Cryptographic primitives are well-established, low-level cryptographic algorithms that are frequently used to build cryptographic protocols for computer security systems.[1] These routines include, but are not limited to, one-way hash functions and encryption functions.
When creating cryptographic systems, designers use cryptographic primitives as their most basic building blocks. Because of this, cryptographic primitives are designed to do one very specific task in a precisely defined and highly reliable fashion.
Since cryptographic primitives are used as building blocks, they must be very reliable, i.e. perform according to their specification. For example, if an encryption routine claims to be only breakable with number of computer operations, and it is broken with significantly fewer than operations, then that cryptographic primitive has failed. If a cryptographic primitive is found to fail, almost every protocol that uses it becomes vulnerable. Since creating cryptographic routines is very hard, and testing them to be reliable takes a long time, it is essentially never sensible (nor secure) to design a new cryptographic primitive to suit the needs of a new cryptographic system. The reasons include:
Cryptographic primitives are one of the building blocks of every cryptosystem, e.g., TLS, SSL, SSH, etc. Cryptosystem designers, not being in a position to definitively prove their security, must take the primitives they use as secure. Choosing the best primitive available for use in a protocol usually provides the best available security. However, compositional weaknesses are possible in any cryptosystem and it is the responsibility of the designer(s) to avoid them.
Cryptographic primitives are not cryptographic systems, as they are quite limited on their own. For example, a bare encryption algorithm will provide no authentication mechanism, nor any explicit message integrity checking. Only when combined in security protocols can more than one security requirement be addressed. For example, to transmit a message that is not only encoded but also protected from tinkering (i.e. it is confidential and integrity-protected), an encoding routine, such as DES, and a hash-routine such as SHA-1 can be used in combination. If the attacker does not know the encryption key, they cannot modify the message such that message digest value(s) would be valid.
Combining cryptographic primitives to make a security protocol is itself an entire specialization. Most exploitable errors (i.e., insecurities in cryptosystems) are due not to design errors in the primitives (assuming always that they were chosen with care), but to the way they are used, i.e. bad protocol design and buggy or not careful enough implementation. Mathematical analysis of protocols is, at the time of this writing, not mature. There are some basic properties that can be verified with automated methods, such as BAN logic. There are even methods for full verification (e.g. the SPI calculus) but they are extremely cumbersome and cannot be automated. Protocol design is an art requiring deep knowledge and much practice; even then mistakes are common. An illustrative example, for a real system, can be seen on the OpenSSL vulnerability news page here.