Relation With Key Entropy and Plaintext Redundancy
The unicity distance can also be defined as the minimum amount of ciphertext-only required to permit a computationally unlimited adversary to recover the unique encryption key.
The expected unicity distance is accordingly:
where U is the unicity distance, H(k) is the entropy of the key space (e.g. 128 for 2128 equiprobable keys, rather less if the key is a memorized pass-phrase).
D is defined as the plaintext redundancy in bits per character.
Now an alphabet of 32 characters can carry 5 bits of information per character (as 32 = 25). In general the number of bits of information is lg N, where N is the number of characters in the alphabet. So for English each character can convey lg 26 = 4.7 bits of information. Remember that lg is meant as the logarithm for base two in this case. See Binary logarithm for details.
However the average amount of actual information carried per character in meaningful English text is only about 1.5 bits per character. So the plain text redundancy is D = 4.7 − 1.5 = 3.2.
Basically the bigger the unicity distance the better. For a one time pad, given the unbounded entropy of the key space, we have, which is consistent with the one-time pad being theoretically unbreakable.
For a simple substitution cipher, the number of possible keys is 26! = 4.0329 × 1026, the number of ways in which the alphabet can be permuted. Assuming all keys are equally likely, H(k) = lg(26!) = 88.4 bits. For English text D = 3.2, thus U = 88.4/3.2 = 28.
So given 28 characters of ciphertext it should be theoretically possible to work out an English plaintext and hence the key.
Read more about this topic: Unicity Distance
Famous quotes containing the words relation with, relation, key and/or entropy:
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)
“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and centripetal forces, so the hours should be instructed by the ages and the ages explained by the hours.”
—Ralph Waldo Emerson (1803–1882)
“Every revolution was first a thought in one man’s mind, and when the same thought occurs in another man, it is the key to that era.”
—Ralph Waldo Emerson (1803–1882)
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)