OKlibrary  0.2.1.6
general.hpp File Reference

Investigating boolean functions representing permutations of {0,1}^n. More...

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## Detailed Description

Investigating boolean functions representing permutations of {0,1}^n.

Todo:
Generalities
• For a natural number n >= 0 one considers the set V_n = {0,1}^n of all bit-vectors.
• The functions to be studied here are permutations f of V_n.
• Permutations are represented as relations by boolean functions f^*: {0,1}^(2n), that is, f^*(x,y) = 1 iff f(x) = y.
• These boolean function have exactly one DNF-representation (using 2^n clauses of length 2*n).
• The CNF-representations are of special interest here.
• In ComputerAlgebra/Satisfiability/Lisp/FiniteFunctions/Permutations.mac we have perm2cnffcs(P), which for a permutation P in list-form creates the full cnf-fcs.
• In ComputerAlgebra/Satisfiability/Lisp/Primality/PrimeImplicatesImplicants.mac we have all_minequiv_bvs_fcs to compute then all minimum CNF's (that is, by all_minequiv_bvs_fcs(perm2cnffcs(P)).
• A simpler function there is rsubsumption_hg_full_fcs: By rsubsumption_hg_full_fcs(FF)[2] we obtain the list of necessary prime implicates (that is, by rsubsumption_hg_full_fcs(perm2cnffcs(P))[2]).
• The permutations are (according to our standard representation) not permutations of bit-vectors, but of the numbers 1,..,2^n; which are considered as bit-vectors using binary representation of numbers.
• So all permutations are created by permutations(setn(2^n)).
• There are 2^(2^(2n)) boolean functions altogether of 2n arguments, while there are (2^n)! such permutations f.
• V_n can be considered as an n-dimensional algebra over F_2 (the two-element field), consisting of the n-dimensional vectorspace over F_2 plus the field-structure of V_n as a field of order 2^n.
• Special permutations of interest are as follows (based on the algebraic structures).
1. Linear automorphisms, of which there are order_gl(n,2).
2. Special linear automorphisms are given by multiplication with non-zero field elements, of which there are 2^n-1.
3. Translations, of which there are 2^n.
4. Affine automorphisms, of which there are order_gl(n,2) * 2^n.
5. The multiplicative inverse x -> x^-1, extended by 0 -> 0.
6. The compositions of inversions with affine automorphisms.
7. For such compositions, is there a fundamental difference between first applying the inversion or first applying the affine automorphism?
We need general tools (including representations and conversions) to handle these objects.
• Of course, besides these "algebraic permutations" we need to study random permutations.
• Isomorphism of boolean functions:
1. Most powerful is to admit permutations of variables and individual flips of variables.
2. A basic question is how this compares with conjugatedness of permutations (equivalent to having the same cycle type)?
Todo:
What to investigate for these boolean functions
• The number of prime implicates is of importance, the number of necessary clauses amongst them, the size of minimum CNF representations, and their number.
• At Maxima-level this is computed as available by investigate_permutations(n) (in Experimentation/Investigations/BooleanFunctions/Permutations.mac).
• Shortest r_k-compressions of the set of prime implicates are of high interest (since we expect them to be most useful for their use in SAT-translations). See rand_rbase_cs(F,r) in ComputerAlgebra/Satisfiability/Lisp/Reductions/RBases.mac.
• Also OBDD-representations are to be studied.
• And shortest circuit-representations.
Todo:
Isomorphism types
Todo:
Trivial cases
• The case n=0: investigate_permutations(0) yields "1 1 1 1".
• The case n=1: investigate_permutations(1) yields two times "2 2 2 1".
Todo:
The case n=2
• Here we have just (2^2)! = 24 permutations altogether, so we can conveniently list them all (by permutations({1,2,3,4})).
• We get
```h2 : investigate_permutations(2)\$
ev_ip(h2);

[4,4,4,1] 8
[10,0,5,2] 16
```
• So we have two cases: One with 4 prime implicates, which all are necessary, and one with 10 prime implicates, none of which are necessary, and having 2 minimum representations, each with 5 clauses.
• The first case is given by the identity, the second case by the permutation [1,2,4,3]:
```last(ev_hm(h2,[4,4,4,1]));
[1,2,3,4]
last(ev_hm(h2,[10,0,5,2]));
[1,2,4,3]

all_minequiv_bvs_fcs(perm2cnffcs([1,2,3,4]));
[{{-4,2},{-3,1},{-2,4},{-1,3}}]
all_minequiv_bvs_fcs(perm2cnffcs([1,2,4,3]));
[{{-4,-3,2},{-4,-2,1},{-3,-2,4},{-1,3},{1,2,4}},
{{-4,-2,3},{-4,-1,2},{-3,1},{-2,-1,4},{2,3,4}}]
```
• The identity is treated in general below. For the second case (a transposition) actually the two minimum clause-sets are disjoined (a partitioning of the set of all prime-clauses).
• The number of linear automorphisms is order_gl(2,2) = 6, while there are 2^2=4 translationen, which makes 24 affine automorphisms altogether.
• So here every permutation is an affine automorphism.
Todo:
The case n=3
• Here we have just (2^3)! = 40320 permutations altogether, so we can still consider them all (algorithmically).
• Experiment:
```oklib_monitor:true;
h3 : investigate_permutations(3)\$
[1,2,3,4,5,6,7,8] [6,6,6,1]
[1,2,3,4,5,6,8,7] [20,4,10,256]
[1,2,3,4,5,7,6,8] [26,2,10,288]
[1,2,3,4,5,7,8,6] [30,2,11,512]
[1,2,3,4,6,5,8,7] [12,2,7,2]
[1,2,3,4,6,7,8,5] [28,0,10,48]
[1,2,3,4,6,8,5,7] [26,0,9,2]
[1,2,3,4,8,7,6,5] [26,0,8,2]
[1,2,3,5,4,6,7,8] [36,0,14,870]
[1,2,3,5,4,6,8,7] [36,0,12,2]
[1,2,3,5,4,7,6,8] [44,0,12,485]
[1,2,3,5,4,7,8,6] [38,0,12,106]
[1,2,3,5,6,4,7,8] [36,0,12,1]
[1,2,3,5,6,4,8,7] [34,0,10,6]
[1,2,3,5,6,7,4,8] [41,0,12,592]
[1,2,3,5,6,7,8,4] [38,0,11,82]
[1,2,3,5,6,8,4,7] [38,0,11,1]
[1,2,3,5,8,4,6,7] [38,0,11,67]
[1,2,3,5,8,4,7,6] [37,0,12,4]
[1,2,3,5,8,7,6,4] [40,0,10,8]
[1,2,3,6,5,4,8,7] [40,0,11,64]
[1,2,3,6,5,7,4,8] [39,0,12,2088]
[1,2,3,6,5,7,8,4] [41,0,11,60]
[1,2,3,6,7,8,5,4] [28,2,10,192]
[1,2,3,6,8,7,4,5] [40,0,10,64]
[1,2,3,6,8,7,5,4] [44,0,11,100]
[1,2,3,8,6,5,7,4] [35,2,12,7680]
[1,2,3,8,6,7,5,4] [45,0,12,576]
[1,2,4,3,6,5,7,8] [36,0,10,4]
[1,2,4,3,6,7,5,8] [40,0,11,70]
[1,2,4,3,7,8,6,5] [30,0,9,6]
[1,2,4,5,6,3,7,8] [48,0,12,722]
[1,2,4,5,6,7,3,8] [47,0,12,694]
[1,2,4,5,7,8,6,3] [42,0,10,3]
[1,2,4,5,8,7,3,6] [44,0,11,84]
[1,2,4,7,6,8,3,5] [48,0,11,2]
[1,2,4,7,8,5,6,3] [51,0,12,1152]
[1,4,6,7,8,5,3,2] [40,0,10,20]
XXX
```
Computation aborted (last output for permutation 1897, and then until permutation 3277 no new cases; unclear whether there might be more cases --- faster computation, at C++ level is needed).
• Some permutations take more than a minute to process, while some take only a few seconds? It can't be the small hash-map? It seems it is just that some permutations are more difficult than others.
• See "Write analyse_all_permutations" in Investigations/BooleanFunctions/plans/general.hpp for a tool at C++ level.
• Some special cases of permutations p:
1. Some transpositions (2-cycles):
```P : [2,1,3,4,5,6,7,8];
evalpermasbf(P);
[20,4,10,256]
P : [1,3,2,4,5,6,7,8];
evalpermasbf(P);
[26,2,10,288]
P : [8,2,3,4,5,6,7,1];
evalpermasbf(P);
[36,0,14,870]
```
2. There are 8*7/2=28 transpositions (in general 2^n*(2^n-1)/2):
```t3 : investigate_transpositions(3);
ev_ip(t3);
[20,4,10,256] 12
[26,2,10,288] 12
[36,0,14,870] 4
```
3. The isomorphism-type of the boolean function should be just determined by the number of bits flipped in the transposition?
4. Then in general there would be exactly n isomorphism types, corresponding to 1 <= f <= n flipped bits.
5. And for f flipped bits there would be 2^n*binom(n,i)/2 permutations belonging to it.
6. So in this case, for n=3: i=1 -> 8*3/2=12, i=2 -> 8*3/2=12, i=3 -> 8*1/2=4.
7. It remains the task to find for these 3 cases (in general n cases) all prime implicates, and to determine their structure (irredundant clauses, minimum representations).
8. Special transitive permutations:
```P : cyclepres2perl({[1,2,3,4,5,6,7,8]});
[2,3,4,5,6,7,8,1]
evalpermasbf(P);
[28,0,10,48]

P : cyclepres2perl({[1,4,7,2,5,8,3,6]});
[4,5,6,7,8,1,2,3]
evalpermasbf(P);
[28,0,10,48]

P : cyclepres2perl({[1,2,6,5,4,3,8,7]});
[2,6,8,3,4,5,1,7]
evalpermasbf(P);
[41,0,12,592]
\verbatim
</li>
<li> There are 7! = 5040 transitive permutations; perhaps this class is
too large? Or can we say something, e.g., what are the most complicated
boolean functions amongst them? </li>
<li> The maximal order of a permutation here is 3*5=15:
\verbatim
P : cyclepres2perl({[1,2,3],[4,5,6,7,8]});
[2,3,1,5,6,7,8,4]
evalpermasbf(P);
[38,0,11,67]

P : cyclepres2perl({[1,8,3],[2,6,4,7,5]});
[8,6,1,7,2,4,5,3]
evalpermasbf(P);
[35,2,12,7680]
```
• The number of linear automorphisms is order_gl(3,2) = 168, while there are 2^3=8 translations, which makes 1344 affine automorphisms altogether.
Todo:
The case n=8
• This case is especially interesting because of AES; see Cryptography/AdvancedEncryptionStandard/plans/FieldMulInvestigations.hpp and Cryptography/AdvancedEncryptionStandard/plans/SboxInvestigations.hpp.
• Here we have (2^8)! ~ 8.578*10^506 permutations altogether (while there are ~ 2.003*10^19728 boolean functions (in 16 variables).
• The number of linear automorphisms is order_gl(8,2) ~ 5.348*10^18.
• Special linear (affine) automorphisms to consider are the ones involved in the S-box and its inverse.
Todo:
Prime implicates of simple permutations
• The identity function:
• For the identity function id over {0,1}^n the prime implicates of id^* are exactly the clauses {v_{1,i},-v_{2,i}} and {-v_{1,i},v_{2,i}} for all 1 <= i <= n.
• These clauses encode the equality of the variables in the input set with those of the output set using binary constraints.
• We have here the case of a parallel and independent composition of n boolean functions, namely v_{1,i} = v_{2,i}.
• In general, for such independent parallel compositions the prime clauses are just taken alltogether.
• The negation:
1. x -> neg x componentwise.
2. Here we have parallel independent composition of v_{i,1} = not v_{i,2}, and so the prime implicates are exactly the clauses {v_{1,i},v_{2,i}} and {-v_{1,i},-v_{2,i}} for all 1 <= i <= n.
Todo:
Sampling random permutations
• The data should be at least the total number of clauses, the number of clauses of each size, and the number of irredundant clauses (easy to compute in this case, since the truth table is given).
• See "Preparations for computing optimum representations" in OKlib/Satisfiability/FiniteFunctions/plans/QuineMcCluskeySubsumptionHypergraph.hpp for discussion of adding output of necessary clauses to the subsumption hypergraph generator.
• DONE (see analyse_random_permutations) A script is to be written, which generates random permutations (simplest via Maxima), computes then (by our C++ program) the set of all prime implicates, computes basic measurements and puts it into a file, for evaluation by R.
Todo:
First considerations of random permutation
• n=8:
```set_random(1)\$
P : random_permutation(create_list(i,i,1,256))\$
output_perm_fullcnf_stdname(P)\$

> QuineMcCluskeySubsumptionHypergraph-n16-O3-DNDEBUG Permutation_full.cnf > SP.cnf

> cat Permutation_full.cnf_primes | ExtendedDimacsFullStatistics-O3-DNDEBUG
n non_taut_c red_l taut_c orig_l comment_count finished_bool
16 140925 1035782 0 1035782 1 1
length count
5 2
6 3556
7 85110
8 51647
9 610

> cat SP.cnf | ExtendedDimacsFullStatistics-O3-DNDEBUG n > SP_stat

n non_taut_c red_l taut_c orig_l comment_count finished_bool
140925 65280 60521472 0 60521472 1 1

> summary(E)
length           count
Min.   : 177.0   Min.   :  1.00
1st Qu.: 575.8   1st Qu.:  9.00
Median : 876.5   Median : 34.00
Mean   : 876.9   Mean   : 54.22
3rd Qu.:1177.2   3rd Qu.:100.00
Max.   :1593.0   Max.   :175.00
> plot(E)
```
• Looks similar to the AES Sbox (see "Basic data" in Investigations/Cryptography/AdvancedEncryptionStandard/plans/Representations/Sbox_8.hpp).
• 1557 experiments for n=8:
```shell> \${OKlib}/Experimentation/Investigations/BooleanFunctions/analyse_random_permutations 8 1
^C
```
and then:
```R> E = read_experiment_dirs("random_perm", list("e","seed"), "Permutation_full.cnf_primes_stats", header=TRUE, skip=2)
R> ET = rows2columns_df(E, "length", "count", list("e","seed"))
R> summary(ET)
5
Min.   : 0.000
1st Qu.: 3.000
Median : 5.000
Mean   : 5.314
3rd Qu.: 7.000
Max.   :20.000
6              7               8               9
Min.   :2824   Min.   :76750   Min.   :39658   Min.   :502.0
1st Qu.:3999   1st Qu.:81310   1st Qu.:45265   1st Qu.:657.0
Median :4272   Median :82419   Median :47247   Median :709.0
Mean   :4289   Mean   :82355   Mean   :47256   Mean   :713.6
3rd Qu.:4563   3rd Qu.:83483   3rd Qu.:49195   3rd Qu.:765.0
Max.   :5737   Max.   :86368   Max.   :58320   Max.   :997.0
10
Min.   :0.0000
1st Qu.:0.0000
Median :0.0000
Mean   :0.1933
3rd Qu.:0.0000
Max.   :4.0000
16    e
Min.   :0   16:1557
1st Qu.:0
Median :0
Mean   :0
3rd Qu.:0
Max.   :0
```
Note the experiment was stopped and restarted with a different seed at one point.
• Note here that not all permutations have prime implicates of length 5 or length 10, but all have length 6,7,8, and 9.
• We can have permutations without any length 5 or length 10 prime implicates:
```R> E2[E2[6] == 0,]
0 1 2 3 4 5    6     7     8   9 10 11 12 13 14 15 16       seed e
392   0 0 0 0 0 0 3522 85462 51359 746  0  0  0  0  0  0  0         12 16
2415  0 0 0 0 0 0 3471 84533 52137 746  0  0  0  0  0  0  0 2147483733 16
4098  0 0 0 0 0 0 3662 83895 52057 603  1  0  0  0  0  0  0 2147483832 16
6036  0 0 0 0 0 0 4143 82732 49082 739  0  0  0  0  0  0  0 2147483946 16
6376  0 0 0 0 0 0 3503 84905 52757 753  0  0  0  0  0  0  0 2147483966 16
6580  0 0 0 0 0 0 3472 85034 51732 649  0  0  0  0  0  0  0 2147483978 16
7141  0 0 0 0 0 0 3839 82370 51682 722  0  0  0  0  0  0  0 2147484011 16
8892  0 0 0 0 0 0 3878 82136 53194 837  0  0  0  0  0  0  0 2147484114 16
12921 0 0 0 0 0 0 3310 83318 57665 749  0  0  0  0  0  0  0 2147484351 16
15624 0 0 0 0 0 0 3842 83251 50269 713  0  0  0  0  0  0  0 2147484510 16
17188 0 0 0 0 0 0 3616 85329 49868 595  0  0  0  0  0  0  0 2147484602 16
17460 0 0 0 0 0 0 3427 83065 55923 770  0  0  0  0  0  0  0 2147484618 16
20962 0 0 0 0 0 0 3611 85249 50021 654  0  0  0  0  0  0  0 2147484824 16
21285 0 0 0 0 0 0 3727 84473 51727 761  0  0  0  0  0  0  0 2147484843 16
23835 0 0 0 0 0 0 4444 83594 45167 634  0  0  0  0  0  0  0 2147484993 16
24804 0 0 0 0 0 0 3702 84340 50569 613  0  0  0  0  0  0  0 2147485050 16
24821 0 0 0 0 0 0 3689 84816 49657 674  1  0  0  0  0  0  0 2147485051 16
25263 0 0 0 0 0 0 4063 83608 47128 604  0  0  0  0  0  0  0         35 16
25297 0 0 0 0 0 0 4327 82446 48222 725  1  0  0  0  0  0  0         37 16
25739 0 0 0 0 0 0 3862 83246 51154 581  0  0  0  0  0  0  0         60 16
```
In fact most of the permutations we tested didn't have a clause of length 10 when it had a clause of length 5 (as it would be expected if the events would be independent).
• For the total number of prime implicates we have:
```R>  min(as.data.frame(addmargins(as.matrix(E2)))\$Sum)
[1] 124400