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83 ../../principles/security.rst:935
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — From her knowledge of :math:`a` and :math:`B`, Alice can compute :math:`Secret=B^{a} \mod p= (g^{b} \mod p) ^{a} \mod p=g^{a \times b} \mod p` 142/10000 · 142
84 ../../principles/security.rst:937
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — From is knowledge of :math:`b` and :math:`A`, Bob can compute :math:`Secret=A^{b} \mod p=(g^{a} \mod p) ^{b} \mod p=g^{a \times b} \mod p` 138/10000 · 138
85 ../../principles/security.rst:940
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — The security of this protocol relies on the difficulty of computing discrete logarithms, i.e. from the knowledge of :math:`A` (resp. :math:`B`), it is very difficult to extract :math:`\log(A)=\log(g^{a} \mod p)=a` (resp. :math:`\log(B)=\log(g^{b} \mod p)=b`). 259/10000 · 259
86 ../../principles/security.rst:945
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — An example of the utilization of the Diffie-Hellman key exchange is shown below. Before starting the exchange, Alice and Bob agree on a modulus (:math:`p=23`) and a base (:math:`g=5`). These two numbers are public. They are typically part of the standard that defines the protocol that uses the key exchange. 308/10000 · 308
87 ../../principles/security.rst:951
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice chooses a secret integer : :math:`a=8` and sends :math:`A= g^{a} \mod p= 5^{8} \mod 23=16` to Bob 103/10000 · 103
88 ../../principles/security.rst:953
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob chooses a secret integer : :math:`b=13` and sends :math:`B= g^{b} \mod p=5^{13} \mod 23=21` to Alice 104/10000 · 104
89 ../../principles/security.rst:955
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice computes :math:`S_{A}=B^{a} \mod p= 21^{8} \mod 23=3` 59/10000 · 59
90 ../../principles/security.rst:956
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob computes :math:`S_{B}=A^{b} \mod p= 16^{13} \mod 23=3` 58/10000 · 58
91 ../../principles/security.rst:958
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice and Bob have agreed on the secret information :math:`3` without having sent it explicitly through the network. If the integers used are large enough and have good properties, then even Eve who can capture all the messages sent by Alice and Bob cannot recover the secret key that they have exchanged. There is no formal proof of the security of the algorithm, but mathematicians have tried to solve similar problems with integers during centuries without finding an efficient algorithm. As long as the integers that are used are random and large enough, the only possible attack for Eve is to test all possible integers that could have been chosen by Alice and Bob. This is computationally very expensive. This algorithm is widely used in security protocols to agree on a secret key. 788/10000 · 788
92 ../../principles/security.rst:970
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Unfortunately, the Diffie-Hellman key exchange alone cannot cope with man-in-the middle attacks. Consider Mallory who sits in the middle between Alice and Bob and can easily capture and modify their messages. The modulus and the base are public. They are thus known by Mallory as well. He could then operate as follows : 320/10000 · 320
93 ../../principles/security.rst:976
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice chooses a secret integer and sends :math:`A= g^{a} \mod p` to Mallory 75/10000 · 75
94 ../../principles/security.rst:977
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Mallory generates a secret integer, :math:`m` and sends :math:`M=g^{m} \mod p` to Bob 85/10000 · 85
95 ../../principles/security.rst:978
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob chooses a secret integer and sends :math:`B=g^{b} \mod p` to Mallory 72/10000 · 72
96 ../../principles/security.rst:979
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Mallory computes :math:`S_{A}=A^{m} \mod p` and :math:`S_{B}=B^{m} \mod p` 74/10000 · 74
97 ../../principles/security.rst:980
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice computes :math:`S_{A}=M^{a} \mod p` and uses this key to communicate with Mallory (acting as Bob) 103/10000 · 103
98 ../../principles/security.rst:981
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob computes :math:`S_{B}=M^{b} \mod p` and uses this key to communicate with Mallory (acting as Alice) 103/10000 · 103
99 ../../principles/security.rst:983
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — When Alice sends a message, she encrypts it with :math:`S_{A}`. Mallory decrypts it with :math:`S_{A}` and encrypts the plaintext with :math:`S_{B}`. When Bob receives the message, he can decrypt it by using :math:`S_{B}`. 222/10000 · 222
100 ../../principles/security.rst:988
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — To safely use the Diffie-Hellman key exchange, Alice and Bob must use an `authenticated` exchange. Some of the information sent by Alice or Bob must be signed with a public key known by the other user. In practice, it is often important for Alice to authenticate Bob. If Bob has a certificated signed by Ted, the authenticated key exchange could be organized as follows. 370/10000 · 370
101 ../../principles/security.rst:995
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice chooses a secret integer : :math:`a` and sends :math:`A= g^{a} \mod p` to Bob 83/10000 · 83
102 ../../principles/security.rst:997
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob chooses a secret integer : :math:`b`, computes :math:`B= g^{b} \mod p` and sends :math:`Cert(Bob,Bob_{pub},Ted), E_p(Bob_{priv},B)` to Alice 144/10000 · 144