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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
103 ../../principles/security.rst:1000
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Alice checks the signature (with :math:`Bob_{pub}`) and the certificate and computes :math:`S_{A}=B^{a} \mod p` 111/10000 · 111
104 ../../principles/security.rst:1002
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Bob computes :math:`S_{B}=A^{b} \mod p` 39/10000 · 39
105 ../../principles/security.rst:1006
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — This prevents the attack mentioned above since Mallory cannot create a fake certificate and cannot sign a value by using Bob's private key. Given the risk of man-in-the-middle attacks, the Diffie-Hellman key exchange mechanism should never be used without authentication. 271/10000 · 271
106 ../../principles/security.rst:1015
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Footnotes 9/10000 · 9
107 ../../principles/security.rst:1016
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — The wikipedia page on passwords provides many of these references : https://en.wikipedia.org/wiki/Password_strength 115/10000 · 115
108 ../../principles/security.rst:1019
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — A detailed explanation of the operation of the RSA algorithm is outside the scope of this e-book. Various tutorials such as the `RSA page <https://en.wikipedia.org/wiki/RSA_(cryptosystem)>`_ on wikipedia provide examples and tutorial information. 246/10000 · 246
109 ../../principles/security.rst:1024
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — A detailed explanation of the ECC cryptosystems is outside the scope of this e-book. A simple introduction may be found on `Andrea Corbellini's blog <http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/>`_. There have been deployments of ECC recently because ECC schemes usually require shorter keys than RSA and consume less CPU. 369/10000 · 369
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