There are two levels to this answer, generic and specific. The specific problem addresses the decryptation of a single encrypted message, the generic the decryptation of all messages produced by a particular encryptation system. The starting point is the presumption that the message actually contains data. That argues for some kind of pattern to it, commonly a language of some sort, which then offers a handle to the decryptation. In fact, this is what we do anyway in reading plaintext, we interpret what is written into semantic meaning our brains can assimilate. What single-use pads do is stop the development of first-level cracks into generic cracks of the entire system, as there should be no common ground between the pads. However, this is not entirely guaranteed, as some kind of seed is needed to generate a pseudo-random sequence. The most important thing about encryptation is not that it preserves the data content eternally, it's that it keeps it hermetic for long enough for its relevance to degrade, so that if, for instance, a military force were to plan a surprise attack, the attack would happen before the code is broken. I'll leave it to others to discuss the maths, however the fact there is a key means that in theory any message is breakable under a brute-force attack. Sometimes double-encryptation actually creates exploitable lacunae, and sometimes just plain sloppy thinking exposes the key. I personally cracked a major system through simple inspection once, working with one of the system managers, so the exploit was rapidly corrected: that was done on a kudos motivation. Then again, I have a genius IQ, was schooled by one of the founding fathers, and actually made systems manager at Dollis Hill, after the code-breakers had left. The inverse of the problem is that the encrypted message must also be decryptable in short order: it's pointless if a time-critical message takes longer to decrypt than the period it is important for. Equally, the decryptation equipment itself must also be secure in all locations, the loss of several cypher machines gave Bletchley Park huge pointers towards a generic crack of Enigma. It's a subset of the approach of the security gurus, whose ideal machine is disconnected from a power supply, encased in lead, covered in a concrete sarcophagus twenty feed thick and dropped into the crater of your nearest volcano (the Frodo solution!). It may be perfectly secure, it's also perfectly unusable. And in the trade-off lies the weakness. A single-use pad is used twice, to code and decode. Lose either and the game is in play. The Man Who Never Was was a case in point, where the code sold the gambit.

There are two levels to this answer: generic and specific. The specific problem addresses the decryption of a single encrypted message, the generic the decryption of all messages produced by a particular encryption system.

The starting point is the presumption that the message actually contains data. That argues for some kind of pattern to it, commonly a language of some sort, which then offers a handle to the decryption. In fact, this is what we do anyway in reading plaintext; we interpret what is written into semantic meaning our brains can assimilate.

What single-use pads do is stop the development of first-level cracks into generic cracks of the entire system, as there should be no common ground between the pads. However, this is not entirely guaranteed, as some kind of seed is needed to generate a pseudo-random sequence.

The most important thing about encryption is not that it preserves the data content eternally, it's that it keeps it hermetic for long enough for its relevance to degrade, so that if, for instance, a military force were to plan a surprise attack, the attack would happen before the code is broken.

I'll leave it to others to discuss the maths, however the fact there is a key means that in theory any message is breakable under a brute-force attack. Sometimes double-encryption actually creates exploitable lacunae, and sometimes just plain sloppy thinking exposes the key. I personally cracked a major system through simple inspection once, working with one of the system managers, so the exploit was rapidly corrected: that was done on a kudos motivation. Then again, I have a genius IQ, was schooled by one of the founding fathers, and actually made systems manager at Dollis Hill, after the code-breakers had left.

The inverse of the problem is that the encrypted message must also be decryptable in short order: it's pointless if a time-critical message takes longer to decrypt than the period it is important for. Equally, the decryption equipment itself must also be secure in all locations, the loss of several cypher machines gave Bletchley Park huge pointers towards a generic crack of Enigma. It's a subset of the approach of the security gurus, whose ideal machine is disconnected from a power supply, encased in lead, covered in a concrete sarcophagus twenty feed thick and dropped into the crater of your nearest volcano (the Frodo solution!). It may be perfectly secure, it's also perfectly unusable. And in the trade-off lies the weakness. A single-use pad is used twice, to code and decode. Lose either and the game is in play. The Man Who Never Was was a case in point, where the code sold the gambit.