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**Additional info for Algorithmic Information Theory: Mathematics of Digital Information**

**Sample text**

Finally, at the end of this ride across the “land of standards”, we will also have looked closely to the hash algorithm SHA-1, which is a sort of twisted parody of our old friend, the DES. 1 The Data Encryption Standard In 1973, the National Bureau of Standards of the United States invited tenders for a cryptosystem, abutting ﬁnally onto the DES – an ampliﬁed version of a former “family cipher” of IBM, called LUCIFER. After a long series of public discussions and controversies, DES was ﬁnally adopted as standard for data encryption in January 1977.

2) Show: every Huﬀman code is a Shannon code. More precisely: let C be a Huﬀman code (given by its binary tree); then there exists a probability distribution p such that the set of code words of C is the associated Shannon code. (3) Let {A, B, C, D} be an alphabet of four letters, with p(A) ≥ p(B) ≥ p(C) ≥ p(D). (a) Find all associated Huﬀman codes. (b) Give an example of a Shannon code (in choosing appropriate probabilities) which is not a Huﬀman code. (4) Is an optimal binary preﬁx code necessarily a Huﬀman code?

Due to the preﬁx property, we would thus get a better code. The Huﬀman codes are optimal: this is an immediate consequence of the following proposition. Proposition Consider a source S of N states, controlled by the probability distribution p = (p0 , p1 , . . , pN −1 ). Replace the two symbols aj1 and aj2 of smallest probabilities by a single symbol a(j1 ,j2 ) with probability p(j1 ,j2 ) = pj1 +pj2 . Let S be the source of N −1 states we get this way. Let C be an optimal binary preﬁx code for S , and let x be the code word of a(j1 ,j2 ) .