Closest vectors, Successive Minima, and Dual HKZ-Bases of Lattices

Download PDF
3 downloads 0 Views 223KB Size Report
Comment
the closest vector problem (Cvp) and the shortest linearly independent vectors ... result in this direction is due to [13], who show that Cvp can not be approxi-.
                  

                     

            !   " "   "# $    !    " %&'#  (      !     "   "     "  "   )    *  ´½µ         +    "# $    ,  "     "(      &  %"

  -./ .0# 1       !  "             "     *   ´½µ     )   "         +   " # $   "   (        "     "  23 4   (    "   "# 5 " "  "         "     "     "   (         " # 





  

                                                                                !      "           # !       !     !                       $        %      #                              "                                   !                                        &  '() *&                                           &   '+,)         &   '- . +-) #           '+-)         &                    / 



6 )   "    7        ) 8$% '7    +"

   ',)                          &      - 0  1        $$$    #         &          2 '+,)           3   '+4) 21                             5    21         326   7 326          8 9      !    : 0 # !              21                         ;     :  ;       &      21   #                              7  326        326       : 0  & ! #            + 7 

 
        #        # !              #                  '? +@) A                A          0                  >        7      =          #  %    326          &   7                              &        #   7>1            &                 &    '+ 0 +0 +B ++)     #                               <    &   '4) 3       &            &        

 '@)         C           3            &      

     /     $$$    &             0  #                      &     D '+() 3         $$$              7     3   '+4)       C        !        # ! ¾                            5

                                 5   7              C        Æ  7        !      7     #                 Æ  7                #           :                                                     +         5                + E   : <         C          :         3                                  : 3                              

                 :                            Æ  /              -  ;     # E              -  # E           % +   #       &             %  3     326 C              Æ    $   F   5         +      %         # %     326 0    %                   -                   .           #            #           %                      G C                G                        326 7         C       '+4)

#    5     : 0               : -                 #                 5          : .



       

    !                                 H                "                 

 I< &     

 

       

H       '       )          '       )     < +      3  <  #        !    I        



A   '       )       %  '       )   

  
            #       =+       '         ) A            C      <  <               

#   '       )    " #       '       ) #   

 I
                   7   L  M   %                     

   <      +  =    A     +             3           ;                    #     5                 &     A 7   L  M             326                         326 3    7   L  M   2N3         326            "                        $                 ;    =                  

+ :     5         5     #  0  326 '       )                    5    A                 326 '       )   7           '       )                            '       )          5    *&    7   L  M    326                     #       $  4  

*    326                          #             21             2 1         =              5    3                         ;               21    



        

  !   

               5                  :                           A       5 $  0  $  -    

 $ *               '       )&  7   Æ         7&        &


.@>=# -0 # 2  # ; 9% 2 ,  / # />/ .# -=0 D# 1"7  D##   9$   ",    (   ; "   ?     1   ; # -0 D# 6# # "       %  &    (?"( .C.# -.0 D# H#   92 G $  $    2"    23 4    I 0 D# H#   2# # G) 92   6   2( F 

 ;   "