcex94  )      H    1    (  xcex94  )      H    2    (  xcex94  )      x    on      F      2    n       , where    xcex94      1      x    2      x    xcexb4     ,    xcexb4  xe2x88x88    F      2    n        with      Tr      2    n      (  xcexb4  )    1    and      H    1    ,    H    2      are the xcoordinate and 1coordinate of h (see  Definition 2.4 ), respectively. A special case with      H    1        H    2      1   , which means    F  (  x  )      H    1     (    1      x    2      x    xcexb4    )     x   , has been already studied  8 ,  16 ,  21 ,  22 . We summarize the known permutations and then construct 8 new classes of permutations of such form, from which one can derive some permutation polynomials of the form    x  h   (    x      2    n    xe2x88x92  1    )     over      F      2    2  n        directly. Finally, we give an asymptotic result about      (    1      x    2      x    xcexb4    )      2    k        x    to be a permutation over      F      2    n        using the HasseWeil bound.