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../../principles/security.rst:354
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AES is widely used as of this writing, but other secret key encryption schemes continue to appear. ChaCha20, proposed by D. Bernstein is now used by several internet protocols :rfc:`7539`. A detailed discussion of encryption schemes is outside the scope of this book. We will consider encryption schemes as black boxes whose operation depends on a single key. A detailed overview of several of these schemes may be found in [MVV2011]_.
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../../principles/security.rst:363
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In the 1970s, Diffie and Hellman proposed in their seminal paper [DH1976]_, a different type of encryption : `public key cryptography`. In public key cryptography, each user has two different keys :
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../../principles/security.rst:367
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a public key (:math:`K_{pub}`) that he can distribute to everyone
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../../principles/security.rst:368
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a private key (:math:`K_{priv}`) that he needs to store in a secure manner and never reveal to anyone
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../../principles/security.rst:371
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These two keys are generated together and they are linked by a complex mathematical relationship that is such that it is computationally difficult to compute :math:`K_{priv}` from :math:`K_{pub}`.
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../../principles/security.rst:375
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A public key cryptographic scheme is a combination of two functions :
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../../principles/security.rst:380
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The public key is used to encrypt a message so that it can only be read by the intended recipient. For example, let us consider two users : Alice and Bob. Alice (resp. Bob) uses the keys :math:`A_{priv}` and :math:`A_{pub}` (resp. :math:`B_{priv}` and :math:`B_{pub}`). To send a secure message `M` to Alice, Bob computes :math:`CM=E_p(A_{pub},M)` and Alice can decrypt it by using :math:`D_p(A_{priv},CM)=D_p(A_{priv},E_p(A_{pub},M))=M`.
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../../principles/security.rst:393
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Several public key encryption schemes have been proposed. Two of them have reached wide deployment :
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../../principles/security.rst:396
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The Rivest Shamir Adleman (RSA) algorithm [#frsa]_ proposed in [RSA1978]_ that relies on modular exponentiation with large integers.
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../../principles/security.rst:398
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The Elliptic Curve Cryptography techniques [#fecc]_ that rely on special properties of elliptic curves.
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../../principles/security.rst:401
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Another interesting property of public key cryptography is its ability to compute `signatures` that can be used to authenticate a message. This capability comes from the utilization of two different keys that are linked together. If Alice wants to sign a message `M`, she can compute :math:`SM=E_p(A_{priv},M)`. Anyone who receives this signed messaged can extract its content as :math:`D_p(A_{pub},SM)=D_p(A_{pub},E_p(A_{priv},M))=M`. Everyone can use :math:`A_{pub}` to check that the message was signed by using Alice's private key (:math:`A_{priv}`). Since this key is only known by Alice, the ability to decrypt `SM` is a proof that the message was signed by Alice herself.
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../../principles/security.rst:412
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In practice, encrypting a message to sign it can be computationally costly, in particular if the message is a large file. A faster solution would be to summarize the document and only sign the summary of the document. A naive approach could be based on a checksum or CRC computed over the message. Alice would then compute :math:`C=Checksum(M)` and :math:`SC=E_p(A_{priv},C)`. She would then send both `M` and `SC` to the recipient of the message who can easily compute `C` from `SC` and verify the authenticity of the message. Unfortunately, this solution does not protect Alice and the message's recipient against a man-in-the-middle attack. If Mallory can intercept the message sent by Alice, he can easily modify Alice's message and tweak it so that it has the same checksum as the original one. The CRCs, although more complex to compute, suffer from the same problem.
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../../principles/security.rst:433
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To efficiently sign messages, Alice needs to be able to compute a summary of her message in a way that makes prohibits an attacker from generating a different message that has the same summary. `Cryptographic hash functions` were designed to solve this problem. The ideal hash function is a function that returns a different number for every possible input. In practice, it is impossible to find such a function. Cryptographic hash functions are an approximation of this perfect summarization function. They compute a summary of a given message in 128, 160, 256 bits or more. They also exhibit the `avalanche effect`. This effect indicates that a small change in the message causes a large change in the hash value. Finally hash functions are very difficult to invert. Knowing a hash value, it is computationally very difficult to find the corresponding input message. Several hash functions have been proposed by cryptographers. The most popular ones are :
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../../principles/security.rst:447
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MD5, originally proposed in :rfc:`1321`. It has been used in a wide range of applications. In 2010, attacks against MD5 were published and this hash function is now deprecated.
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../../principles/security.rst:450
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SHA-1 is a cryptographic hash function that was standardized by the NIST in 1995. It outputs 160 bits results. It is now used in a variety of network protocols.
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../../principles/security.rst:453
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SHA-2 is another family of cryptographic hash functions designed by the NIST. Different variants of SHA-2 can produce has values of 224, 256, 384 or 512 bits.
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../../principles/security.rst:458
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Another important point about cryptographic algorithms is that often these algorithms require random numbers to operate correctly (e.g. to generate keys). Generating good random numbers is difficult and any implementation of cryptographic algorithms should also include a secure random number generator. :rfc:`4086` provides useful recommendations.
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../../principles/security.rst:469
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Cryptographic protocols
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../../principles/security.rst:471
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We can now combine the cryptographic operations described in the previous section to build some protocols to securely exchange information. Let us first go back to the problem of authenticating Alice on Bob's computer. We have shown earlier that using a simple password for this purpose is insecure in the presence of attackers.
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../../principles/security.rst:478
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A naive approach would be to rely on hash functions. Since hash functions are non-invertible, Alice and Bob could decide to use them to exchange Alice's password in a secure manner. Then, Alice could be authenticated by using the following exchange.
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