Cryptographic Primitive Method

Question

Is there any cryptographic primitive bijective (one-to-one and onto) function for creating cryptographic tools like symmetric encryption/decryption, Hash code generator, MAC, HMAC and Random number generator?

Answer 1

As CodesInChaos notes, a secure block cipher satisfies all the criteria in your question:

• An $n$-bit block cipher is a (keyed family of) bijective function(s) on the set of $n$-bit bitstrings.
• A block cipher can be used for symmetric encryption in any of several modes of operation. In some of these modes (notably OFB and CTR) the block cipher effectively acts as a keystream generator for a synchronous stream cipher.
• A block cipher can be used in several ways to construct a one-way compression function, which in turn can be used to construct a cryptographic hash function. Indeed, most commonly used cryptographic hash functions are based on block ciphers.
• A hash function constructed from a block cipher can certainly be used in the HMAC construction, but there are also several ways (such as CBC-MAC, OMAC and PMAC) to construct a MAC directly from a block cipher.
• As noted above, a block cipher in CTR or OFB mode becomes a stream cipher, which is essentially the same thing as a cryptographically secure pseudorandom number generator.

Indeed, insofar as there exists a "universal building block" in modern cryptography, block ciphers are it. Pretty much the only thing they cannot do is public-key cryptography.

That said, block ciphers certainly aren't the only possible building block. For example, a cryptographic hash or pseudorandom function can be used in the Feistel / Luby–Rackoff construction to construct a block cipher, which can then be used to construct all the other things described above.