Good cryptographic systems should always be designed so that they are as difficult to break as possible. It is possible to build systems that cannot be broken in practice (though this cannot usually be proved). This does not significantly increase system implementation effort; however, some care and expertise is required. There is no excuse for a system designer to leave the system breakable. Any mechanisms that can be used to circumvent security must be made explicit, documented, and brought into the attention of the end users.
In theory, any cryptographic method with a key can be broken by trying all possible keys in sequence. If using brute force to try all keys is the only option, the required computing power increases exponentially with the length of the key. A 32 bit key takes 232 (about 109) steps. This is something anyone can do on his/her home computer. A system with 40 bit keys takes 240 steps - this kind of computation requires something like a week (depending on the efficiency of the algorithm) on a modern home computer. A system with 56 bit keys (such as DES) takes a substantial effort (with a large number of home computers using distributed effort, it has been shown to take just a few months), but is easily breakable with special hardware. The cost of the special hardware is substantial but easily within reach of organized criminals, major companies, and governments. Keys with 64 bits are probably breakable now by major governments, and within reach of organized criminals, major companies, and lesser governments in few years. Keys with 80 bits appear good for a few years, and keys with 128 bits will probably remain unbreakable by brute force for the foreseeable future. Even larger keys are sometimes used.
However, key length is not the only relevant issue. Many ciphers can be broken without trying all possible keys. In general, it is very difficult to design ciphers that could not be broken more effectively using other methods. Designing your own ciphers may be fun, but it is not recommended for real applications unless you are a true expert and know exactly what you are doing.
One should generally be very wary of unpublished or secret algorithms. Quite often the designer is then not sure of the security of the algorithm, or its security depends on the secrecy of the algorithm. Generally, no algorithm that depends on the secrecy of the algorithm is secure. Particularly in software, anyone can hire someone to disassemble and reverse-engineer the algorithm. Experience has shown that the vast majority of secret algorithms that have become
public knowledge later have been pitifully weak in reality.
The key lengths used in public-key cryptography are usually much longer than those used in symmetric ciphers. This is caused by the extra structure that is available to the cryptanalyst. There the problem is not that of guessing the right key, but deriving the matching secret key from the public key. In the case of RSA, this could be done by factoring a large integer that has two large prime factors. In the case of some other cryptosystems it is equivalent to computing the discrete logarithm modulo a large integer (which is believed to be roughly comparable to factoring when the moduli is a large prime number). There are public key cryptosystems based on yet other problems.
To give some idea of the complexity for the RSA cryptosystem, a 256 bit modulus is easily factored at home, and 512 bit keys can be broken by university research groups within a few months. Keys with 768 bits are probably not secure in the long term. Keys with 1024 bits and more should be safe for now unless major cryptographical advances are made against RSA; keys of 2048 bits are considered by many to be secure for decades.
It should be emphasized that the strength of a cryptographic system is usually equal to its weakest link. No aspect of the system design should be overlooked, from the choice algorithms to the key distribution and usage policies.