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












8












$begingroup$


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









$endgroup$

















    8












    $begingroup$


    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









    $endgroup$















      8












      8








      8


      2



      $begingroup$


      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 9 hours ago









      JohnGaltJohnGalt

      1775




      1775






















          1 Answer
          1






          active

          oldest

          votes


















          8












          $begingroup$


          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
            8 hours 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
            8 hours ago










          • $begingroup$
            The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
            $endgroup$
            – Myria
            7 hours ago












          • $begingroup$
            @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
            $endgroup$
            – JohnGalt
            6 hours ago










          • $begingroup$
            @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
            $endgroup$
            – Myria
            5 hours ago













          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "281"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: false,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: null,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcrypto.stackexchange.com%2fquestions%2f66688%2fwhich-properties-of-a-group-are-used-in-the-steps-of-diffie-hellman%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          8












          $begingroup$


          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
            8 hours 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
            8 hours ago










          • $begingroup$
            The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
            $endgroup$
            – Myria
            7 hours ago












          • $begingroup$
            @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
            $endgroup$
            – JohnGalt
            6 hours ago










          • $begingroup$
            @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
            $endgroup$
            – Myria
            5 hours ago


















          8












          $begingroup$


          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
            8 hours 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
            8 hours ago










          • $begingroup$
            The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
            $endgroup$
            – Myria
            7 hours ago












          • $begingroup$
            @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
            $endgroup$
            – JohnGalt
            6 hours ago










          • $begingroup$
            @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
            $endgroup$
            – Myria
            5 hours ago
















          8












          8








          8





          $begingroup$


          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 8 hours ago

























          answered 8 hours ago









          ponchoponcho

          91.1k2142236




          91.1k2142236












          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            8 hours 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
            8 hours ago










          • $begingroup$
            The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
            $endgroup$
            – Myria
            7 hours ago












          • $begingroup$
            @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
            $endgroup$
            – JohnGalt
            6 hours ago










          • $begingroup$
            @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
            $endgroup$
            – Myria
            5 hours ago




















          • $begingroup$
            @fgrieu: thanks...
            $endgroup$
            – poncho
            8 hours 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
            8 hours ago










          • $begingroup$
            The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
            $endgroup$
            – Myria
            7 hours ago












          • $begingroup$
            @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
            $endgroup$
            – JohnGalt
            6 hours ago










          • $begingroup$
            @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
            $endgroup$
            – Myria
            5 hours ago


















          $begingroup$
          @fgrieu: thanks...
          $endgroup$
          – poncho
          8 hours ago




          $begingroup$
          @fgrieu: thanks...
          $endgroup$
          – poncho
          8 hours 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
          8 hours 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
          8 hours ago












          $begingroup$
          The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
          $endgroup$
          – Myria
          7 hours ago






          $begingroup$
          The selection of group is critical for security. Using DH with the group $mod p$ for many primes $p$ with group operation $times$ is secure as far as we know, yet using DH with the group $mod {p-1}$ with group operation $+$ is not secure (discrete log is simply modular division), even though the two groups are isomorphic.
          $endgroup$
          – Myria
          7 hours ago














          $begingroup$
          @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
          $endgroup$
          – JohnGalt
          6 hours ago




          $begingroup$
          @Myria Is the selection of the group critical for security or the selection of the modulus? For DHKE over $Z_p^*$ it would appear from the answer above that the critical factor is choosing a large enough prime so that the sqrt(p) is large enough to make brute forcing the DLP computationally infeasible?
          $endgroup$
          – JohnGalt
          6 hours ago












          $begingroup$
          @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
          $endgroup$
          – Myria
          5 hours ago






          $begingroup$
          @JohnGalt For $mathbb{Z}^*_p$, $p$ needs to be a prime such that $p-1$ has at least one large prime; optimally $p$ is a Sophie-Germain prime. Otherwise, the discrete logarithm is easy (Pohlig-Hellman with Baby-Step Giant-Step). But even with such a $p$, $mathbb{Z}^+_{p-1}$ is still insecure, despite being isometric to $mathbb{Z}^*_p$. Thus, you really need both the group representation and its modulus to be selected well.
          $endgroup$
          – Myria
          5 hours ago




















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Cryptography Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcrypto.stackexchange.com%2fquestions%2f66688%2fwhich-properties-of-a-group-are-used-in-the-steps-of-diffie-hellman%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          How to make a Squid Proxy server?

          Image
          1 I want to set up squid proxy server for monitoring user activity in a network and Internet Web URL filtering for user. I also want to configure the user login, user log, user access group, and block unwanted web sites in my network. Please help me and guide me the full process to do this. server proxy squid share | improve this question edited Jul 31 '12 at 14:45 Jorge Castro 37.1k 107 422 617 asked Jul 31 '12 at 5:16 Goivind Tiwari Goivind Tiwari ...
          Read more

          Is this a new Fibonacci Identity?

          Image
          6 1 $begingroup$ I have found the following Fibonacci Identity (and proved it). If $F_n$ denotes the nth Fibonacci Number, we have the following identity begin{equation} F_{n-r+h}F_{n+k+g+1} - F_{n-r+g}F_{n+k+h+1} = (-1)^{n+r+h+1} F_{g-h}F_{k+r+1} end{equation} where $F_1 = F_2 = 1$ , $r leq n$ , $h leq g$ , and $n, g, k in mathbb{N}$ . It is not too hard to show that this identity subsumes Cassini's Identity, Catalan's Identity, Vajda's Idenity, and d'Ocagne's identity to name a few. I have done a pretty thorough literature review, and I have not found anything like this, but I am still wondering if anyone has seen this identity before? I found this by accident after noticing some patterns in some analysis work I was doing, so if this is already known I would be curious to see w...
          Read more

          19世紀

          Image
          この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。 出典を追加して記事の信頼性向上にご協力ください。 （ 2012年3月 ） 千年紀: 2千年紀 世紀: 18世紀 - 19世紀 - 20世紀 十年紀: 1800年代 1810年代 1820年代 1830年代 1840年代 1850年代 1860年代 1870年代 1880年代 1890年代 19世紀に君臨した大英帝国。 19世紀 （じゅうきゅうせいき）は、西暦1801年から西暦1900年までの100年間を指す世紀。 目次 1 19世紀の歴史 1.1 国民国家の成立 1.2 帝国主義の興隆 1.3 列強の植民地争奪戦 2 できごと 2.1 1800年代 2.2 1810年代 2.3 1820年代 2.4 1830年代 2.5 1840年代 2.6 1850年代 2.7 1860年代 2.8 1870年代 2.9 1880年代 2.10 1890年代 2.11 1900年代 3 文化 3.1 文学 3.2 音楽 3.3 思想 3.4 科学 3.5 技術 4 人物 4.1 ヨーロッパ 4.1.1 政治家・王族 4.1.1.1 フランス 4.1.1.2 オーストリア＝ハンガリー 4.1.1.3 ロシア 4.1.1.4 イギリス 4.1.1.5 ドイツ（プロイセンほかドイツ領邦を含む） 4.1.1.6 北欧 4.1.1.7 イタリア 4.1.1.8 スペイン 4.1.1.9 ベルギー 4.1.1.10 ギリシア 4.1.2 軍人 4.1.3 実業家 4.1.4 科学と技術 4.1.5 思想と哲学・人文諸学 4.1.6 宗教 4.1.7 文学 4.1.8 美術 4.1.9 音楽 4.1.10 社会事業家 4.1.11 探検家・旅行家 4.1.12 料...
          Read more