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What does this paraphrase of the birthday problem mean?


What are the differences between collision attack and birthday attack?k(k-1)/2: Combinations and the Birthday boundIs there any function that does not suffers birthday problem?Why k-lists generalized birthday problem when $k=2$ is classical birthday problem?Time complexity of birthday attack type problemHow does hashing twice protect against birthday attacks?What is a wide block cipher and why does it avoid birthday bound problems?Can the birthday attack be extended in this case?On a lower bound for the birthday problemWhat is the error in this collision probability approximation?













6












$begingroup$


The following is an excerpt from A Generalized Birthday Problem - David Wagner:




One of the best-known combinatorial tools in cryptology is the birthday problem:

Problem 1. Given two lists $L_1, space L_2$ of elements drawn uniformly and independently at random from $0, 1^n$, find $x_1 in L_1$ and $x_2 in L_2$ such that $x_1 oplus x_2 = 0$.




It's not so intuitive for me to understand. In my understanding, the birthday problem is about the probability that at least 2 people in a room have the same birthday. How does the birthday problem transfers to this? Please give me some hints.










share|improve this question











$endgroup$
















    6












    $begingroup$


    The following is an excerpt from A Generalized Birthday Problem - David Wagner:




    One of the best-known combinatorial tools in cryptology is the birthday problem:

    Problem 1. Given two lists $L_1, space L_2$ of elements drawn uniformly and independently at random from $0, 1^n$, find $x_1 in L_1$ and $x_2 in L_2$ such that $x_1 oplus x_2 = 0$.




    It's not so intuitive for me to understand. In my understanding, the birthday problem is about the probability that at least 2 people in a room have the same birthday. How does the birthday problem transfers to this? Please give me some hints.










    share|improve this question











    $endgroup$














      6












      6








      6


      2



      $begingroup$


      The following is an excerpt from A Generalized Birthday Problem - David Wagner:




      One of the best-known combinatorial tools in cryptology is the birthday problem:

      Problem 1. Given two lists $L_1, space L_2$ of elements drawn uniformly and independently at random from $0, 1^n$, find $x_1 in L_1$ and $x_2 in L_2$ such that $x_1 oplus x_2 = 0$.




      It's not so intuitive for me to understand. In my understanding, the birthday problem is about the probability that at least 2 people in a room have the same birthday. How does the birthday problem transfers to this? Please give me some hints.










      share|improve this question











      $endgroup$




      The following is an excerpt from A Generalized Birthday Problem - David Wagner:




      One of the best-known combinatorial tools in cryptology is the birthday problem:

      Problem 1. Given two lists $L_1, space L_2$ of elements drawn uniformly and independently at random from $0, 1^n$, find $x_1 in L_1$ and $x_2 in L_2$ such that $x_1 oplus x_2 = 0$.




      It's not so intuitive for me to understand. In my understanding, the birthday problem is about the probability that at least 2 people in a room have the same birthday. How does the birthday problem transfers to this? Please give me some hints.







      birthday-attack






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited Mar 22 at 13:51









      Squeamish Ossifrage

      22.2k132100




      22.2k132100










      asked Mar 22 at 3:20









      Cedric SunCedric Sun

      314




      314




















          2 Answers
          2






          active

          oldest

          votes


















          5












          $begingroup$

          $x_1 oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).



          Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.



          Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.






          share|improve this answer











          $endgroup$




















            3












            $begingroup$

            Note that $x_1=x_2$, i.e., there is a birthday collision in $0,1^n$ if and only if $x_1oplus x_2=0.$
            In a general additive group $G$, $x_1=x_2$, i.e., there is a birthday collision in $G$ if and only if $x_1-x_2=0.$



            If you have two lists $L_1,L_2,$ then with probability roughly
            $$expleft-frac2^n+1right$$
            there will be no collisions.



            In the birthday paradox, for $N=2^n$ bins, the probability of no collisions after $m$ balls is roughly
            $$expleft-fracm^22Nright$$
            while here we have $|L_1||L_2|$ pairs to consider so $m=|L_1||L_2|.$



            Wagner's paper is about finding efficient algorithms for vectors adding to zero for higher numbers (e.g., 4) of lists.






            share|improve this answer











            $endgroup$













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              2 Answers
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              active

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              active

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              5












              $begingroup$

              $x_1 oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).



              Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.



              Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.






              share|improve this answer











              $endgroup$

















                5












                $begingroup$

                $x_1 oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).



                Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.



                Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.






                share|improve this answer











                $endgroup$















                  5












                  5








                  5





                  $begingroup$

                  $x_1 oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).



                  Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.



                  Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.






                  share|improve this answer











                  $endgroup$



                  $x_1 oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).



                  Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.



                  Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.







                  share|improve this answer














                  share|improve this answer



                  share|improve this answer








                  edited Mar 23 at 11:31

























                  answered Mar 22 at 7:22









                  fgrieufgrieu

                  82k7178350




                  82k7178350





















                      3












                      $begingroup$

                      Note that $x_1=x_2$, i.e., there is a birthday collision in $0,1^n$ if and only if $x_1oplus x_2=0.$
                      In a general additive group $G$, $x_1=x_2$, i.e., there is a birthday collision in $G$ if and only if $x_1-x_2=0.$



                      If you have two lists $L_1,L_2,$ then with probability roughly
                      $$expleft-frac2^n+1right$$
                      there will be no collisions.



                      In the birthday paradox, for $N=2^n$ bins, the probability of no collisions after $m$ balls is roughly
                      $$expleft-fracm^22Nright$$
                      while here we have $|L_1||L_2|$ pairs to consider so $m=|L_1||L_2|.$



                      Wagner's paper is about finding efficient algorithms for vectors adding to zero for higher numbers (e.g., 4) of lists.






                      share|improve this answer











                      $endgroup$

















                        3












                        $begingroup$

                        Note that $x_1=x_2$, i.e., there is a birthday collision in $0,1^n$ if and only if $x_1oplus x_2=0.$
                        In a general additive group $G$, $x_1=x_2$, i.e., there is a birthday collision in $G$ if and only if $x_1-x_2=0.$



                        If you have two lists $L_1,L_2,$ then with probability roughly
                        $$expleft-frac2^n+1right$$
                        there will be no collisions.



                        In the birthday paradox, for $N=2^n$ bins, the probability of no collisions after $m$ balls is roughly
                        $$expleft-fracm^22Nright$$
                        while here we have $|L_1||L_2|$ pairs to consider so $m=|L_1||L_2|.$



                        Wagner's paper is about finding efficient algorithms for vectors adding to zero for higher numbers (e.g., 4) of lists.






                        share|improve this answer











                        $endgroup$















                          3












                          3








                          3





                          $begingroup$

                          Note that $x_1=x_2$, i.e., there is a birthday collision in $0,1^n$ if and only if $x_1oplus x_2=0.$
                          In a general additive group $G$, $x_1=x_2$, i.e., there is a birthday collision in $G$ if and only if $x_1-x_2=0.$



                          If you have two lists $L_1,L_2,$ then with probability roughly
                          $$expleft-frac2^n+1right$$
                          there will be no collisions.



                          In the birthday paradox, for $N=2^n$ bins, the probability of no collisions after $m$ balls is roughly
                          $$expleft-fracm^22Nright$$
                          while here we have $|L_1||L_2|$ pairs to consider so $m=|L_1||L_2|.$



                          Wagner's paper is about finding efficient algorithms for vectors adding to zero for higher numbers (e.g., 4) of lists.






                          share|improve this answer











                          $endgroup$



                          Note that $x_1=x_2$, i.e., there is a birthday collision in $0,1^n$ if and only if $x_1oplus x_2=0.$
                          In a general additive group $G$, $x_1=x_2$, i.e., there is a birthday collision in $G$ if and only if $x_1-x_2=0.$



                          If you have two lists $L_1,L_2,$ then with probability roughly
                          $$expleft-frac2^n+1right$$
                          there will be no collisions.



                          In the birthday paradox, for $N=2^n$ bins, the probability of no collisions after $m$ balls is roughly
                          $$expleft-fracm^22Nright$$
                          while here we have $|L_1||L_2|$ pairs to consider so $m=|L_1||L_2|.$



                          Wagner's paper is about finding efficient algorithms for vectors adding to zero for higher numbers (e.g., 4) of lists.







                          share|improve this answer














                          share|improve this answer



                          share|improve this answer








                          edited Mar 22 at 6:19

























                          answered Mar 22 at 4:14









                          kodlukodlu

                          9,30811331




                          9,30811331



























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