# Applied Cryptography

If you think cryptography is the answer to your problem, then you don't know what your problem is. —Peter G. Neumann

Crypto is an important building block for security. But there's much more to security than just crypto.

• Crypto is not the solution. It can help and harm security. Used incorrectly, crypto makes systems less secure.
• Crypto is not easy. Don't invent it yourself. Use well-studied solutions (and standards, though they sometimes have problems).
• Crypto is not cheap, but who cares? It may incur a performance hit, but what do you want: a fast system, or a secure system? We have enough fast, insecure systems.

This material is dangerous. You won't know enough about crypto when we're done, but you'll go off and use it anyway. Be very suspicious of yourself. Take further courses in cryptography if you really want to play in this space.

There are two branches of crypto: modern and applied.

• Modern: we prove it's secure, mathematically, but the algorithms are typically inefficient.
• Applied: we think it's secure, in practice, and the algorithms are typically efficient.

We're exclusively covering applied crypto.

# Part 1: Confidentiality

Threat: An attacker who controls the communication network. This attacker can arbitrarily read, modify, and delete messages. Think of this communication model as one in which messages are always sent to the attacker, never to the intended recipient. The attacker can then forward the message along if he chooses, redirect the message, save it for later replay, etc. This kind of threat is called a Dolev–Yao attacker.

Harm: Messages containing secret information could be disclosed to the adversary, thus violating confidentiality.

Vulnerability: The communication channel between sender and receiver can be read by untrusted principals.

Countermeasure: Encryption.

Shared-key Encryption:

1. Alice: c = Enc(m; k)
2. Alice → Bob: c
3. Bob: m = Dec(c; k)

(The format we use above is a protocol narration: each step is numbered and is either a computation or a message. We identify the principal(s) involved at each step by writing their names followed by a colon.)

Enc is the encryption algorithm; Dec is decryption. Alice and Bob must somehow share a key k that has previously been generated:

1. k = Gen(len) // len is length of key
2. ...

Together, (Gen,Enc,Dec) constitute an encryption scheme or cryptosystem. Well known examples of encryption schemes include AES (which uses shared keys) and RSA (which does not).

What makes an encryption scheme secure?

• Kerckhoffs's Principle: Secrecy should depend only upon key being secret—not the algorithms. You might see "proprietary encryption" algorithms touted as a good thing. They're not.
• For each key, Enc behaves like a random function. I.e., the map from m to c should be random. If it isn't, then there's structure in encrypted ciphertexts that could be exploited to break the encryption.

There is a provably perfectly secure encryption scheme called the one-time pad. Gen must generate a uniformly random sequence of bits of the same length as the message to be encrypted. Enc simply xors those random bits with the message, and Dec is identical to Enc. There are practical problems to deploying this scheme: (i) the keys must be really long (as long as the messages), (ii) you may never re-use a key (because doing so would reveal relationships between messages: (m1 ⊕ k) ⊕ (m2 ⊕ k) = m1 ⊕ m2), (iii) hence distributing the keys is difficult. Practical schemes instead rely on one short key that can be reused for many messages.

### Block Ciphers

Efficient encryption schemes usually operate on fixed-size messages called blocks. Such schemes are called block ciphers.

Well-known examples:

• DES (Data Encryption Standard). Block size: 64 bits; key size: 56 bits. DES was designed by IBM in 1973-4, tweaked by the NSA, then became the US standard for encryption. International adoption followed.
• 3DES (Triple DES). Block size: 64 bits; key size: 112 or 168 bits. 3DES is a strengthening of DES introduced in 1998, because 56 bit keys had become feasible to brute force. 3DES is simply three DES encryptions with two different keys, for an effective 112 bit key; or with three different keys, for an effective 168 bit key.
• AES (Advanced Encryption Standard). Block size: 128 bits; key size: 128, 192, or 256 bits. AES resulted from a public competition held by NIST, ending in 2001. It's now the US standard, approved by the NSA for Top Secret information. In 2009, new theoretical attacks were discovered that, if ever made practical, would break AES.

When a block cipher has multiple key lengths available, we indicate the particular length being used by appending it to the name of the cipher. AES-192, for example, means AES with 192 bit keys.

### Breaking a Cryptosystem

Assume the attacker doesn't know the key under which a ciphertext was encrypted. A brute force or exhaustive search means trying every possible key to decrypt a ciphertext (e.g., for AES-128, 2^128 tries). A break of a cryptosystem is an attack that succeeds in fewer steps than brute force. (e.g., only 2^99.5 tries for AES-256, which is what one theoretical, impractical attack already achieves).

If 2^X is the number of tries necessary to find the plaintext, then X is the security level of an encryption scheme. In the best case, the security level equals the key length. In practice, the security level goes down as attacks are discovered. E.g., 3DES-168 has a known attack that requires only 2^112 tries, reducing its security level from 168 to 112. Currently no practical attacks are known for AES, so—for now—its security level remains at the key length.

### Recommended Key Lengths

Various entities publish recommendations for security levels based on known attacks, hardware capabilities, and predicted advances. This website summarizes NIST's recommendations, as well as others: http://www.keylength.com/en/4/.

### Block Cipher Modes

If block ciphers work only on fixed length blocks, how can we send longer messages than the block length? A block cipher mode is an algorithm that uses a fixed-length block cipher to send an arbitrary-length message.

Strawman idea: chunk message into blocks; encrypt each block individually. Ciphertext block number i, written c_i, is thus Enc(m_i; k), where m_i is plaintext block number i. This algorithm is called electronic codebook mode (ECB).

ECB:
c_i = Enc(m_i; k)

ECB is a BAD IDEA that unfortunately gets invented over and over again, especially by students of crypto. Why is it bad? Because any two blocks that are same in plaintext will the be same in ciphertext. (Wikipedia has a nice graphical illustration of how ECB fails to provide confidentiality.) Do not use ECB. Unfortunately, it is still the default in Java, but you don't have to settle for the default.

One of best-known, good block cipher modes is cipher block chaining (CBC). With it, every ciphertext block depends on all previous ciphertext blocks, which avoids repetition problems like we observed with ECB.

CBC:
c_i = Enc(m_i XOR c_{i-1}; k)

If the first plaintext block is m_1, what is c_0? It can't be the encryption of a plaintext block; it has to be somehow invented from scratch for each new encryption. Block c_0 is, therefore, called the initialization vector (IV). It must be unpredictable to attackers for CBC to be secure. The best practice is to choose a new IV randomly for each (multi-block) message. The IV is sent in the clear, without encryption, because there is no meaningful information in it.

Another good block cipher mode is counter mode (CTR):

CTR:
k_i = Enc(n, i; k)
c_i = m_i XOR k_i

CTR uses Enc to encrypt a nonce n and a counter i in the same plaintext block. Observe the notation we use for that: commas between the parts of the plaintext that we're combining into one message, and a semi-colon before the key. In an implementation, we could use bit concatenation to combine message parts. Like the IV in CBC, nonce n should be randomly chosen for each new message (but stays the same for each block in the stream for a given message), and can be sent in the clear as ciphertext block c0.

An advantage of CTR over CBC is that each block in CTR can be computed in parallel, whereas CBC must process the blocks sequentially.

### Nonces

Both n and the IV in the modes above are examples of a nonce: a number used once. Nonces show up a lot in crypto. A nonce must always be

• Unique, meaning that it has never been used before in the lifetime of the system. (A synonym for "unique" is "fresh".) Nonces may also be
• Unpredictable, meaning that it isn't possible to predict the next nonce, even given knowledge of all the nonces that have been used for far in the lifetime of a system.

Nonces can come from several sources:

• Keeping a counter is simple to implement, but requires principals to keep state. Counters are always unique—unless the counter overflows. (Which can happen!) Counters are highly predictable.
• Using a clock value amount to keeping a counter that tracks time. If the clock can roll back, its value might not be unique. Clocks are highly predictable.
• Generating random nonces with a random number generator (RNG) yields unpredictability, if a good RNG is used. Such nonces are not necessarily unique, if random numbers are drawn from small space. But by making the space large enough, the probability of collisions can be made very low.

### Generating Random Numbers

How can software generate good, cryptographically strong random numbers? Frankly, without hardware support, it's a black art. In Java, use `java.security.SecureRandom`. Do not use `java.util.Random`, and do not use `Math.random()`, which itself uses `java.util.Random`. These latter two are predictable.

When using a block cipher mode, what should the mode do if the last block of plaintext message isn't full? Fill with 0's? No: there's no way to unambiguously remove the padding.

PKCS5 padding: Suppose B is the number of bytes that need to be added to the final plaintext block to fill it out completely. Then pad with B copies of the byte representing integer B. In the worst case, when the plaintext block is already filled, this requires adding one extra block to the message.

### Public-Key Cryptography

There's a big problem with the encryption schemes we've examined so far: the shared keys have to be distributed. For each pair of principals who want to communicate, a key needs to be shared. If there are n principals, that's O(n^2) keys. That sharing costs time and money. It used to be a big problem for militaries; they had to physically distribute code books containing the keys throughout the world.

This problem motivated the invention of another kind of encryption scheme: asymmetric or public key cryptography. RSA is the most famous example. The name "asymmetric" comes from the fact that different keys are used for encryption vs. decryption. In symmetric schemes like AES, the same key is used both for encryption and decryption.

In a public-key cryptosystem, every principal has its own key pair, comprising a

• public key, which is published for the world to see, and a
• private key, which is kept secret and never shared with anyone.

N.B. Our usually fastidious terminology breaks down here. Some people call symmetric schemes "secret-key schemes", even though in both symmetric and asymmetric schemes there is a key that is kept secret. And "private" key here doesn't necessarily mean that the key is personally-identifying information.

With public-key schemes, key distribution becomes much easier. We need only to publish a "phonebook" of public keys, which contains just O(n) keys. Thus we reduce from a quadratic problem to a linear problem.

Public-key Encryption:

1. Bob: (K_B, k_B) = Gen(len)
2. Alice: c = Enc(m; K_B)
3. Alice → Bob: c
4. Bob: m = Dec(c; k_B)

Note how we use upper-case K for public keys and lower-case k for private keys.

### Big Integers

Asymmetric encryption schemes are usually implemented in terms of really, really big integers—not the byte arrays that symmetric schemes use. The integers used for asymmetric encryption are far too big to fit in a standard `int` data type. In Java, the implementation is in terms of `BigInteger`. However, the Java crypto library interfaces conveniently let you pass in byte arrays and handle the conversion for you.

This use of big integers might seem like a minor implementation detail, but it's important. The lesser reason it's important is that the maximum size value you can encrypt is always bounded by the key size. Try to pass in too much plaintext, and Java will throw an exception.

The bigger reason it's important is that computation on big integers is much, much slower than computation on byte arrays. How much? In Java, anywhere from one to three orders of magnitude, based on simple experiments we did comparing AES to RSA in a past semester of this course.

Since asymmetric schemes use big integers, not byte arrays, padding works differently than for symmetric encryption. With RSA, the common practice is to use a padding function called OAEP: optimal asymmetric encryption padding. OAEP actually does much more than just padding, despite its name. It even takes extra precautions to improve the security of plain RSA encryption.

### Block Modes for Asymmetric Encryption

Since asymmetric encryption limits the maximum size of the plaintext, you might think that we should use block modes to encrypt arbitrary-length messages. In fact, this can be done. You could use CBC or CTR. You still should not use ECB, for the same reason as before.

In practice, though, block modes don't get used with asymmetric encryption, because encrypting many blocks with an asymmetric scheme would be really slow. Instead, the typical practice is to use a combination of both asymmetric and symmetric encryption, as discussed next.

### Hybrid Encryption

To efficiently encrypt a long message using public-key crypto, we use a mash-up of asymmetric and symmetric encryption called hybrid encryption.

Hybrid encryption uses a symmetric encryption scheme (Gen_S, Enc_S, Dec_S) and an asymmetric scheme (Gen_A, Enc_A, Dec_A), as well as a block cipher mode if necessary.

Hybrid Encryption:

1. Bob: (K_B, k_B) = Gen_A(len_A)
2. Alice:
k_s = Gen_S(len_S)
c1 = Enc_A(k_s; K_B)
c2 = Enc_S(m; k_s) // using a block cipher mode
3. Alice → Bob: c1, c2
4. Bob:
k_s = Dec_A(c1; k_B)
m = Dec_S(c2; k_s)

Key k_s is an example of a session key: a key that is for a limited time then discarded. If the session key is later compromised, only those messages it protected are vulnerable—unlike if a long-term symmetric key were used. The session key in hybrid encryption is valid only for one encryption from Alice to Bob; it shouldn't be reused for future encryptions from Alice to Bob, and Bob shouldn't use it to encrypt messages to Alice.

There's still one big problem we haven't solved: how can we distribute the "phonebook" containing everyone's public key? We'll delay discussion of that until we get to authentication of machines later in the course.

# Part 2: Integrity

Threat: A Dolev–Yao attacker.

Harm: The information contained in messages could be modified, thus violating integrity.

Harm: The purported sender of a message could be changed, thus violating integrity.

Vulnerability: Messages sent on the communication channel between the sender and receiver can be modified by untrusted principals.

Countermeasure: MACs and digital signatures.

NOT a Countermeasure: Encryption does not, in general, protect integrity. Encryption protects only confidentiality!

The usual mistake is to reason as follows: "The message is encrypted. If attacker changes the ciphertext, it will decrypt to nonsense. That nonsense can be detected." But that reasoning is not valid. For example:

• Attackers can be smart about changes. A ciphertext from another execution of the same protocol might decrypt just fine.
• The plaintext block could itself be a random number, and recipient would have no way of determining whether it's the right one if the attacker substitutes a different random number.
• In CTR mode (or any stream cipher), it's easy to flip individual bits. E.g., change "admin=0" to "admin=1" just by knowing position of that bit in stream.
• In CBC mode, it's easy to truncate blocks from the beginning of a message.
• For more examples, see section 9.6.5.i, "Encryption alone does not guarantee data integrity", in [Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, Boca Raton, 1997.].

There are some block modes designed in last decade to protect both confidentiality and integrity. But these aren't yet widely supported, including in Java.

Encryption does not protect integrity.

Like encryption, there are symmetric and asymmetric algorithms for protecting integrity. The symmetric version is called a message authentication codes (MAC). The asymmetric version is called a digital signature. Both use another primitive, hash functions, which we'll cover first.

### Cryptographic Hash Functions

A cryptographic hash function, also called a message digest, takes an arbitrary size input m and produces a fixed length output H(m). The output length is typically 128–1024 bits.

The goal of a cryptographic hash is to produce a compact representation of an original object. That representation should behave much like a fingerprint:

• It's hard to find 2 people with same fingerprint. That's true whether you get to pick pairs of people, or whether you are given one person then must find another. That means fingerprints are collision resistant.
• Given a person, it's easy to get their fingerprint. But given a fingerprint, it's hard to find the person it came from. (Which is why law enforcement invests money in building databases to do just that.) That means fingerprints are one way.

Likewise, cryptographic hash functions must be collision resistant and one way. What makes a hash function secure? It should behave like a random function.

Cryptographic hash functions are not the same as the ordinary hash functions that are used to implement hash tables, even though both compress their inputs. Collision resistance and one way-ness are not required of ordinary hash functions.

The security level of a hash function is, in the absence of any clever attacks, half the function's output length. E.g., if the output length is 256 bits, then the security level is at most 128 bits. Why? There's a generic attack that works on all hash functions that halves the security level. It's called the birthday attack.

MD5 and SHA-1 used to be the most commonly used hash functions. But:

• The collision resistance of MD5 (invented by Ron Rivest in 1991) was broken in 2004–8. It's now possible to find collisions in mere seconds. Moreover, the collisions can even be engineered to be (maliciously) useful, for example, generating rogue CA certificates.
• The collision resistance of SHA-1 (released by the NSA in 1995) has been broken in ongoing work since 2005. Attacks are known that reduce its security level to only 63 bits or fewer.

SHA-2, released by the NSA in 2001. is actually a whole family of algorithms, SHA-{224,256,384,512}. The name indicates the output size in bits. Each should have security level equal to its output size halved. But these are based on similar ideas to SHA-1, so there's concern that they might one day turn out to be vulnerable to similar attacks.

Next will be SHA-3. NIST held a public competition for the new algorithm. There were five finalists, all based on different ideas than SHA-1 and SHA-2, and all developed openly and peer reviewed. The winner was announced in October 2012; the name of the winning algorithm is Keccak. It is in the process of being standardized. The output size can be 224, 256, 384, or 512 bits; or a variable-length output can be produced using a variant called SHAKE.

### Message Authentication Codes (MACs)

A message authentication code is an algorithm for detecting modification of messages based on a shared key.

MAC:

1. k = Gen(len) // A and B somehow share key k
2. A: t = MAC(m; k) // t is called the "tag"
3. A→B: m, t
4. B: verify t = MAC(m; k)

The length of input m to MAC may be arbitrary. The output length of MAC is fixed and depends upon the particular MAC algorithm.

When is a MAC secure? It should behave like a random function, for each key. Especially, it shouldn't be possible to predict new (m,t) pairs if you don't know k.

There are many examples of MACs. HMAC (a hash-based MAC) is one of the most common.

HMAC(m; k) = H(f1(k), H(f2(k), m))

Function H is a cryptographic hash function. It can be instantiated, for example, by any of the SHA-2 family. Functions f1 and f2 are specially designed to prevent certain attacks. Their exact details aren't important here—you can look them up if you're curious.

Another example is CBC-MAC, which uses CBC mode encryption to produce a tag.

Note that MACs do not protect confidentiality, at least not necessarily. Some happen to do so, but it's easy to construct MACs that don't.

### Digital Signatures

A digital signature scheme is a set of algorithms for detecting modification of messages based on a public–private key pair. The public key for principal A, written K_A, is used to verify A's signatures. The private key for principal A, written k_A, is used by A to create signatures.

Digital Signature:

1. (K_A, k_A) = Gen(len)
2. A: s = Sign(m; k_A)
3. A → B: m, s
4. B: accept if Ver(m; s; K_A)

The digital signature scheme is the triple (Gen, Sign, Ver) of algorithms. Note that Ver takes three inputs: the message to verify, the purported signature on that message, and the verification key of the signer.

As with MACs, we want to be able to sign arbitrary length messages. But these Sign and Ver are public-key algorithms, which operate on big integers. So, as with public-key encryption, they are constrained to a limited input size.

In practice, messages are therefore hashed before being signed:

Digital Signature with Hashing:

1. (K_A, k_A) = Gen(len)
2. A: s = Sign(H(m); k_A)
3. A → B: m, s
4. B: accept if Ver(H(m); s; K_A)

Hashing is such a pervasive practice with signatures that, henceforth, we'll just assume the message is hashed without bothering to write that down as part of the protocol.

When is a digital signature scheme secure? It should work like hand-written signatures. In fact, it should be even better: an adversary shouldn't be able to forge signatures on new messages, even if given samples of other signed messages. Mathematically, the signature algorithm should behave, for each key, like a random function from messages to signatures.

Well-known examples of digital signature schemes include the following:

• Digital Signature Algorithm (DSA). Released by NIST in 1991 as part of the Digital Signature Standard (DSS). There is no proof of security for DSA, but it's been used for decades now without any serious attacks being discovered. Originally, the standard required SHA-1 as the hash, but that's since been updated to SHA-2. Also, the keys were originally short, but have since been lengthened.
• RSA. The RSA encryption scheme can be adapted for use as a digital signature scheme. The Gen algorithm stays the same. Sign is roughly equivalent to Dec, and Ver to Enc. However, they aren't exactly the same. Sometimes people will say that you should "do an RSA decryption to sign". That's partially correct, but also partially wrong. Here is a set of notes explaining why RSA decryption is not the same as RSA signing. Also, never use the same key pair for both signing and encryption. There's a provably secure variant of RSA signatures called RSA-PSS (probabilistic signature scheme). Unfortunately, it's not yet implemented in Java's standard provider.

### Digital Certificates

Here's one important use for digital signatures. Operating systems and browsers come preinstalled with digital certificates for companies such as Verisign. A digital certificate is means of associating a public key with a principal's identity. Let id_S be a string encoding the identity of a subject, let K_S be the subject's public (verification or encryption) key, and let k_I be the signing key of an issuer.

Digital Certificate:
I<<S>> = Sign(id_S, K_S; k_I)

I<<S>> is a digital certificate issued by I for S. It binds id_S to K_S according to I.

In practice, the most common format for certificates is X.509, an international standard. An X.509 certificate includes additional information, including a serial number for the certificate, a validity interval, etc.

We'll discuss digital certificates further when we cover authentication of machines.

# Part 3: Confidentiality and Integrity

### Authenticated Encryption

Suppose you want to protect both confidentiality and integrity. The result is called authenticated encryption. There are three generic ways of constructing authenticated encryption out of a standard block cipher and MAC. All three are used in real-world protocols.

• Encrypt and MAC. Encrypt the message. Separately MAC the message under a different shared key. Send both the ciphertext and the tag. This algorithm is probably the worst of the three, because the tag could reveal information about the plaintext. Secure Shell (ssh) uses this algorithm in a safe way.
• Encrypt then MAC. Encrypt the message. MAC the resulting ciphertext. Send both the ciphertext and the tag. This is provably the most secure of the three algorithms; IPsec uses it.
• MAC then Encrypt. MAC the plaintext message. Encrypt the message and the tag together. Send the resulting ciphertext. As long as the MAC algorithm is strong enough (and HMAC is), this algorithm is just as secure as the previous one. SSL uses this algorithm.

There are also block cipher modes that are specifically designed to achieve both confidentiality and integrity. Galois/Counter Mode (GCM) is a popular choice, because it has high performance and isn't encumbered by patents.

### Secure Sockets Layer (SSL)

Authenticated encryption is such a massively useful thing that it's long been available as part of libraries and other software distributions. Netscape introduced a protocol for it back in 1996 called Secure Sockets Layer (SSL) v3. SSL essentially provides authenticated encryption on top of TCP. SSL is used widely—for example, HTTPS is just HTTP run over SSL. SSL was standardized under the name Transport Layer Security (TLS), so you'll see it referred to by either name in the literature.

TLS manages sessions, which are bi-directional communication between a client and a server. The communication is optionally secured for both confidentiality and integrity against a Dolev–Yao attacker. Sessions are logical: there can be many sessions between any two physical hosts, and each host could be either client or server in any given session.

Each message sent during a session is called a record. Records are protected by MAC-then-Encrypt. The MAC used is HMAC. The hash function and encryption scheme used can be negotiated by the client and server for each SSL session. Digital signatures and certificates can used to negotiate the shared encryption and MAC keys for each session. We'll look more at the details of this negotiation when we discuss authentication of machines.

SSL/TLS in Java: Java provides JSSE (Java Secure Socket Extension). It's mostly very easy to use as a drop-in-replacement of standard network sockets. The hard part typically is management of digital certificates. Using JSSE in your course project is usually a Very Good Idea, assuming it satisfies your security goals. It's much better to reuse crypto code than to implement everything yourself.