Symmetric algorithms are used for traditional encryption, where the same key is used for encryption and decryption. For a well-designed symmetric algorithm, the strength of the cryptographic algorithm should depend exponentially on the size of the key used. Thus if a key is only 4 bits long, there are 24 or 16 possible keys. A brute-force attacker who tried all 16 keys would break the encryption; but on average the attacker would have to try only half of them, or 8 keys, before finding the correct key. Because the numbers get large quickly via exponentiation, the strength of algorithms is generally quoted in bits.

A well-designed symmetric algorithm has a strength equal to the number of bits in its key. In a TPM, the symmetric algorithms usually have key sizes of 128, 192, or 256 bits.

A secure hash algorithm is sort of like an algorithm that encrypts but cannot decrypt. This may sound non-useful, but it has a number of very interesting uses. A secure hash algorithm always produces the same size output, independent of the input. A wonderful property of a secure hash algorithm is that given the input, you always get the same output; but given just the output, you can't calculate the input. The strength of such an algorithm can be calculated two ways:

• The number of tries that would have to be attempted to guarantee finding an input that produces a given hash output. For a welldesigned hash algorithm, this is assumed to be the size of the output in bits.

• The number of tries that would have to be attempted to have a 50% chance of finding two inputs with exactly the same output. For a well-designed hash algorithm, this is half the size of the output in bits. ^{[1]}

Depending on how the secure hash is used, either can be correct; but because cryptographers tend to be P3 people (Paid Professional Paranoids), the latter is generally used for the strength of a well-designed secure hash algorithm.

Asymmetric algorithms are strange at first introduction: The encryption algorithm is different from the decryption algorithm, and the two use different keys, which together form a public and private key pair. There are two asymmetric algorithms you should be concerned about and that are described later in this chapter: RSA (named after its inventors, Rivest, Shamir, and Adleman) and Elliptic Curve Cryptography (ECC).

For asymmetric algorithms, it is difficult to calculate strength corresponding to a particular key size. For RSA, there are tables you can consult. Those tables say that 2048 bits in an RSA asymmetric key corresponds to 112 bits in a symmetric key; 3,076 bits corresponds to 128 bits; and 15,360 bits corresponds to 256 bits of key strength. For ECC, the strength is considered to be half the number of bits of its key's size. Therefore, a 256-bit ECC key is the same strength as a 128-bit symmetric key, and a 384-bit ECC key corresponds to a 192-bit symmetric key.

If brute-force attacks are infeasible due to a large key size, the attacker may seek to analyze the mathematics of the cryptographic algorithm or protocol with the hope of finding a shortcut.

[1] This kind of attack is generally called a birthday attack, because of an old party trick. If there are 23 (which is close to the square root of 365) people in a room, the chances of 2 of them having the same birthday is 50%. If there are substantially more people in the room, the probability rises accordingly. If there are 40 people, the probability is almost 90%.

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