Which properties of a group are used in the steps of Diffie Hellman?












4












$begingroup$


I’m trying to understand which properties of a group are used in DHKE at each step.



For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e.g. $k_{pubA}$ = $A^{k_{prA}}$ (mod p)?



When creating the shared key Alice and Bob appear to also use the associative property of a group $k_{AB}$ = $B^{k_{prA}}$ (mod p)?



So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?






diffie-hellman number-theory group-theory







share|improve this question









$endgroup$

















    4












    $begingroup$


    I’m trying to understand which properties of a group are used in DHKE at each step.



    For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e.g. $k_{pubA}$ = $A^{k_{prA}}$ (mod p)?



    When creating the shared key Alice and Bob appear to also use the associative property of a group $k_{AB}$ = $B^{k_{prA}}$ (mod p)?



    So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?






    diffie-hellman number-theory group-theory







    share|improve this question









    $endgroup$















      4












      4








      4


      1



      $begingroup$


      I’m trying to understand which properties of a group are used in DHKE at each step.



      For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e.g. $k_{pubA}$ = $A^{k_{prA}}$ (mod p)?



      When creating the shared key Alice and Bob appear to also use the associative property of a group $k_{AB}$ = $B^{k_{prA}}$ (mod p)?



      So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?






      diffie-hellman number-theory group-theory







      share|improve this question









      $endgroup$




      I’m trying to understand which properties of a group are used in DHKE at each step.



      For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e.g. $k_{pubA}$ = $A^{k_{prA}}$ (mod p)?



      When creating the shared key Alice and Bob appear to also use the associative property of a group $k_{AB}$ = $B^{k_{prA}}$ (mod p)?



      So to perform both main steps of DHKE the multiplicative inverse property does not seem to be used at all?






      diffie-hellman number-theory group-theory




      diffie-hellman number-theory group-theory






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 1 hour ago









      JohnGaltJohnGalt

      1575




      1575






















          1 Answer
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          4












          $begingroup$


          I’m trying to understand which properties of a group are used in DHKE at each step.




          Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = Atimes A times A = (A times A) times A = A times (A times A)$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).



          We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.



          Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...






          share|improve this answer











          $endgroup$













          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            1 hour ago










          • $begingroup$
            I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
            $endgroup$
            – JohnGalt
            51 mins ago











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          active

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          4












          $begingroup$


          I’m trying to understand which properties of a group are used in DHKE at each step.




          Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = Atimes A times A = (A times A) times A = A times (A times A)$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).



          We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.



          Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...






          share|improve this answer











          $endgroup$













          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            1 hour ago










          • $begingroup$
            I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
            $endgroup$
            – JohnGalt
            51 mins ago
















          4












          $begingroup$


          I’m trying to understand which properties of a group are used in DHKE at each step.




          Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = Atimes A times A = (A times A) times A = A times (A times A)$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).



          We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.



          Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...






          share|improve this answer











          $endgroup$













          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            1 hour ago










          • $begingroup$
            I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
            $endgroup$
            – JohnGalt
            51 mins ago














          4












          4








          4





          $begingroup$


          I’m trying to understand which properties of a group are used in DHKE at each step.




          Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = Atimes A times A = (A times A) times A = A times (A times A)$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).



          We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.



          Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...






          share|improve this answer











          $endgroup$




          I’m trying to understand which properties of a group are used in DHKE at each step.




          Actually, you can implement a DH-style operation in any semigroup; you need closure, and you need associativity (so $A^3 = Atimes A times A = (A times A) times A = A times (A times A)$ is well defined), but other than that, you really don't need anything. You don't need an identity, you don't need the semigroup to be abelian (although the sub-semigroup generated by a single element will always be abelian), it doesn't have to be finite (although infinite semigroups would cause practical problems during implementation) and you don't need inverses (which, if you don't have an identity, aren't well-defined anyways).



          We typically don't talk about doing DH in a semigroup mostly because (AFAIK) no one has found a semigroup (that's not also a group) that has any particular advantage over a true group.



          Now, what's a more interesting (and considerably harder) question is "what properties do you need for DHKE to be secure?" We do have assumptions such as the CDH assumption ("given $g, g^a, g^b$, it's hard to compute $g^{ab}$), however we don't know what semigroup properties ensure that...







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 1 hour ago

























          answered 1 hour ago









          ponchoponcho

          91.1k2142236




          91.1k2142236












          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            1 hour ago










          • $begingroup$
            I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
            $endgroup$
            – JohnGalt
            51 mins ago


















          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            1 hour ago










          • $begingroup$
            I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
            $endgroup$
            – JohnGalt
            51 mins ago
















          $begingroup$
          @fgrieu: thanks...
          $endgroup$
          – poncho
          1 hour ago




          $begingroup$
          @fgrieu: thanks...
          $endgroup$
          – poncho
          1 hour ago












          $begingroup$
          I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
          $endgroup$
          – JohnGalt
          51 mins ago




          $begingroup$
          I appreciate the answer and am going to mark it as answered. Regarding, "what properties do you need for DHKE to be secure?" that was my intended question but that definitely wasn't clear. I'm glad, however, that they're two questions now, one with the minimum properties necessary for DHKE implementation (regardless of security) and this other question regarding the secure-ness are hardness I guess. Should I ask a separate question are could you expand on it here?
          $endgroup$
          – JohnGalt
          51 mins ago


















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