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Research on the VMPC Algorithms

1. VMPC One-Way Function
2. VMPC Stream Cipher
3. VMPC Key Scheduling Algorithm
4. VMPC-MAC scheme

 





1. VMPC One-Way Function

1) Description of the algorithm of inverting a k-level VMPC function for n-element permutations, with example average running times; by Bartosz Zoltak, more here

2) Observation that for a random permutation P of n>1 elements and 1-level VMPC function (Q[x]=P[P[P[x]]+1]), the event Q[x]=P[x+1] occurs with probability 2/n (rather than 1/n); reported by Francois Grieu, more here





2. VMPC Stream Cipher

1) Average computational effort required to break the VMPC Stream Cipher - by recovering the cipher's internal state from the generated output - is approximated at over 2^900 computational operations; by Bartosz Zoltak, more here

2) Single-outputs probabilities. Frequencies of occurrence of each of the 256 possible output-values showed no statistically significant deviation from the expected value of 1 / 2^8; by Greg Rose, more here

3) Single-outputs on given positions probabilities. Frequencies of occurrence of each possible output-value on each of the possible 256 positions (for each possible value of n) showed no statistically significant deviation from the expected value of 1 / 2^16; by Greg Rose, more here

4) Digraph probabilities. Frequencies of occurrence of each possible pair of neighboring outputs showed no statistically significant deviation from the expected value of 1 / 2^16; by Bartosz Zoltak, more here

5) Digraphs on given positions probabilities. Frequencies of occurrence of each possible pair of neighboring outputs on each of the possible 256 positions (for each possible value of n) showed no statistically significant deviation from the expected value of 1 / 2^24; by Bartosz Zoltak and Greg Rose, more here

6) Trigraph probabilities. Frequencies of occurrence of each possible set of three consecutive outputs showed no statistically significant deviation from the expected value of 1 / 2^24; by Bartosz Zoltak and Greg Rose, more here

7) First outputs probabilities. Frequencies of occurrence of each possible value of each of the first 256 bytes generated directly after running the VMPC Key Scheduling Algorithm showed no statistically significant deviation from the expected frequency of 1 / 2^8, by Bartosz Zoltak, more here

8) Equal neighboring outputs probabilities. Frequencies of occurrence of situations where there occurs a given number (0,1,2,3,4,5 and over 5) of direct (generated consecutively) and indirect (separated by one more output) equal neighboring outputs in the consecutive 256-byte sub-streams of the cipher's output and the average total number of direct and indirect equal neighbors - showed no statistically significant deviation from their expected values, by Bartosz Zoltak, more here

9) Short cycles. Analysis of scaled-down variants of the cipher show that probability of entering a short cycle for the full cipher is negligibly low.

9.1) Probability of entering a cycle shorter than 2^1000 is estimated to be about 1 / 2^700, by Bartosz Zoltak, more here

9.2) No short cycles analogous to the Finney states defined for the alleged RC4 algorithm can occur in the VMPC Stream Cipher; by Bartosz Zoltak, more here

10) Binary derivatives of bit output sequences probabilities. Distributions of first, second and third binary derivatives of all 7 bits output sequences (for 7-bit words) were tested. None of the measured frequencies showed a statistically significant deviation from its expected value of 1 / 2, by Bartosz Zoltak, more here

11) The DIEHARD battery of statistical tests by George Marsaglia reported no bias in the ouput generated by the Cipher, more here

12) The NIST statistical tests suite reported no bias in the output generated by the Cipher, more here

13) Battery of statistical tests by David Sexton reported no bias in the ouput generated by the Cipher, more here





3. VMPC Key Scheduling Algorithm

Diffusion of changes of the cryptographic key. A change of one bit of the cryptographic key of size 128, 256 and 512 bits shows to cause a random-like change in the generated permutation. Points 1,2,3. Tests were run without the Initialization Vector.

1) Frequencies of occurrence of situations where in two permutations, generated from keys differing in one byte, there occurs a given number (0,1,2,3,4,5) of equal elements in the corresponding positions and the average number of equal elements in the corresponding positions - showed no statistically significant deviation from their expected values; by Bartosz Zoltak, more here

2) Frequencies of occurrence of situations where in two 256-byte streams generated by the VMPC Stream Cipher directly after running the VMPC KSA for keys differing in one byte, there occurs a given number (0,1,2,3,4,5) of equal elements in the corresponding positions and the average number of equal elements in the corresponding positions - showed no statistically significant deviation from their expected values; by Bartosz Zoltak, more here

3) Frequencies of occurrence of situations where the elements in each of the corresponding positions of the permutations, generated from keys differing in one byte, are equal - showed no statistically significant deviation from their expected values, by Bartosz Zoltak, more here





4. VMPC-MAC scheme

1) Description of the best known forgery attack against the VMPC-MAC scheme; by Bartosz Zoltak, more here


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