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23 ../../principles/security.rst:301
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Unfortunately, it is difficult to use this cipher in practice since the key must be as long as the message that needs to be transmitted. If the key is smaller than the message and the message is divided into blocks that have the same length as the key, then the scheme becomes less secure since the same key is used to decrypt different parts of the message. In practice, `XOR` is often one of the basic operations used by encryption schemes. To be usable, the deployed encryption schemes use keys that are composed of a small number of bits, typically 56, 64, 128, 256, ... 574/10000 · 574
24 ../../principles/security.rst:310
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — A secret key encryption scheme is a perfectly reversible functions, i.e. given an encryption function `E`, there is an associated decryption function `D` such that :math:`\forall k \forall M : D(K, E(M,K))=M`. 209/10000 · 209
25 ../../principles/security.rst:316
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Various secret key cryptographic functions have been proposed, implemented and deployed. The most popular ones are : 116/10000 · 116
26 ../../principles/security.rst:319
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — DES, the Data Encryption Standard that became a standard in 1977 and has been widely used by industry. It uses 56 bits keys that are not considered sufficiently secure nowadays since attackers can launch brute-force attacks by testing all possible keys. Triple DES combines three 56 bits keys, making the brute force attacks more difficult. 340/10000 · 340
27 ../../principles/security.rst:324
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — RC4 is an encryption scheme defined in the late 1980s by Ron Rivest for RSA Security. Given the speed of its software implementation, it has been included in various protocols and implementations. However, cryptographers have identified several weaknesses in this algorithm. It is now deprecated and should not be used anymore :rfc:`7465`. 339/10000 · 339
28 ../../principles/security.rst:329
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — AES or the Advanced Encryption Standard is an encryption scheme that was designed by the Belgian cryptographers Joan Daemen and Vincent Rijmen in 2001 [DR2002]_. This algorithm has been standardized by the U.S. National Institute of Standards and Technology (NIST). It is now used by a wide range of applications and various hardware and software implementations exist. Many microprocessors include special instructions that ease the implementation of AES. AES divides the message to be encrypted in blocks of 128 bits and uses keys of length 128, 192 or 256 bits. The block size and the key length are important parameters of an encryption scheme. The block size indicates the smallest message that can be encrypted and forces the sender to divide each message in blocks of the supported size. If the message is larger than an integer number of blocks, then the message must be padded before being encrypted and this padding must be removed after decryption. The key size indicates the resistance of the encryption scheme against brute force attacks, i.e. attacks where the attacker tries all possible keys to find the correct one. 1133/10000 · 1133
29 ../../principles/security.rst:354
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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]_. 435/10000 · 435
30 ../../principles/security.rst:363
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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 : 198/10000 · 198
31 ../../principles/security.rst:367
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — a public key (:math:`K_{pub}`) that he can distribute to everyone 65/10000 · 65
32 ../../principles/security.rst:368
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — a private key (:math:`K_{priv}`) that he needs to store in a secure manner and never reveal to anyone 101/10000 · 101
33 ../../principles/security.rst:371
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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}`. 196/10000 · 196
34 ../../principles/security.rst:375
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — A public key cryptographic scheme is a combination of two functions : 69/10000 · 69
35 ../../principles/security.rst:377
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — an encryption function, :math:`E_{p}`, that takes a key and a message as parameters 83/10000 · 83
36 ../../principles/security.rst:378
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — a decryption function, :math:`D_{p}`, that takes a key and a message as parameters 82/10000 · 82
37 ../../principles/security.rst:380
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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`. 438/10000 · 438
38 ../../principles/security.rst:393
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Several public key encryption schemes have been proposed. Two of them have reached wide deployment : 100/10000 · 100
39 ../../principles/security.rst:396
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — The Rivest Shamir Adleman (RSA) algorithm [#frsa]_ proposed in [RSA1978]_ that relies on modular exponentiation with large integers. 132/10000 · 132
40 ../../principles/security.rst:398
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — The Elliptic Curve Cryptography techniques [#fecc]_ that rely on special properties of elliptic curves. 103/10000 · 103
41 ../../principles/security.rst:401
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 678/10000 · 678
42 ../../principles/security.rst:412
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 873/10000 · 873