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| The '''computational Diffie–Hellman (CDH assumption)''' is the assumption that a certain [[computational problem]] within a [[cyclic group]] is hard.
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| Consider a cyclic group ''G'' of order ''q''. The CDH assumption states that, given
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| :<math>(g,g^a,g^b) \, </math>
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| for a randomly chosen generator ''g'' and random
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| :<math>a,b \in \{0, \ldots, q-1\},\,</math>
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| it is [[computationally intractable]] to compute the value
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| :<math>g^{ab}. \,</math>
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| The security of many [[cryptosystem]]s is based on the CDH assumption. Also, the confidentiality of [[ElGamal encryption]] is equivalent to the CDH assumption (though the [[semantic security]] of the scheme is based on the [[decisional Diffie–Hellman assumption]]).
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| The CDH assumption is related to the [[discrete logarithm assumption]], which holds that computing the [[discrete logarithm]] of a value base a generator <math>g</math> is hard. If taking discrete logs in <math>{\mathbb G}</math> were easy, then the CDH assumption would be false: given
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| :<math>(g,g^a,g^b), \, </math>
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| one could efficiently compute <math>g^{ab}</math> in the following way:
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| * compute <math>a</math> by taking the discrete log of <math>g^a</math> to base <math>g</math>;
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| * compute <math>g^{ab}</math> by exponentiation: <math>g^{ab} = (g^b)^a</math>;
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| It is an open problem to determine whether the discrete log assumption is equivalent to CDH, though in certain special cases this can be shown to be the case.
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| The CDH assumption is also related to the [[decisional Diffie–Hellman assumption]] (DDH), which holds that it is hard to distinguish tuples of the form <math>(g,g^a,g^b,g^{ab})</math> from random tuples. If computing <math>g^{ab}</math> from <math>(g,g^a,g^b)</math> were easy, then one could detect DDH tuples trivially. It is believed that CDH is a '''weaker''' assumption than DDH: there are groups for which detecting DDH tuples is easy, but solving CDH problems is believed to be hard.
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| ==See also==
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| * [[Diffie–Hellman problem]]
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| * [[Diffie–Hellman key exchange]]
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| ==References==
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| #Variations of the Diffie–Hellman Problem ([http://www.i2r.a-star.edu.sg/icsd/publications/Baofeng_2003_Variations%20of%20Diffie%20Hellman%20problems.pdf pdf file])
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| #Towards the Equivalence of Breaking the Diffie–Hellman Protocol and Computing Discrete Logarithms ([http://dsns.csie.nctu.edu.tw/research/crypto/HTML/PDF/C94/271.PDF pdf file])
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| {{DEFAULTSORT:Computational Diffie-Hellman assumption}}
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| [[Category:Computational hardness assumptions]]
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