By Ming Li

With this ebook, the authors try to give in a unified remedy an advent to the vital rules and their functions of the Kolmogorov Complexity, the idea facing the amount of knowledge in person gadgets. This e-book is acceptable for both a one- or two-semester introductory path in departments of desktop technological know-how, arithmetic, physics, chance conception and facts, synthetic intelligence, and philosophy. even though the mathematical idea of Kolmogorov complexity comprises subtle arithmetic, the quantity of math one must be aware of to use the notions in commonly divergent parts, is especially little. The authors' goal is to enhance the idea intimately and description quite a lot of illustrative functions. This publication is an try to grab the mass of fragmented wisdom of this interesting thought. bankruptcy 1 is a compilation of fabric at the different notations and disciplines we draw upon which will make the ebook self-contained. The mathematical thought of Kolmogorov complexity is taken care of in chapters 2-4; the purposes are handled in chapters 4-8.

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**Extra info for An Introduction to Kolmogorov Complexity and Its Applications**

**Example text**

Once k3 is found, how do you recover kl and k2? What is the complexity of the whole attack? 6). This time, we are going to mount a chosen-ciphertext attack. The ciphertext C we choose, is the concatenation of four n-bit blocks such that C = (A, A, B , B) (where A, B denote arbitrary blocks of n bits). The four blocks of the corresponding plaintext are denoted Pl to P4. 4 Find a relation between kl, k3, IV1, IV2, PI, P2 and A. Similarly, find a relation between kl, k3, IV1, P3,Pq,A, and B. 5 Deduce a (smart) attack that recovers kl and k3.

Show how a collision on encrypted blocks in CBC mode can leak some information on the plaintexts. What is the complexity of this attack when the block cipher used is DES? What is the complexity if we replace DES by 3DES? How can we protect ourselves against this attack? We now try to transform DES into a block cipher with 128-bit plaintext blocks, that we denote ExtDES. We use a 112-bit key which is split into two DES keys K1 and K2. For this, we define the encryption of a 128-bit block x as follows: rn we split x into two 64-bit halves xr, and rn we let u~ = DESK,(xL) and UR = XR such that x = X L ~ ~ X R DESK, (XR) 21 Conventional Cryptography rn rn we split uLlluR into four 32-bit quarters u l , u2, us, u4 such that UL = u111u2 and UR = u311u4 we let VL = DES;(:(U~ IIu4) and VR = D E S K : ( U ~ ~ ~ U ~ ) we split v ~ l l vinto ~ four 32-bit quarters v1,v2,v3,v4 such that VL = v111v2 and VR = v311v4 we let YL = DESK, (vlllv4) and y~ = DESK, (v311va) , ~of~ x~ , we define y = yL[lyRas the encryption E x ~ D E S ~(x) 4 Draw a diagram of ExtDES.

19 Conventional Cryptography the standard DES decryption of the message 0 under all 256 keys. Then we use a chosen-plaintext attack to build a second table containing the 256 ciphertexts resulting from box encryptions of the elements of the first table. Given these two tables, one can find both K1 and K2 used by the encryption box. Explain how one may proceed. The whole attack should take no more than 260 DES encryptions (or decryptions) and no more than 261 bytes of memory. D Exercise 6 Solution on page 37 *Exhaustive Search on 3DES We consider 3DES with three independent keys.