# Mathematical cryptology by Keijo Ruohonen

By Keijo Ruohonen

(translation via Jussi Kangas and Paul Coughlan)

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Extra resources for Mathematical cryptology

Sample text

And a possible eavesdropper won’t get anything out of it without the key. During the decrypting we correspondingly add the same vector b to the encrypted message, since 2b ≡ 0 mod 2. In this way we get the so-called one-time-pad cryptosystem ONE-TIME-PAD. 4 Cryptanalysis The purpose of cryptanalysis is to break the cryptosystem, in other words, to find the decrypting key or encrypting key, or to at least produce a method which will let us get some information out of encrypted messages. In this case it is usually assumed that the cryptanalyzer is an eavesdropper or some other hostile party and that the cryptanalyzer knows which cryptosystem is being used but does not know the key being used.

By applying frequency analysis some KP data can in principle be found, especially if d is relatively small. In a KP attack it is sufficient to find message-blockcryptoblock pairs (i1 , j1 ), . . , (id , jd ) such that the matrices     i1 j1  ..   ..  S =  .  and R =  .  id jd are invertible modulo M. Note that in fact it is sufficient to know one of these matrices is invertible, the other will then also be invertible. Of course S can be directly chosen in a CP attack and R in a CC attack.

Using the trap door information the knapsack problem (a, c) can be restored to its original easily solved form, and in this way the encrypted message can be decrypted. But this does not lead to a strong cryptosystem, in other words, by using the trap door we don’t obtain a disguised knapsack system, whose cryptanalysis would be N Pcomplete, or even very difficult. In fact different variants of KNAPSACK have been noticed to be dangerously weak and so they are not used anymore. g. S ALOMAA . 4 Problems Suitable for Public-Key Encryption As the knapsack problem, the types of problems found useful in public-key encryption are usually problems of number theory or algebra, often originally of merely theoretical interest and quite abstract.