From e944a6e5c1d0ba117954365dfedad16f183cf681 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Wed, 22 Jun 2016 15:44:41 +0200
Subject: [PATCH]  intro principalement

---
 biblio.bib                            |  17 +-
 chaos.tex                             |   3 +-
 evalPRNG/calculeStopTime.py           |  27 +-
 evalPRNG/compareFonctionMixingTime.py |  14 +-
 evalPRNG/texput.log                   |  16 +-
 generating.tex                        |  19 +-
 hamilton.tex                          | 166 ++++-
 intro.tex                             | 195 +++---
 main.aux                              |  75 ++-
 main.bbl                              |  12 +-
 main.blg                              |  66 +-
 main.log                              | 891 ++++++++++----------------
 main.pdf                              | Bin 613533 -> 495003 bytes
 main.tex                              |   9 +-
 preliminaries.tex                     |   4 +-
 stopping.tex                          | 133 ++--
 16 files changed, 809 insertions(+), 838 deletions(-)

diff --git a/biblio.bib b/biblio.bib
index 0857ae7..74701ae 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -17,22 +17,6 @@
   OPTmonth =     {},
 }
 
-@inproceedings{chgw14oip,
-inhal = {no},
-domainehal = {INFO:INFO_DC, INFO:INFO_CR, INFO:INFO_MO, INFO:INFO_SE},
-equipe = {ie},
-classement = {COM},
-author = {Couchot, Jean-Fran\c{c}ois and H\'eam, Pierre-Cyrille and Guyeux, Christophe and Wang, Qianxue and Bahi, Jacques},
-title = {Pseudorandom Number Generators with Balanced Gray Codes},
-booktitle = {Secrypt 2014, 11th Int. Conf. on Security and Cryptography},
-pages = {469--475},
-address = {Vienna, Austria},
-month = aug,
-date = {28-30 aout},
-year = 2014,
-note = {Position short paper},
-
-}
 
 @Misc{GridComp,
   OPTkey =       {},
@@ -917,6 +901,7 @@ year = 2013,
   pages =	"267--272",
   URL =  	"http://dx.doi.org/10.1016/j.ipl.2008.10.015",
 }
+
 @inproceedings{DBLP:conf/secrypt/CouchotHGWB14,
   author    = {Jean{-}Fran{\c{c}}ois Couchot and
                Pierre{-}Cyrille H{\'{e}}am and
diff --git a/chaos.tex b/chaos.tex
index d5ab5f9..122c0ad 100644
--- a/chaos.tex
+++ b/chaos.tex
@@ -525,7 +525,8 @@ and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$ is strongly connected.
   and thus $\Gamma_{\{b\}}(f_0)$ is not strongly connected.
 \end{proof}
 
-The next section shows how to generate functions and a iteration number $b$
+The next section recalls a general scheme to produce
+functions and a iteration number $b$
 such that $\Gamma_{\{b\}}$ is strongly connected.
 
 
diff --git a/evalPRNG/calculeStopTime.py b/evalPRNG/calculeStopTime.py
index c9f59c3..e011564 100644
--- a/evalPRNG/calculeStopTime.py
+++ b/evalPRNG/calculeStopTime.py
@@ -24,7 +24,7 @@ def dec(ch,n):
             acc = acc + 2**(n-i-1)        
     return acc
 
-"""
+
 def MarkovMatrixUnPas(fbin,n,p2n,lp2nm1):
     MM = np.zeros(p2n*p2n).reshape((p2n,p2n))
     # diagonal
@@ -36,9 +36,10 @@ def MarkovMatrixUnPas(fbin,n,p2n,lp2nm1):
             ipb =[_ for _ in ib]
             ipb[j] = image[j]
             MM[i, dec(ipb,n)] +=float(1)/(2*n)
+    print MM
     return MM
-"""
 
+"""
 def MarkovMatrixUnPas(fbin,n,p2n,lp2nm1):
     MM = np.zeros(p2n*p2n).reshape((p2n,p2n))
     # diagonal
@@ -90,8 +91,8 @@ def MarkovMatrixSaut(fbin,n,p2n,lp2nm1):
     return MM
 
                 
-
-    
+"""
+"""    
 
     # f is the binary function
     # b is the number to iterate
@@ -107,24 +108,28 @@ def MarkovMatrixSaut(fbin,n,p2n,lp2nm1):
             xp[st] = image[st]
         #print "xp     ",xp
     return xp
-    
+"""    
         
 
 ## n'est pas un circuit hmiltonien
 lf= []
 #lf +=[ [13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]] #4
-#lf += [[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4],[29, 22, 21, 30, 19, 27, 24, 28, 7, 20, 5, 4, 23, 26, 25, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 1, 6, 11, 18, 0,16]] # essai des 2 elements à 5
-
-
+#lf+=[[29, 22, 21, 24, 31, 30, 27, 26, 7, 23, 5, 28, 3, 19, 25, 17, 13, 12, 9, 8, 15, 2, 1, 10, 6, 14, 4, 20, 11, 18, 0, 16]]
 
+#lf += [[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4],[29, 22, 21, 30, 19, 27, 24, 28, 7, 20, 5, 4, 23, 26, 25, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 1, 6, 11, 18, 0,16]] # essai des 2 elements a 5
 
+lf +=[
+    [29, 22, 21, 30, 19, 27, 24, 28, 7, 20, 5, 4, 23, 26, 25, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 1, 6, 11, 18, 0, 16],#sylvain 1
+    [29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4],#sylvain 2
+    [29, 22, 21, 24, 31, 30, 27, 26, 7, 23, 5, 28, 3, 19, 25, 17, 13, 12, 9, 8, 15, 2, 1, 10, 6, 14, 4, 20, 11, 18, 0, 16], #mini avec 7
+    [29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4]] #5
+    
 
 
 
-#f += [[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4]] #5
 lf +=  [[55, 60, 45, 44, 58, 62, 61, 48, 53, 50, 52, 36, 59, 34, 33, 49, 15, 42, 47, 46, 35, 10, 57, 56, 7, 54, 39, 37, 51, 2, 1, 40, 63, 26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13, 12, 29, 8, 43, 14, 41, 0, 5, 38, 4, 6, 11, 3, 9, 32]] #6
 
-
+"""
 lf += [[111, 94, 93, 116, 122, 90, 125, 88, 115, 126, 119, 84, 123, 98, 81, 120, 109, 106, 105, 110, 99, 107, 104, 72, 71, 118, 117, 96, 103, 102, 113, 64, 79, 86, 95, 124, 83, 91, 121, 24, 85, 22, 69, 20, 19, 114, 17, 112, 77, 76, 13, 108, 74, 10, 9, 73, 67, 66, 101, 100, 75, 82, 97, 0, 127, 54, 57, 62, 51, 59, 56, 48, 53, 38, 37, 60, 55, 58, 33, 49, 63, 44, 47, 40, 42, 46, 45, 41, 35, 34, 39, 52, 43, 50, 32, 36, 29, 28, 61, 92, 26, 18, 89, 25, 87, 30, 23, 4, 27, 2, 16, 80, 31, 78, 15, 14, 3, 11, 8, 12, 5, 70, 21, 68, 7, 6, 65, 1]] #7
 lf += [[223, 190, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227, 178, 240, 248, 237, 236, 253, 172, 203, 170, 201, 168, 229, 166, 165, 244, 163, 242, 241, 192, 215, 220, 205, 216, 218, 222, 221, 208, 213, 210, 212, 214, 219, 211, 217, 209, 239, 202, 207, 140, 139, 234, 193, 204, 135, 196, 199, 132, 194, 130, 225, 200, 159, 62, 185, 252, 59, 250, 169, 56, 191, 246, 245, 52, 243, 50, 176, 48, 173, 238, 189, 44, 235, 42, 137, 184, 231, 38, 37, 228, 35, 226, 177, 224, 151, 156, 141, 152, 154, 158, 157, 144, 149, 146, 148, 150, 155, 147, 153, 145, 175, 206, 143, 136, 11, 142, 129, 8, 7, 198, 197, 4, 195, 2, 161, 160, 255, 124, 109, 108, 122, 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106, 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65, 104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75, 67, 73, 96, 55, 58, 45, 188, 51, 186, 61, 40, 119, 182, 181, 53, 179, 54, 33, 49, 15, 174, 47, 60, 171, 46, 57, 32, 167, 6, 36, 164, 43, 162, 1, 0, 63, 26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13, 10, 29, 14, 3, 138, 41, 12, 39, 134, 133, 5, 131, 34, 9, 128]]#8
 
@@ -185,7 +190,7 @@ def traite_f(f):
 
 
 for f in lf: 
-    dev = 1E-8
+    dev = 1E-12
     n,cpt1,cpt2 = traite_f(f)
     print f, n,cpt1,cpt2
     print 8*n*n + 4*n*log(n+1)#,8*n*n + (n+2)*(log(n)+2)
diff --git a/evalPRNG/compareFonctionMixingTime.py b/evalPRNG/compareFonctionMixingTime.py
index d5dc782..faaebd8 100644
--- a/evalPRNG/compareFonctionMixingTime.py
+++ b/evalPRNG/compareFonctionMixingTime.py
@@ -25,7 +25,7 @@ def dec(ch,n):
             acc = acc + 2**(n-i-1)        
     return acc
 
-
+"""
 def MarkovMatrixUnPas(fbin,n,p2n,lp2nm1):
     print p2n
     MM = np.zeros((p2n,p2n))
@@ -93,7 +93,7 @@ def MarkovMatrixSaut(fbin,n,p2n,lp2nm1):
             MM[i, dec(ipb,n)] +=float(1)/(2**n)
     return MM
 
-"""                
+               
 
     
 
@@ -107,12 +107,16 @@ lf= []
 
 #lf += [[111, 94, 93, 116, 122, 90, 125, 88, 115, 126, 119, 84, 123, 98, 81, 120, 109, 106, 105, 110, 99, 107, 104, 72, 71, 118, 117, 96, 103, 102, 113, 64, 79, 86, 95, 124, 83, 91, 121, 24, 85, 22, 69, 20, 19, 114, 17, 112, 77, 76, 13, 108, 74, 10, 9, 73, 67, 66, 101, 100, 75, 82, 97, 0, 127, 54, 57, 62, 51, 59, 56, 48, 53, 38, 37, 60, 55, 58, 33, 49, 63, 44, 47, 40, 42, 46, 45, 41, 35, 34, 39, 52, 43, 50, 32, 36, 29, 28, 61, 92, 26, 18, 89, 25, 87, 30, 23, 4, 27, 2, 16, 80, 31, 78, 15, 14, 3, 11, 8, 12, 5, 70, 21, 68, 7, 6, 65, 1]] #7
 
-lf += [[223, 190, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227, 178, 240, 248, 237, 236, 253, 172, 203, 170, 201, 168, 229, 166, 165, 244, 163, 242, 241, 192, 215, 220, 205, 216, 218, 222, 221, 208, 213, 210, 212, 214, 219, 211, 217, 209, 239, 202, 207, 140, 139, 234, 193, 204, 135, 196, 199, 132, 194, 130, 225, 200, 159, 62, 185, 252, 59, 250, 169, 56, 191, 246, 245, 52, 243, 50, 176, 48, 173, 238, 189, 44, 235, 42, 137, 184, 231, 38, 37, 228, 35, 226, 177, 224, 151, 156, 141, 152, 154, 158, 157, 144, 149, 146, 148, 150, 155, 147, 153, 145, 175, 206, 143, 136, 11, 142, 129, 8, 7, 198, 197, 4, 195, 2, 161, 160, 255, 124, 109, 108, 122, 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106, 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65, 104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75, 67, 73, 96, 55, 58, 45, 188, 51, 186, 61, 40, 119, 182, 181, 53, 179, 54, 33, 49, 15, 174, 47, 60, 171, 46, 57, 32, 167, 6, 36, 164, 43, 162, 1, 0, 63, 26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13, 10, 29, 14, 3, 138, 41, 12, 39, 134, 133, 5, 131, 34, 9, 128]]#8 pas totally
-
-lf +=[[223, 250, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227, 178, 240, 248, 237, 236, 173, 172, 171, 238, 201, 168, 229, 166, 228, 244, 235, 242, 241, 192, 215, 220, 205, 216, 218, 222, 153, 152, 151, 210, 212, 214, 219, 146, 217, 209, 239, 142, 141, 206, 195, 234, 193, 136, 231, 196, 199, 197, 194, 226, 225, 200, 63, 188, 253, 252, 59, 190, 189, 176, 191, 246, 245, 164, 243, 162, 161, 177, 143, 170, 45, 44, 43, 138, 185, 184, 135, 38, 167, 165, 179, 34, 129, 224, 31, 154, 221, 158, 147, 26, 25, 156, 159, 22, 213, 149, 211, 150, 144, 208, 207, 14, 13, 204, 203, 202, 169, 8, 133, 198, 132, 4, 139, 131, 1, 160, 255, 124, 109, 108, 122, 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106, 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65, 104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75, 67, 73, 96, 55, 58, 57, 62, 51, 186, 41, 40, 119, 182, 181, 53, 35, 54, 48, 56, 175, 174, 61, 60, 11, 46, 9, 32, 37, 6, 36, 52, 163, 50, 49, 0, 23, 28, 157, 24, 155, 30, 29, 16, 21, 18, 20, 148, 27, 19, 145, 17, 47, 10, 15, 140, 3, 42, 137, 12, 39, 134, 7, 5, 2, 130, 33, 128]] #8 totally
+#lf += [[223, 190, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227, 178, 240, 248, 237, 236, 253, 172, 203, 170, 201, 168, 229, 166, 165, 244, 163, 242, 241, 192, 215, 220, 205, 216, 218, 222, 221, 208, 213, 210, 212, 214, 219, 211, 217, 209, 239, 202, 207, 140, 139, 234, 193, 204, 135, 196, 199, 132, 194, 130, 225, 200, 159, 62, 185, 252, 59, 250, 169, 56, 191, 246, 245, 52, 243, 50, 176, 48, 173, 238, 189, 44, 235, 42, 137, 184, 231, 38, 37, 228, 35, 226, 177, 224, 151, 156, 141, 152, 154, 158, 157, 144, 149, 146, 148, 150, 155, 147, 153, 145, 175, 206, 143, 136, 11, 142, 129, 8, 7, 198, 197, 4, 195, 2, 161, 160, 255, 124, 109, 108, 122, 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106, 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65, 104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75, 67, 73, 96, 55, 58, 45, 188, 51, 186, 61, 40, 119, 182, 181, 53, 179, 54, 33, 49, 15, 174, 47, 60, 171, 46, 57, 32, 167, 6, 36, 164, 43, 162, 1, 0, 63, 26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13, 10, 29, 14, 3, 138, 41, 12, 39, 134, 133, 5, 131, 34, 9, 128]]#8 pas totally
 
+#lf +=[[223, 250, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227, 178, 240, 248, 237, 236, 173, 172, 171, 238, 201, 168, 229, 166, 228, 244, 235, 242, 241, 192, 215, 220, 205, 216, 218, 222, 153, 152, 151, 210, 212, 214, 219, 146, 217, 209, 239, 142, 141, 206, 195, 234, 193, 136, 231, 196, 199, 197, 194, 226, 225, 200, 63, 188, 253, 252, 59, 190, 189, 176, 191, 246, 245, 164, 243, 162, 161, 177, 143, 170, 45, 44, 43, 138, 185, 184, 135, 38, 167, 165, 179, 34, 129, 224, 31, 154, 221, 158, 147, 26, 25, 156, 159, 22, 213, 149, 211, 150, 144, 208, 207, 14, 13, 204, 203, 202, 169, 8, 133, 198, 132, 4, 139, 131, 1, 160, 255, 124, 109, 108, 122, 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106, 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65, 104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75, 67, 73, 96, 55, 58, 57, 62, 51, 186, 41, 40, 119, 182, 181, 53, 35, 54, 48, 56, 175, 174, 61, 60, 11, 46, 9, 32, 37, 6, 36, 52, 163, 50, 49, 0, 23, 28, 157, 24, 155, 30, 29, 16, 21, 18, 20, 148, 27, 19, 145, 17, 47, 10, 15, 140, 3, 42, 137, 12, 39, 134, 7, 5, 2, 130, 33, 128]] #8 totally
 
 
+lf +=[
+    [29, 22, 21, 30, 19, 27, 24, 28, 7, 20, 5, 4, 23, 26, 25, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 1, 6, 11, 18, 0, 16],#sylvain 1
+    [29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4],#sylvain 2
+    [29, 22, 21, 24, 31, 30, 27, 26, 7, 23, 5, 28, 3, 19, 25, 17, 13, 12, 9, 8, 15, 2, 1, 10, 6, 14, 4, 20, 11, 18, 0, 16], #mini avec 7
+    [29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, 31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4]] #5
     
 
 # 14 = (1,1,1,0) = f(0,0,0,0)
diff --git a/evalPRNG/texput.log b/evalPRNG/texput.log
index fe09a01..32d2e01 100644
--- a/evalPRNG/texput.log
+++ b/evalPRNG/texput.log
@@ -1,20 +1,20 @@
-This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=pdflatex 2015.4.24)  1 SEP 2015 14:12
+This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2016.5.31)  20 JUN 2016 22:12
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
-**president.tex
+**main.tex
 
 ! Emergency stop.
-<*> president.tex
-                 
+<*> main.tex
+            
 End of file on the terminal!
 
  
 Here is how much of TeX's memory you used:
- 3 strings out of 494991
- 118 string characters out of 6180055
- 46069 words of memory out of 5000000
- 3329 multiletter control sequences out of 15000+600000
+ 3 strings out of 494924
+ 106 string characters out of 6179708
+ 45807 words of memory out of 5000000
+ 3399 multiletter control sequences out of 15000+600000
  3640 words of font info for 14 fonts, out of 8000000 for 9000
  14 hyphenation exceptions out of 8191
  0i,0n,0p,1b,6s stack positions out of 5000i,500n,10000p,200000b,80000s
diff --git a/generating.tex b/generating.tex
index 354058a..ace2f7a 100644
--- a/generating.tex
+++ b/generating.tex
@@ -7,12 +7,11 @@ if and only if its Markov matrix is a doubly stochastic matrix.
 
 
 In~\cite[Section 4]{DBLP:conf/secrypt/CouchotHGWB14},
-we have presented an efficient
-approach which generates 
+we have presented a general scheme which generates 
 function with strongly connected iteration graph $\Gamma(f)$ and
 with doubly stochastic Markov probability matrix.
 
-Basically, let consider the ${\mathsf{N}}$-cube. Let us next 
+Basically, let us consider the ${\mathsf{N}}$-cube. Let us next 
 remove one Hamiltonian cycle in this one. When an edge $(x,y)$ 
 is removed, an edge $(x,x)$ is added. 
 
@@ -62,6 +61,18 @@ It has been shown in~\cite[Lemma 3]{bcgr11:ip}  that $M$ is regular.
 There exists thus $b$ such there is an arc between any $x$ and $y$.
 \end{proof}
 
-The next section presents how to build hamiltonian cycles in the 
+This section ends with the idea of removing a Hamiltonian cycle in the 
+$\mathsf{N}$-cube. 
+In such a context, the Hamiltonian cycle is equivalent to a Gray code.
+Many approaches have been proposed a way to build such codes, for instance 
+the Reflected Binary Code. In this one, one of the bits is switched 
+exactly $2^{\mathsf{N}-}$ for a $\mathsf{N}$-length cycle. 
+
+%%%%%%%%%%%%%%%%%%%%%
+
+The function that is built 
+from the 
+
+The next section presents how to build balanced Hamiltonian cycles in the 
 $\mathsf{N}$-cube with the objective to embed them into the 
 pseudorandom number generator.
diff --git a/hamilton.tex b/hamilton.tex
index 5764995..17d9380 100644
--- a/hamilton.tex
+++ b/hamilton.tex
@@ -17,11 +17,12 @@ and an algorithm that establishes locally balanced Gray codes is given.
 The current context is to provide a function 
 $f:\Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ by removing a Hamiltonian 
 cycle in the $\mathsf{N}$ cube. Such a function is going to be iterated
-$b$ time to produce a pseudo random number, \textit{i.e.} a vertex in the 
+$b$ times to produce a pseudo random number,
+\textit{i.e.} a vertex in the 
 $\mathsf{N}$ cube.
 Obviously, the number of iterations $b$ has to be sufficiently large 
 to provide a uniform output distribution.
-To reduced the number of iterations, the provided Gray code
+To reduce the number of iterations, the provided Gray code
 should ideally possess the both balanced and locally balanced properties.
 However, none of the two algorithms is compatible with the second one:
 balanced Gray codes that are generated by state of the art works~\cite{ZanSup04,DBLP:journals/combinatorics/BhatS96} are not locally balanced. Conversely,
@@ -35,16 +36,16 @@ the number of switches of each element is as uniform as possible.
 \subsection{Analysis of the Robinson-Cohn extension algorithm}
 As far as we know three works, 
 namely~\cite{Robinson:1981:CS},~\cite{DBLP:journals/combinatorics/BhatS96}, 
-and~\cite{ZanSup04} have adressed the probem of providing an approach
+and~\cite{ZanSup04} have addressed the problem of providing an approach
 to produce balanced gray code.
 The authors of~\cite{Robinson:1981:CS} introduced an inductive approach
 aiming at producing balanced Gray codes, provided the user gives 
 a special subsequence of the transition sequence at each induction step.
 This work have been strengthened in~\cite{DBLP:journals/combinatorics/BhatS96}
-where the authors have explicitely shown how to construct such a subsequence.
+where the authors have explicitly shown how to construct such a subsequence.
 Finally the authors of~\cite{ZanSup04} have presented 
 the \emph{Robinson-Cohn extension} 
-algorithm. There rigourous presentation of this one 
+algorithm. There rigorous presentation of this one 
 have mainly allowed them to prove two properties.
 The former states that if 
 $\mathsf{N}$ is a 2-power, a balanced Gray code is always totally balanced.
@@ -67,16 +68,16 @@ which produces a $n$-bits Gray code.
 $u_1, u_2, \dots , u_{l-2}, v$ (maybe empty) subsequences of $S_{\mathsf{N}-2}$ 
 such that $S_{\mathsf{N}-2}$ is the concatenation of 
 $$
-s_{i_1}, u_0, s_{i_2}, u_1, s_{i_3}, u_2, . . . , s_{i_l-1}, u_{l-2}, s_{i_l}, v
+s_{i_1}, u_0, s_{i_2}, u_1, s_{i_3}, u_2, \dots , s_{i_l-1}, u_{l-2}, s_{i_l}, v
 $$
 where $i_1 = 1$, $i_2 = 2$, and $u_0 = \emptyset$ (the empty sequence).
-\item Replace in $S_{\mathsf{N}-2}$ the sequences $u_0, u_1, u_2, \ldots, u_{l-2}$ 
+\item\label{item:u'}Replace in $S_{\mathsf{N}-2}$ the sequences $u_0, u_1, u_2, \ldots, u_{l-2}$ 
   by 
   $\mathsf{N} - 1,  u'(u_1,\mathsf{N} - 1, \mathsf{N}) , u'(u_2,\mathsf{N}, \mathsf{N} - 1), u'(u_3,\mathsf{N} - 1,\mathsf{N}), \dots, u'(u_{l-2},\mathsf{N}, \mathsf{N} - 1)$
   respectively, where $u'(u,x,y)$ is the sequence $u,x,u^R,y,u$ such that 
   $u^R$ is $u$ in reversed order. 
   The obtained sequence is further denoted as $U$.
-\item Construct the sequences $V=v^R,\mathsf{N},v$, $W=\mathsf{N}-1,S_{\mathsf{N}-2},\mathsf{N}$, and let $W'$ be $W$ where the first 
+\item\label{item:VW} Construct the sequences $V=v^R,\mathsf{N},v$, $W=\mathsf{N}-1,S_{\mathsf{N}-2},\mathsf{N}$, and let $W'$ be $W$ where the first 
 two elements have been exchanged.
 \item The transition sequence $S_{\mathsf{N}}$ is thus the concatenation $U^R, V, W'$.
 \end{enumerate} 
@@ -87,10 +88,10 @@ if $S_{\mathsf{N}-2}$ is.
 However, the step~(\ref{item:nondet}) is not a constructive 
 step that precises how to select the subsequences which ensures that 
 yielded Gray code is balanced.
-Next section shows how to choose the sequence $l$ to have the balancy property.
+Next section shows how to choose the sequence $l$ to have the balance property.
 
 \subsection{Balanced Codes}
-Let us first recall how to formalize the balancy property of a Gray code.
+Let us first recall how to formalize the balance property of a Gray code.
 Let  $L = w_1, w_2, \dots, w_{2^\mathsf{N}}$ be the sequence 
 of a $\mathsf{N}$-bits cyclic Gray code. 
 The  transition sequence 
@@ -131,7 +132,7 @@ the code is neither balanced nor totally balanced.
 
 
 \begin{thrm}\label{prop:balanced}
-Let $\mathsf{N}$ in $\Nats$, and $a_{\mathsf{N}}$ be defined by
+Let $\mathsf{N}$ in $\Nats^*$, and $a_{\mathsf{N}}$ be defined by
 $a_{\mathsf{N}}= 2 \lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \rfloor$. 
 There exists then a sequence $l$ in 
 step~(\ref{item:nondet}) of the \emph{Robinson-Cohn extension} algorithm
@@ -144,7 +145,7 @@ for any $i$, $1 \le i \le \mathsf{N}$.
 
 
 
-The proof is done by induction on $\mathsf{N}$. Let us imadialty verify 
+The proof is done by induction on $\mathsf{N}$. Let us immediately verify 
 that it is established for both odd and even smallest values, \textit{i.e.} 
 $3$ and $4$.
 For the initial case where $\mathsf{N}=3$, \textit{i.e.} $\mathsf{N-2}=1$ we successively have:  $S_1=1,1$, $l=2$,  $u_0 = \emptyset$, and $v=\emptyset$.
@@ -166,7 +167,7 @@ odd and even initial values.
 
 For the inductive case, let us first define some variables.
 Let $c_{\mathsf{N}}$ (resp. $d_{\mathsf{N}}$) be the number of elements 
-whose transistion count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$).
+whose transition count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$).
 These two variables are defined by the system 
  
 $$
@@ -181,18 +182,149 @@ c_{\mathsf{N}}a_{\mathsf{N}} + d_{\mathsf{N}}(a_{\mathsf{N}}+2) & = & 2^{\mathsf
 \qquad 
 \left\{
 \begin{array}{lcl}
-d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N}} -n.a_{\mathsf{N}}}{2} \\
+d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N}} -\mathsf{N}.a_{\mathsf{N}}}{2} \\
 c_{\mathsf{N}} &= &\mathsf{N} -  d_{\mathsf{N}}
 \end{array}
 \right.
 $$
 
-Since $a_{\mathsf{N}$ is even, $d_{\mathsf{N}}$ is defined.
-Moreover, both $c_{\mathsf{N}}$ and  $d_{\mathsf{N}}$ are obviously positves.
+Since $a_{\mathsf{N}}$ is even, $d_{\mathsf{N}}$ is an integer.
+Let us first proove that both $c_{\mathsf{N}}$ and  $d_{\mathsf{N}}$ are positive
+integers.
+Let $q_{\mathsf{N}}$ and $r_{\mathsf{N}}$, respectively, be  
+the quotient and the remainder in the Euclidean disvision
+of $2^{\mathsf{N}}$ by $2\mathsf{N}$, \textit{i.e.} 
+$2^{\mathsf{N}} = q_{\mathsf{N}}.2\mathsf{N} + r_{\mathsf{N}}$, with $0 \le r_{\mathsf{N}} <2\mathsf{N}$.
+First of all, the integer $r$ is even since $r_{\mathsf{N}} = 2^{\mathsf{N}} - q_{\mathsf{N}}.2\mathsf{N}= 2(2^{\mathsf{N}-1} - q_{\mathsf{N}}.\mathsf{N})$. 
+Next,  $a_{\mathsf{N}}$ is $\frac{2^{\mathsf{N}}-r_{\mathsf{N}}}{\mathsf{N}}$. Consequently 
+$d_{\mathsf{N}}$ is $r_{\mathsf{N}}/2$ and is thus a positive integer s.t. 
+$0 \le d_{\mathsf{N}} <\mathsf{N}$.
+The proof for $c_{\mathsf{N}}$ is obvious.
 
 
+For any $i$, $1 \le i \le \mathsf{N}$, let $zi_{\mathsf{N}}$ (resp. $ti_{\mathsf{N}}$ and $bi_{\mathsf{N}}$) 
+be the occurence number of element $i$ in the sequence $u_0, \dots, u_{l-2}$ 
+(resp. in the sequences $s_{i_1}, \dots , s_{i_l}$ and $v$)
+in step (\ref{item:nondet}) of the algorithm.
+
+Due to the definition of $u'$ in step~(\ref{item:u'}), 
+$3.zi_{\mathsf{N}} + ti_{\mathsf{N}}$ is the
+ number of element $i$ in the sequence $U$.
+It is clear that the number of element $i$ in the sequence $V$ is 
+$2bi_{\mathsf{N}}$ due to step (\ref{item:VW}). 
+We thus have the following system:
+$$
+\left\{
+\begin{array}{lcl}
+3.zi_{\mathsf{N}} + ti_{\mathsf{N}} + 2.bi_{\mathsf{N}} + \textit{TC}_{\mathsf{N}-2}(i) &= &\textit{TC}_{\mathsf{N}}(i) \\
+zi_{\mathsf{N}} + ti_{\mathsf{N}}  + bi_{\mathsf{N}} & =& \textit{TC}_{\mathsf{N}-2}(i)
+\end{array}
+\right.
+\qquad 
+\Leftrightarrow 
+$$
+
+\begin{equation}
+\left\{
+\begin{array}{lcl}
+zi_{\mathsf{N}} &= & 
+\dfrac{\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i) - bi_{\mathsf{N}}}{2}\\
+ti_{\mathsf{N}} &= & \textit{TC}_{\mathsf{N}-2}(i)-zi_{\mathsf{N}}-bi_{\mathsf{N}}
+\end{array}
+\right.
+\label{eq:sys:zt1}
+\end{equation}
+
+In this set of 2 equations with 3 unknown variables, let $b_i$ be set with 0.
+In this case, since $\textit{TC}_{\mathsf{N}}$ is even (equal to $a_{\mathsf{N}}$ 
+or to $a_{\mathsf{N}}+2$), the variable  $zi_{\mathsf{N}}$ is thus an integer.
+Let us now prove that the resulting system has always positive integer 
+solutions $z_i$, $t_i$, $0 \le z_i, t_i \le \textit{TC}_{\mathsf{N}-2}(i)$ 
+and s.t. their sum is equal to $\textit{TC}_{\mathsf{N}-2}(i)$. 
+This latter consraint is obviously established if the system has a solution.
+We thus have the following system.
+
+
+
+\begin{equation}
+\left\{
+\begin{array}{lcl}
+zi_{\mathsf{N}} &= & 
+\dfrac{\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i) }{2}\\
+ti_{\mathsf{N}} &= & \textit{TC}_{\mathsf{N}-2}(i)-zi_{\mathsf{N}}
+\end{array}
+\right.
+\label{eq:sys:zt2}
+\end{equation}
+
+The definition of $\textit{TC}_{\mathsf{N}}(i)$ depends on the value of $\mathsf{N}$.
+When $3 \le N \le 7$, values are defined as follows:
+\begin{eqnarray*}
+\textit{TC}_{3} & = & [2,2,4] \\
+\textit{TC}_{5} & = & [6,6,8,6,6] \\
+\textit{TC}_{7} & = & [18,18,20,18,18,18,18] \\
+\\
+\textit{TC}_{4} & = & [4,4,4,4] \\
+\textit{TC}_{6} & = & [10,10,10,10,12,12] \\
+\end{eqnarray*}
+It is not hard to verify that all these instanciations verify the aformentioned contraints. 
+
+When  $N  \ge 8$, $\textit{TC}_{\mathsf{N}}(i)$ is defined as follows:
+\begin{equation}
+\textit{TC}_{\mathsf{N}}(i) = \left\{
+\begin{array}{l} 
+a_{\mathsf{N}} \textrm{ if } 1 \le i \le c_{\mathsf{N}} \\
+a_{\mathsf{N}}+2 \textrm{ if } c_{\mathsf{N}} +1 \le i \le c_{\mathsf{N}} + d_{\mathsf{N}} 
+\end{array}
+\right.
+\label{eq:TCN:def}
+\end{equation}
+
+
+We thus  have
+$$
+\begin{array}{rcl} 
+\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i) 
+&\ge& 
+a_{\mathsf{N}} - 2(a_{\mathsf{N}-2}+2) \\
+&\ge& 
+\frac{2^{\mathsf{N}}-r_{\mathsf{N}}}{\mathsf{N}}
+-2 \left( \frac{2^{\mathsf{N-2}}-r_{\mathsf{N-2}}}{\mathsf{N-2}}+2\right)\\
+&\ge& 
+\frac{2^{\mathsf{N}}-2N}{\mathsf{N}}
+-2 \left( \frac{2^{\mathsf{N-2}}}{\mathsf{N-2}}+2\right)\\
+&\ge& 
+\dfrac{(\mathsf{N} -2).2^{\mathsf{N}}-2N.2^{\mathsf{N-2}}-6N(N-2)}{\mathsf{N.(N-2)}}\\
+\end{array}
+$$
+
+A simple variation study of the function $t:\R \rightarrow \R$ such that 
+$x \mapsto t(x) = (x -2).2^{x}-2x.2^{x-2}-6x(x-2)$ shows that 
+its derivative is strictly postive if $x \ge 6$ and $t(8)=224$.
+The integer $\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i)$ is thus positive 
+for any $\mathsf{N} \ge 8$ and the proof is established.
+
+
+% \begin{table}[ht]
+% $$
+% \begin{array}{|l|l|l|l|l|l|}
+% \hline
+% \mathsf{N}     & 3 & 4 & 5 & 6  & 7\\
+% \hline
+% a_{\mathsf{N}} & 2 & 4 & 6 & 10 & 18\\
+% \hline
+% \end{array}
+% $$
+% \label{tab:an}
+% \caption{First values of  $a_{\mathsf{N}}$}
+% \end{table}
+
+For each element $i$, we are then left to choose $zi_{\mathsf{N}}$ positions 
+among $\textit{TC}_{\mathsf{N}}(i)$, which leads to 
+${\textit{TC}_{\mathsf{N}}(i) \choose zi_{\mathsf{N}} }$ possibilities.
+Notice that all such choices lead to a hamiltonian path.
+
 
 
 
 
-\subsection{Toward a local uniform distribution of switches}
diff --git a/intro.tex b/intro.tex
index c0b6e7d..df2a0ee 100644
--- a/intro.tex
+++ b/intro.tex
@@ -1,102 +1,113 @@
-The exploitation of chaotic systems to generate pseudorandom sequences is
-an hot topic~\cite{915396,915385,5376454}. Such systems are fundamentally 
-chosen due to their unpredictable character and their sensitiveness to initial conditions.
-In most cases, these generators simply consist in iterating a chaotic function like 
-the logistic map~\cite{915396,915385} or the Arnold's one~\cite{5376454}\ldots
-It thus remains to find optimal parameters in such functions so that attractors are
-avoided, hoping by doing so that the generated numbers follow a uniform distribution.
-In order to check the quality of the produced outputs, it is usual to test the 
-PRNGs   (Pseudo-Random  Number   Generators) with statistical batteries like
-the so-called DieHARD~\cite{Marsaglia1996}, NIST~\cite{Nist10}, or TestU01~\cite{LEcuyerS07} ones.
+The exploitation of chaotic systems to generate pseudorandom sequences
+is  an   hot  topic~\cite{915396,915385,5376454}.  Such   systems  are
+fundamentally chosen  due to  their unpredictable character  and their
+sensitiveness to initial conditions.   In most cases, these generators
+simply  consist in  iterating  a chaotic  function  like the  logistic
+map~\cite{915396,915385} or  the Arnold's  one~\cite{5376454}\ldots It
+thus  remains to  find optimal  parameters in  such functions  so that
+attractors are avoided, hoping by  doing so that the generated numbers
+follow a uniform  distribution.  In order to check the  quality of the
+produced outputs, it is usual  to test the PRNGs (Pseudo-Random Number
+Generators)   with   statistical    batteries   like   the   so-called
+DieHARD~\cite{Marsaglia1996},          NIST~\cite{Nist10},          or
+TestU01~\cite{LEcuyerS07} ones.
 
-In its general understanding, chaos notion is often reduced to the strong
-sensitiveness to the initial conditions (the well known ``butterfly effect''):
-a continuous function $k$ defined on a metrical space is said 
-\emph{strongly sensitive to the initial conditions} if for each point 
-$x$ and each positive value $\epsilon$, it is possible to find another 
-point $y$ as close as possible to $x$, and an integer $t$ such that the distance
-between the $t$-th iterates of $x$ and $y$, denoted by $k^t(x)$ and $k^t(y)$,
-are larger than $\epsilon$. However, in his definition of chaos, Devaney~\cite{Devaney} 
-imposes to the chaotic function two other properties called
-\emph{transitivity} and \emph{regularity}. Functions evoked above have
-been studied according to these properties, and they have been proven as chaotic on $\R$.
-But nothing guarantees that such properties are preserved when iterating the functions
-on floating point numbers, which is the domain of interpretation of real numbers $\R$ on
-machines.
+In its  general understanding,  chaos notion is  often reduced  to the
+strong  sensitiveness  to  the  initial  conditions  (the  well  known
+``butterfly effect''): a continuous function $k$ defined on a metrical
+space is said  \emph{strongly sensitive to the  initial conditions} if
+for each point $x$ and each  positive value $\epsilon$, it is possible
+to find another point $y$ as close  as possible to $x$, and an integer
+$t$ such that the distance between the $t$-th iterates of $x$ and $y$,
+denoted by $k^t(x)$ and $k^t(y)$, are larger than $\epsilon$. However,
+in  his definition  of  chaos, Devaney~\cite{Devaney}  imposes to  the
+chaotic function  two other properties called  \emph{transitivity} and
+\emph{regularity}. Functions evoked above  have been studied according
+to these  properties, and they  have been  proven as chaotic  on $\R$.
+But  nothing  guarantees  that  such  properties  are  preserved  when
+iterating the functions on floating point numbers, which is the domain
+of interpretation of real numbers $\R$ on machines.
 
-To avoid this lack of chaos, we have previously presented some PRNGs that iterate
-continuous functions $G_f$ on a discrete domain  $\{ 1, \ldots,  n \}^{\Nats}
- \times  \{0,1\}^n$, where $f$  is a Boolean function (\textit{i.e.},  $f  :
- \{0,1\}^n      \rightarrow      \{0,1\}^n$).       These generators are 
-$\textit{CIPRNG}_f^1(u)$ \cite{guyeuxTaiwan10,bcgr11:ip},
-$\textit{CIPRNG}_f^2(u,v)$ \cite{wbg10ip}                                     and
-$\chi_{\textit{14Secrypt}}$ \cite{chgw14oip}     where    \textit{CI}    means
-\emph{Chaotic Iterations}.
-We have firstly proven in~\cite{bcgr11:ip} that, to establish the chaotic nature
-of algorithm $\textit{CIPRNG}_f^1$, it is necessary and sufficient that the 
-asynchronous iterations are strongly connected. We then have proven that it is necessary
-and sufficient that the Markov matrix associated to this graph is doubly stochastic,
-in order to have a uniform distribution of the outputs. We have finally established 
-sufficient conditions to guarantee the first property of connectivity. Among the 
-generated functions, we thus have considered for further investigations only the ones that
-satisfy the second property too. In~\cite{chgw14oip}, we have proposed an algorithmic 
-method allowing to directly obtain a strongly connected iteration graph having a doubly
-stochastic Markov matrix. 
+To avoid this  lack of chaos, we have previously  presented some PRNGs
+that iterate  continuous functions $G_f$  on a discrete domain  $\{ 1,
+\ldots,  n  \}^{\Nats}  \times  \{0,1\}^n$, where  $f$  is  a  Boolean
+function  (\textit{i.e.},  $f  :  \{0,1\}^{\mathsf{N}}
+\rightarrow  \{0,1\}^{\mathsf{N}}$).
+These         generators          are         $\textit{CIPRNG}_f^1(u)$
+\cite{guyeuxTaiwan10,bcgr11:ip},            $\textit{CIPRNG}_f^2(u,v)$
+\cite{wbg10ip}             and             $\chi_{\textit{14Secrypt}}$
+\cite{DBLP:conf/secrypt/CouchotHGWB14}    where   \textit{CI}    means
+\emph{Chaotic Iterations}.  We have firstly proven in~\cite{bcgr11:ip}
+that,    to    establish    the   chaotic    nature    of    algorithm
+$\textit{CIPRNG}_f^1$,  it  is  necessary   and  sufficient  that  the
+asynchronous iterations  are strongly  connected. We then  have proven
+that it is necessary and  sufficient that the Markov matrix associated
+to  this graph  is  doubly  stochastic, in  order  to  have a  uniform
+distribution of  the outputs.  We have finally  established sufficient
+conditions to guarantee the first  property of connectivity. Among the
+generated   functions,   we   thus   have   considered   for   further
+investigations  only  the  ones   that  satisfy  the  second  property
+too.
 
+However,      it     cannot      be     directly      deduced     that
+$\chi_{\textit{14Secrypt}}$ is chaotic since we  do not output all the
+successive values of iterating $G_f$.   This algorithm only displays a
+subsequence $x^{b.n}$  of a whole  chaotic sequence $x^{n}$ and  it is
+indeed not correct  that the chaos property is preserved for any
+subsequence of  a chaotic sequence.  This  article presents conditions
+to preserve this property.
 
-However, it cannot be directly deduced
-that $\chi_{\textit{14Secrypt}}$ is chaotic 
-since we do not output all the successive
-values of iterating $F_f$. 
-This algorithm only displays a
-subsequence $x^{b.n}$ of a whole chaotic sequence $x^{n}$ and
-it is indeed  definitively false that the chaos property is 
-preserved for any subsequence of a chaotic sequence.
-This article presents conditions to preserve this property.
 
-An approach to generate a large class of chaotic functions has 
-been presented in~\cite{chgw14oip}.
-It is basically fourfold:
-first build a $\mathsf{N}$-cube, next remove an Hamiltonian cycle, further add self-loop
-on each vertex and finally, translate this into a Boolean map.
-We are then left to check whether this approach proposes maps with the required conditions 
-for the chaos. 
-The answer is indeed positive. The pseudorandom number generation can thus be seen as a
-random walk in a $\mathsf{N}$-cube without a Hamiltonian cycle.
+Finding a Boolean function which provides a  strongly connected
+iteration graph having a doubly stochastic Markov matrix is however
+not an easy task.
+We have firstly proposed in~\cite{bcgr11:ip} a  generate-and-test based
+approach that solves this issue. However, this one was not efficient enough.
+Thus, a second scheme has been further presented
+in~\cite{DBLP:conf/secrypt/CouchotHGWB14} by remarking that
+a  $\mathsf{N}$-cube where an Hamiltonian cycle (or equivalently a Gray code)
+has been removed is strongly connected and has
+a doubly stochastic Markov matrix.
 
-In the PRNG context, there remains to find which subsequence
-is theoretically and practically 
-sufficient to extract.
-A uniform distribution is indeed awaited and this 
-cannot be obtained in a walk in the hypercube
-with paths of short length $b$.
-However, the higher 
-is $b$  the slower is the 
-algorithm to generate pseudorandom
-numbers. 
-The  time  until the
-corresponding  Markov chain is close 
-to the uniform distribution is a metric
-that should be theoretically and practically studied.
-Finally, the ability of the approach to face classical 
-tests suite has to be evaluated.
+However, the removed Hamiltonian cycle  
+has a great influence in the quality of the output.
+For instance, if this one one is not balanced (\textit{i.e.},
+the number of changes in different bits are completely different),
+some bits would be hard to switch.
+This article shows an effective algorithm that efficiently 
+implements the previous scheme and provides thus functions issued
+from removing in the $\mathsf{N}$-cube a \emph{balanced} Hamiltonian cycle. 
 
+The length $b$ of the walk to reach a
+distribution close to the uniform one would be dramatically long.
+This article theoretically and practically
+studies the length $b$ until the corresponding Markov
+chain is close to the uniform distribution.
+Finally, the ability of the approach to face classical tests
+suite is evaluated.
 
-%A upper bound of this time  quadratic in the number of 
-%generated bits.
 
 
-The remainder of this article is organized as follows. The next section is devoted to 
-preliminaries, basic notations, and terminologies regarding Boolean map iterations.
-Then, in Section~\ref{sec:proofOfChaos}, Devaney's definition of chaos is recalled
-while the proofs of chaos of our most general PRNGs is provided. 
-This is the first major contribution.
-Section~\ref{sec:SCCfunc} shows how to generate functions with required properties 
-making the PRNG chaotic.
-The next section (Sect.~\ref{sec:hypercube}) defines the theoretical framework 
-to study the stopping-time, \textit{i.e.}, time until reaching
-a uniform distribution.
-This is the second major contribution.
-The Section~\ref{sec:prng} gives practical results on evaluating the PRNG against the NIST suite.
-This research work ends by a conclusion section, where the contribution is summarized and
-intended future work is outlined.
+
+The  remainder of  this  article  is organized  as  follows. The  next
+section   is   devoted   to  preliminaries,   basic   notations,   and
+terminologies   regarding   Boolean    map   iterations.    Then,   in
+Section~\ref{sec:proofOfChaos},  Devaney's  definition   of  chaos  is
+recalled  while the  proofs  of chaos  of our  most  general PRNGs  is
+provided.      This    is     the     first    major     contribution.
+Section~\ref{sec:SCCfunc}  recalls a general scheme
+to obtain functions with awaited behavior. Main theorems are recalled
+to make the document self-content. 
+The  next  section (Sect.~\ref{sec:hamilton}) presents an algorithm that
+implements this scheme and proves it always produces a solution.  
+This  is   the   second   major  contribution
+The  later section
+(Sect.~\ref{sec:hypercube}) defines the theoretical framework to study
+the  mixing-time,  \textit{i.e.},  time  until  reaching  a  uniform
+distribution. It proves that this one is at worth quadratic in the number
+of elements. Experiments show that the bound is practically largely much
+lower. This  is   the   third   major  contribution.    The
+Section~\ref{sec:prng} gives practical results  on evaluating the PRNG
+against  the NIST  suite.  This  research  work ends  by a  conclusion
+section, where the contribution is summarized and intended future work
+is outlined.
diff --git a/main.aux b/main.aux
index 0bd4d61..0665af7 100644
--- a/main.aux
+++ b/main.aux
@@ -13,15 +13,14 @@
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}}
 \citation{guyeuxTaiwan10,bcgr11:ip}
 \citation{wbg10ip}
-\citation{chgw14oip}
+\citation{DBLP:conf/secrypt/CouchotHGWB14}
+\citation{bcgr11:ip}
 \citation{bcgr11:ip}
-\citation{chgw14oip}
-\citation{chgw14oip}
 \citation{DBLP:conf/secrypt/CouchotHGWB14}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Preliminaries}}{3}}
 \newlabel{sec:preliminaries}{{2}{3}}
 \newlabel{eq:asyn}{{1}{3}}
-\citation{bcgr11:ip}
+\citation{DBLP:conf/secrypt/CouchotHGWB14}
 \citation{bcgr11:ip}
 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Iteration Graph $\Gamma (f^*)$ of the function $f^*$\relax }}{4}}
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
@@ -30,6 +29,7 @@
 \newlabel{CI Algorithm}{{1}{4}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Proof Of Chaos}}{4}}
 \newlabel{sec:proofOfChaos}{{3}{4}}
+\citation{bcgr11:ip}
 \citation{Devaney}
 \citation{Banks92}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Devaney's Chaotic Dynamical Systems}}{5}}
@@ -38,27 +38,27 @@
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{A Metric Space for PRNG Iterations}}{5}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A metric on $\mathcal  {X}_{\mathsf  {N},\mathcal  {P}}$}}{6}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{$\Gamma _{\mathcal  {P}}(f)$ as an extension of $\Gamma (f)$}}{8}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{8}}
 \newlabel{graphe1}{{2a}{9}}
 \newlabel{sub@graphe1}{{a}{9}}
 \newlabel{graphe2}{{2b}{9}}
 \newlabel{sub@graphe2}{{b}{9}}
 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Iterating $f_0:(x_1,x_2) \DOTSB \mapstochar \rightarrow (\overline  {x_1}, \overline  {x_2})$\relax }}{9}}
 \newlabel{fig:itg}{{2}{9}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{9}}
 \newlabel{prop:trans}{{3.8}{9}}
 \citation{bcgr11:ip}
 \citation{DBLP:conf/secrypt/CouchotHGWB14}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Functions with Strongly Connected $\Gamma _{\{b\}}(f)$}}{10}}
-\newlabel{sec:SCCfunc}{{4}{10}}
 \citation{bcgr11:ip}
 \citation{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04,Bykov2016}
 \citation{DBLP:journals/combinatorics/BhatS96,ZanSup04}
 \citation{Bykov2016}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Functions with Strongly Connected $\Gamma _{\{b\}}(f)$}}{11}}
+\newlabel{sec:SCCfunc}{{4}{11}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Balanced Hamiltonian Cycle}}{11}}
+\newlabel{sec:hamilton}{{5}{11}}
 \citation{ZanSup04,DBLP:journals/combinatorics/BhatS96}
 \citation{Bykov2016}
 \citation{ZanSup04}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{(Locally) Balanced Hamiltonian Cycle}}{11}}
-\newlabel{sec:hamilton}{{5}{11}}
 \citation{Robinson:1981:CS}
 \citation{DBLP:journals/combinatorics/BhatS96}
 \citation{ZanSup04}
@@ -66,39 +66,52 @@
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 \newlabel{item:nondet}{{1}{12}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{12}}
+\citation{ZanSup04}
+\newlabel{item:u'}{{2}{13}}
+\newlabel{item:VW}{{3}{13}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{13}}
 \newlabel{prop:balanced}{{5.1}{13}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.3}{Toward a local uniform distribution of switches}}{13}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{13}}
-\newlabel{sec:hypercube}{{6}{13}}
+\newlabel{eq:sys:zt1}{{3}{14}}
+\newlabel{eq:sys:zt2}{{4}{14}}
+\newlabel{eq:TCN:def}{{5}{15}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{15}}
+\newlabel{sec:hypercube}{{6}{15}}
 \citation{LevinPeresWilmer2006}
-\newlabel{eq:Markov:rairo}{{3}{15}}
-\newlabel{lm:h}{{6.2}{15}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.1}{Formalizing the Random Walk}}{16}}
+\newlabel{sub:stop:formal}{{6.1}{16}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.2}{Upper bound of Stopping Time}}{17}}
+\newlabel{sub:stop:bound}{{6.2}{17}}
+\newlabel{eq:Markov:rairo}{{6}{17}}
+\newlabel{lm:h}{{6.2}{17}}
+\newlabel{prop:stop}{{6.4}{18}}
+\newlabel{prop:lambda}{{6.5}{18}}
 \citation{proba}
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-\newlabel{prop:lambda}{{6.5}{16}}
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-\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{19}}
-\newlabel{table:nc}{{1}{19}}
+\newlabel{lm:stopprime}{{6.6}{19}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.3}{Practical Evaluation of Stopping Times}}{20}}
+\newlabel{sub:stop:exp}{{6.3}{20}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Pseudo Code of the stoping time calculus\relax }}{20}}
+\newlabel{algo:stop}{{2}{20}}
+\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Average Stopping Time\relax }}{21}}
+\newlabel{table:stopping:moy}{{1}{21}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Experiments}}{21}}
+\newlabel{sec:prng}{{7}{21}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {3}{\ignorespaces Pseudo Code of the $\chi _{\textit  {15Rairo}}$ PRNG\relax }}{21}}
+\newlabel{CI Algorithm:2}{{3}{21}}
+\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{22}}
+\newlabel{table:nc}{{2}{22}}
 \bibstyle{alpha}
 \bibdata{biblio}
 \bibcite{Banks92}{BBCS92}
-\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces NIST SP 800-22 test results ($\mathbb  {P}_T$)\relax }}{20}}
-\newlabel{The passing rate}{{2}{20}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{20}}
+\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces NIST SP 800-22 test results ($\mathbb  {P}_T$)\relax }}{23}}
+\newlabel{The passing rate}{{3}{23}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{23}}
 \bibcite{bcgr11:ip}{BCGR11}
 \bibcite{Nist10}{BR10}
 \bibcite{DBLP:journals/combinatorics/BhatS96}{BS96}
 \bibcite{Bykov2016}{Byk16}
-\bibcite{chgw14oip}{CHG{$^{+}$}14a}
-\bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14b}
+\bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14}
 \bibcite{5376454}{CMZ09}
 \bibcite{Devaney}{Dev89}
 \bibcite{guyeuxTaiwan10}{GWB10}
@@ -111,9 +124,9 @@
 \bibcite{915396}{SPK01}
 \bibcite{ZanSup04}{SZ04}
 \bibcite{wbg10ip}{WBGF10}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{21}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.77466pt}
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+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{24}}
diff --git a/main.bbl b/main.bbl
index e2aaabb..1bd49a8 100644
--- a/main.bbl
+++ b/main.bbl
@@ -1,5 +1,5 @@
 \newcommand{\etalchar}[1]{$^{#1}$}
-\begin{thebibliography}{CHG{\etalchar{+}}14b}
+\begin{thebibliography}{WBGF10}
 
 \bibitem[BBCS92]{Banks92}
 J.~Banks, J.~Brooks, G.~Cairns, and P.~Stacey.
@@ -30,15 +30,7 @@ I.~S. Bykov.
 \newblock {\em Journal of Applied and Industrial Mathematics}, 10(1):78--85,
   2016.
 
-\bibitem[CHG{\etalchar{+}}14a]{chgw14oip}
-Jean-Fran\c{c}ois Couchot, Pierre-Cyrille H\'eam, Christophe Guyeux, Qianxue
-  Wang, and Jacques Bahi.
-\newblock Pseudorandom number generators with balanced gray codes.
-\newblock In {\em Secrypt 2014, 11th Int. Conf. on Security and Cryptography},
-  pages 469--475, Vienna, Austria, August 2014.
-\newblock Position short paper.
-
-\bibitem[CHG{\etalchar{+}}14b]{DBLP:conf/secrypt/CouchotHGWB14}
+\bibitem[CHG{\etalchar{+}}14]{DBLP:conf/secrypt/CouchotHGWB14}
 Jean{-}Fran{\c{c}}ois Couchot, Pierre{-}Cyrille H{\'{e}}am, Christophe Guyeux,
   Qianxue Wang, and Jacques~M. Bahi.
 \newblock Pseudorandom number generators with balanced gray codes.
diff --git a/main.blg b/main.blg
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 subscribe by emailing sympa@lirmm.fr with 'subscribe <list> <firstname name>'
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+- Author: Christophe Fiorio (christophe.fiorio@umontpellier.fr)
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+   defining Unicode char U+0135 (decimal 309)
+   defining Unicode char U+0136 (decimal 310)
+   defining Unicode char U+0137 (decimal 311)
    defining Unicode char U+0139 (decimal 313)
    defining Unicode char U+013A (decimal 314)
+   defining Unicode char U+013B (decimal 315)
+   defining Unicode char U+013C (decimal 316)
    defining Unicode char U+013D (decimal 317)
    defining Unicode char U+013E (decimal 318)
    defining Unicode char U+0141 (decimal 321)
    defining Unicode char U+0142 (decimal 322)
    defining Unicode char U+0143 (decimal 323)
    defining Unicode char U+0144 (decimal 324)
+   defining Unicode char U+0145 (decimal 325)
+   defining Unicode char U+0146 (decimal 326)
    defining Unicode char U+0147 (decimal 327)
    defining Unicode char U+0148 (decimal 328)
    defining Unicode char U+014A (decimal 330)
    defining Unicode char U+014B (decimal 331)
+   defining Unicode char U+014C (decimal 332)
+   defining Unicode char U+014D (decimal 333)
+   defining Unicode char U+014E (decimal 334)
+   defining Unicode char U+014F (decimal 335)
    defining Unicode char U+0150 (decimal 336)
    defining Unicode char U+0151 (decimal 337)
    defining Unicode char U+0152 (decimal 338)
    defining Unicode char U+0153 (decimal 339)
    defining Unicode char U+0154 (decimal 340)
    defining Unicode char U+0155 (decimal 341)
+   defining Unicode char U+0156 (decimal 342)
+   defining Unicode char U+0157 (decimal 343)
    defining Unicode char U+0158 (decimal 344)
    defining Unicode char U+0159 (decimal 345)
    defining Unicode char U+015A (decimal 346)
    defining Unicode char U+015B (decimal 347)
+   defining Unicode char U+015C (decimal 348)
+   defining Unicode char U+015D (decimal 349)
    defining Unicode char U+015E (decimal 350)
    defining Unicode char U+015F (decimal 351)
    defining Unicode char U+0160 (decimal 352)
@@ -437,10 +485,22 @@ File: t1enc.dfu 2014/09/29 v1.1m UTF-8 support for inputenc
    defining Unicode char U+0163 (decimal 355)
    defining Unicode char U+0164 (decimal 356)
    defining Unicode char U+0165 (decimal 357)
+   defining Unicode char U+0168 (decimal 360)
+   defining Unicode char U+0169 (decimal 361)
+   defining Unicode char U+016A (decimal 362)
+   defining Unicode char U+016B (decimal 363)
+   defining Unicode char U+016C (decimal 364)
+   defining Unicode char U+016D (decimal 365)
    defining Unicode char U+016E (decimal 366)
    defining Unicode char U+016F (decimal 367)
    defining Unicode char U+0170 (decimal 368)
    defining Unicode char U+0171 (decimal 369)
+   defining Unicode char U+0172 (decimal 370)
+   defining Unicode char U+0173 (decimal 371)
+   defining Unicode char U+0174 (decimal 372)
+   defining Unicode char U+0175 (decimal 373)
+   defining Unicode char U+0176 (decimal 374)
+   defining Unicode char U+0177 (decimal 375)
    defining Unicode char U+0178 (decimal 376)
    defining Unicode char U+0179 (decimal 377)
    defining Unicode char U+017A (decimal 378)
@@ -448,6 +508,31 @@ File: t1enc.dfu 2014/09/29 v1.1m UTF-8 support for inputenc
    defining Unicode char U+017C (decimal 380)
    defining Unicode char U+017D (decimal 381)
    defining Unicode char U+017E (decimal 382)
+   defining Unicode char U+01CD (decimal 461)
+   defining Unicode char U+01CE (decimal 462)
+   defining Unicode char U+01CF (decimal 463)
+   defining Unicode char U+01D0 (decimal 464)
+   defining Unicode char U+01D1 (decimal 465)
+   defining Unicode char U+01D2 (decimal 466)
+   defining Unicode char U+01D3 (decimal 467)
+   defining Unicode char U+01D4 (decimal 468)
+   defining Unicode char U+01E2 (decimal 482)
+   defining Unicode char U+01E3 (decimal 483)
+   defining Unicode char U+01E6 (decimal 486)
+   defining Unicode char U+01E7 (decimal 487)
+   defining Unicode char U+01E8 (decimal 488)
+   defining Unicode char U+01E9 (decimal 489)
+   defining Unicode char U+01EA (decimal 490)
+   defining Unicode char U+01EB (decimal 491)
+   defining Unicode char U+01F0 (decimal 496)
+   defining Unicode char U+01F4 (decimal 500)
+   defining Unicode char U+01F5 (decimal 501)
+   defining Unicode char U+0218 (decimal 536)
+   defining Unicode char U+0219 (decimal 537)
+   defining Unicode char U+021A (decimal 538)
+   defining Unicode char U+021B (decimal 539)
+   defining Unicode char U+01E02 (decimal 7682)
+   defining Unicode char U+01E03 (decimal 7683)
    defining Unicode char U+200C (decimal 8204)
    defining Unicode char U+2013 (decimal 8211)
    defining Unicode char U+2014 (decimal 8212)
@@ -467,9 +552,11 @@ Now handling font encoding OT1 ...
 ... processing UTF-8 mapping file for font encoding OT1
 
 (/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu
-File: ot1enc.dfu 2014/09/29 v1.1m UTF-8 support for inputenc
+File: ot1enc.dfu 2015/12/03 v1.1r UTF-8 support for inputenc
+   defining Unicode char U+00A0 (decimal 160)
    defining Unicode char U+00A1 (decimal 161)
    defining Unicode char U+00A3 (decimal 163)
+   defining Unicode char U+00AD (decimal 173)
    defining Unicode char U+00B8 (decimal 184)
    defining Unicode char U+00BF (decimal 191)
    defining Unicode char U+00C5 (decimal 197)
@@ -487,6 +574,14 @@ File: ot1enc.dfu 2014/09/29 v1.1m UTF-8 support for inputenc
    defining Unicode char U+0142 (decimal 322)
    defining Unicode char U+0152 (decimal 338)
    defining Unicode char U+0153 (decimal 339)
+   defining Unicode char U+0174 (decimal 372)
+   defining Unicode char U+0175 (decimal 373)
+   defining Unicode char U+0176 (decimal 374)
+   defining Unicode char U+0177 (decimal 375)
+   defining Unicode char U+0218 (decimal 536)
+   defining Unicode char U+0219 (decimal 537)
+   defining Unicode char U+021A (decimal 538)
+   defining Unicode char U+021B (decimal 539)
    defining Unicode char U+2013 (decimal 8211)
    defining Unicode char U+2014 (decimal 8212)
    defining Unicode char U+2018 (decimal 8216)
@@ -498,7 +593,7 @@ Now handling font encoding OMS ...
 ... processing UTF-8 mapping file for font encoding OMS
 
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-File: omsenc.dfu 2014/09/29 v1.1m UTF-8 support for inputenc
+File: omsenc.dfu 2015/12/03 v1.1r UTF-8 support for inputenc
    defining Unicode char U+00A7 (decimal 167)
    defining Unicode char U+00B6 (decimal 182)
    defining Unicode char U+00B7 (decimal 183)
@@ -526,19 +621,21 @@ Package: fontenc 2005/09/27 v1.99g Standard LaTeX package
 
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 File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file
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+Package: babel 2016/02/24 3.9q The Babel package
 
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 Language: english 2012/08/20 v3.3p English support from the babel system
 
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-File: babel.def 2014/09/25 3.9l Babel common definitions
+File: babel.def 2016/02/24 3.9q Babel common definitions
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 \U@D=\dimen129
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+\l@british = a dialect from \language\l@english 
+\l@UKenglish = a dialect from \language\l@english 
 \l@canadian = a dialect from \language\l@american 
 \l@australian = a dialect from \language\l@british 
 \l@newzealand = a dialect from \language\l@british 
@@ -551,7 +648,7 @@ LaTeX Font Info:    Redeclaring math symbol \boxdot on input line 44.
 Package: latexsym 1998/08/17 v2.2e Standard LaTeX package (lasy symbols)
 \symlasy=\mathgroup7
 LaTeX Font Info:    Overwriting symbol font `lasy' in version `bold'
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+Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK)
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 (/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex
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 \XKV@tempa@toks=\toks29
 )
 \XKV@depth=\count114
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+`PSTricks' v2.64b  <2015/14/11> (tvz)
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+File: pstricks.tex 2015/14/11 v2.64b `PSTricks' (tvz,hv)
 
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 Package: pst-node 2012/09/18 v1.01 LaTeX wrapper for `pst-node' (HV)
 Package: pst-node 2010/04/22 package wrapper for pst-node.tex
@@ -738,8 +835,8 @@ Package: pst-node 2010/04/22 package wrapper for pst-node.tex
 \psrow=\count145
 \pscol=\count146
 \psmatrixcnt=\count147
-\psrowsep=\skip67
-\pscolsep=\skip68
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 \num@pts=\count149
 \pst@argcnt=\count150
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 (/usr/share/texlive/texmf-dist/tex/generic/pst-coil/pst-coil.tex
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+File: pst-coil.tex 2015/05/13 v1.07 `PST-coil' (tvz,hv)
 )
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 \Urlmuskip=\muskip11
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 Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS)
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 (/usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
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 (/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)
 \pgf@x=\dimen170
@@ -807,7 +904,7 @@ File: pgf.cfg 2008/05/14  (rcs-revision 1.7)
 Driver file for pgf: pgfsys-pdftex.def
 
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
-File: pgfsys-pdftex.def 2013/07/18  (rcs-revision 1.33)
+File: pgfsys-pdftex.def 2014/10/11  (rcs-revision 1.35)
 
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.de
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@@ -824,7 +921,7 @@ tex
 File: pgfsysprotocol.code.tex 2006/10/16  (rcs-revision 1.4)
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-Package: pgfcore 2010/04/11 v3.0.0 (rcs-revision 1.7)
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 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.te
@@ -853,13 +950,13 @@ File: pgfcorepathconstruct.code.tex 2013/10/07  (rcs-revision 1.29)
 )
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code
 .tex
-File: pgfcorepathusage.code.tex 2013/12/13  (rcs-revision 1.23)
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 \pgf@shorten@start@additional=\dimen195
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 x
-File: pgfcorescopes.code.tex 2013/10/09  (rcs-revision 1.44)
+File: pgfcorescopes.code.tex 2015/05/08  (rcs-revision 1.46)
 \pgfpic=\box49
 \pgf@hbox=\box50
 \pgf@layerbox@main=\box51
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 )
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.c
 ode.tex
-File: pgfcoregraphicstate.code.tex 2013/09/19  (rcs-revision 1.11)
+File: pgfcoregraphicstate.code.tex 2014/11/02  (rcs-revision 1.12)
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 )
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformation
 s.code.tex
-File: pgfcoretransformations.code.tex 2013/10/10  (rcs-revision 1.17)
+File: pgfcoretransformations.code.tex 2015/08/07  (rcs-revision 1.20)
 \pgf@pt@x=\dimen197
 \pgf@pt@y=\dimen198
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 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
 File: pgfcorequick.code.tex 2008/10/09  (rcs-revision 1.3)
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 )
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 x
-File: pgfcorearrows.code.tex 2013/11/07  (rcs-revision 1.40)
-\pgfarrowsep=\dimen200
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 )
 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
 File: pgfcoreshade.code.tex 2013/07/15  (rcs-revision 1.15)
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 (/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
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 tex
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 .code.tex
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-
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+) (./chaos.tex [4 <./images/iter_f0c.pdf>] [5]
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 mr/m/n/10 dig-its aim at mea-sur-ing how much $\OML/cmm/m/it/10 u[]; u[]; [] ; 
@@ -1344,13 +1367,13 @@ Overfull \hbox (2.48117pt too wide) in paragraph at lines 374--377
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  []
 
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-<graphe2.pdf, id=74, 194.7275pt x 173.64874pt>
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 File: graphe2.pdf Graphic file (type pdf)
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 Package pdftex.def Info: graphe2.pdf used on input line 409.
@@ -1369,94 +1392,15 @@ r/m/n/10 ))\OML/cmm/m/it/10 ; [] \OT1/cmr/m/n/10 = ([]\OML/cmm/m/it/10 ; \OT1/c
 mr/m/n/10 ([]\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 2
  []
 
-
+[9 <./graphe1.pdf> <./graphe2.pdf>]
 Underfull \hbox (badness 10000) in paragraph at lines 472--477
 []\T1/cmr/m/n/10 Let $\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 e;
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-[9 <./graphe1.pdf> <./graphe2.pdf>]) (./generating.tex
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+) (./generating.tex [10]) (./hamilton.tex [11] [12]
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 []\T1/cmr/m/it/10 Let now $\OML/cmm/m/it/10 L[] \OT1/cmr/m/n/10 = 0000\OML/cmm/
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 []\T1/cmr/m/it/10 On the con-trary, for the stan-dard $\OT1/cmr/m/n/10 4$\T1/cm
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@@ -1479,170 +1423,21 @@ t/10 ; \OT1/cmr/m/n/10 0100\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1100\OML/cmm/m/it
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  []
 
-Runaway argument?
-{2^{\mathsf {N} -n.a_{\mathsf {N}}}{2} \\ c_{\mathsf {N}} = \mathsf {\ETC.
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-
-! LaTeX Error: Something's wrong--perhaps a missing \item.
-
-See the LaTeX manual or LaTeX Companion for explanation.
-Type  H <return>  for immediate help.
- ...                                              
-                                                  
-l.194 \subsection
-                 {Toward a local uniform distribution of switches}
-Try typing  <return>  to proceed.
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-
-
-! LaTeX Error: Something's wrong--perhaps a missing \item.
-
-See the LaTeX manual or LaTeX Companion for explanation.
-Type  H <return>  for immediate help.
- ...                                              
-                                                  
-l.194 \subsection
-                 {Toward a local uniform distribution of switches}
-Try typing  <return>  to proceed.
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-! Missing } inserted.
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-                }
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-I've inserted something that you may have forgotten.
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-! Missing \cr inserted.
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-)
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-I've inserted something that you may have forgotten.
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-      
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diff --git a/main.tex b/main.tex
index 773caee..0ea729e 100644
--- a/main.tex
+++ b/main.tex
@@ -86,10 +86,11 @@
 
 \begin{document}
 
-\author{Jean-François Couchot, Christophe Guyeux, Pierre-Cyrille Heam}
-\address{FEMTO-ST Institute, University of Franche-Comté, Belfort, France}
+\author{Sylvain Contassot-Vivier, Jean-François Couchot, Christophe Guyeux, Pierre-Cyrille Heam}
+\address{LORIA, Université de Lorraine, Nancy, France\\
+FEMTO-ST Institute, University of Franche-Comté, Belfort, France}
 
-\keywords{Pseudorandom Number Generator, Theory of Chaos, Markov Matrice, Hamiltonian Path, Mixing Time, Stopping Time, Statistical Test}
+\keywords{Pseudorandom Number Generator, Theory of Chaos, Markov Matrice, Hamiltonian Path,  Stopping Time, Statistical Test}
 
 \subjclass{34C28, 37A25,11K45}
 
@@ -117,7 +118,7 @@ the classical statistical tests.
 \section{Functions with Strongly Connected $\Gamma_{\{b\}}(f)$}\label{sec:SCCfunc}
 \input{generating}
 
-\section{(Locally) Balanced Hamiltonian Cycle}\label{sec:hamilton}
+\section{Balanced Hamiltonian Cycle}\label{sec:hamilton}
 \input{hamilton}
 
 
diff --git a/preliminaries.tex b/preliminaries.tex
index 1200bc5..7c6b050 100644
--- a/preliminaries.tex
+++ b/preliminaries.tex
@@ -94,7 +94,9 @@ return $x$\;
 
 
 
-With all this material, we can study the chaos properties of these 
+
+Based on this setup,
+we can study the chaos properties of these 
 function.
 This is the aims of the next section. 
 
diff --git a/stopping.tex b/stopping.tex
index 3a07e06..6632a23 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -1,11 +1,6 @@
-
-
-
-Let thus be given such kind of map.
-This article focuses on studying its iterations according to
-the equation~(\ref{eq:asyn}) with a given strategy.
-First of all, this can be interpreted as walking into its iteration graph 
-where the choice of the edge to follow is decided by the strategy.
+This section considers functions $f: \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}} $ 
+issued from an hypercube where an Hamiltonian path has been removed
+as described in previous section.
 Notice that the iteration graph is always a subgraph of 
 ${\mathsf{N}}$-cube augmented with all the self-loop, \textit{i.e.}, all the 
 edges $(v,v)$ for any $v \in \Bool^{\mathsf{N}}$. 
@@ -43,50 +38,19 @@ P=\dfrac{1}{6} \left(
 \end{xpl}
 
 
-% % Let us first recall the  \emph{Total Variation} distance $\tv{\pi-\mu}$,
-% % which is defined for two distributions $\pi$ and $\mu$ on the same set 
-% % $\Bool^n$  by:
-% % $$\tv{\pi-\mu}=\max_{A\subset \Bool^n} |\pi(A)-\mu(A)|.$$ 
-% % It is known that
-% % $$\tv{\pi-\mu}=\frac{1}{2}\sum_{x\in\Bool^n}|\pi(x)-\mu(x)|.$$
-
-% % Let then $M(x,\cdot)$ be the
-% % distribution induced by the $x$-th row of $M$. If the Markov chain
-% % induced by
-% % $M$ has a stationary distribution $\pi$, then we define
-% % $$d(t)=\max_{x\in\Bool^n}\tv{M^t(x,\cdot)-\pi}.$$
-% Intuitively $d(t)$ is the largest deviation between
-% the distribution $\pi$ and $M^t(x,\cdot)$, which 
-% is the result of iterating $t$ times the function.
-% Finally, let $\varepsilon$ be a positive number, the \emph{mixing time} 
-% with respect to $\varepsilon$ is given by
-% $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
-% It defines the smallest iteration number 
-% that is sufficient to obtain a deviation lesser than $\varepsilon$.
-% Notice that the upper and lower bounds of mixing times cannot    
-% directly be computed with eigenvalues formulae as expressed 
-% in~\cite[Chap. 12]{LevinPeresWilmer2006}. The authors of this latter work  
-% only consider reversible Markov matrices whereas we do no restrict our 
-% matrices to such a form.
-
-
-
-
-
-
-
-This section considers functions $f: \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}} $ 
-issued from an hypercube where an Hamiltonian path has been removed.
 A specific random walk in this modified hypercube is first 
-introduced. We further detail
-a theoretical study on the length of the path 
-which is sufficient to follow to get a uniform distribution.
+introduced (See section~\ref{sub:stop:formal}). We further 
+theoretical study this random walk to 
+provide a upper bound of fair sequences 
+(See section~\ref{sub:stop:bound}).
+We finally complete these study with experimental
+results that reduce this bound (Sec.~\ref{sub:stop:stop}).
 Notice that for a general references on Markov chains
-see~\cite{LevinPeresWilmer2006}
-, and particularly Chapter~5 on stopping times.  
-
+see~\cite{LevinPeresWilmer2006}, 
+and particularly Chapter~5 on stopping times.  
 
 
+\subsection{Formalizing the Random Walk}\label{sub:stop:formal}
 
 First of all, let $\pi$, $\mu$ be two distributions on $\Bool^{\mathsf{N}}$. The total
 variation distance between $\pi$ and $\mu$ is denoted $\tv{\pi-\mu}$ and is
@@ -104,6 +68,13 @@ $$d(t)=\max_{X\in\Bool^{\mathsf{N}}}\tv{P^t(X,\cdot)-\pi}.$$
 and
 
 $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
+
+Intuitively speaking, $t_{\rm mix}$ is a mixing time 
+\textit{i.e.}, is the time until the matrix $X$ of a Markov chain  
+is $\epsilon$-close to a stationary distribution.
+
+
+
 One can prove that
 
 $$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
@@ -140,6 +111,9 @@ randomized stopping time (possibly depending on the starting position $X$),
 such that  the distribution of $X_\tau$ is $\pi$:
 $$\P_X(X_\tau=Y)=\pi(Y).$$
 
+\subsection{Upper bound of Stopping Time}\label{sub:stop:bound}
+
+
 A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is
 independent of $\tau$. 
 
@@ -250,8 +224,11 @@ $b=1$ with probability $\frac{1}{2}$ and $b=0$ with probability
 $\frac{1}{2}$. Since $h(X_{\tau_\ell-1})\neq\ell$ the value of the $\ell$-th
 bit of $X_{\tau_\ell}$ 
 is $0$ or $1$ with the same probability ($\frac{1}{2}$).
+This probability is independent of the value of the other bits.
+
 
- Moving next in the chain, at each step,
+
+Moving next in the chain, at each step,
 the $l$-th bit  is switched from $0$ to $1$ or from $1$ to $0$ each time with
 the same probability. Therefore,  for $t\geq \tau_\ell$, the
 $\ell$-th bit of $X_t$ is $0$ or $1$ with the same probability, proving the
@@ -374,4 +351,60 @@ there are one ongoing arc and one outgoing arc that are removed.
 The calculus does not consider (balanced) Hamiltonian cycles, which 
 are more regular and more binding than this constraint.
 In this later context, we claim that the upper bound for the stopping time 
-should be reduced.
+should be reduced. This fact is studied in the next section.
+
+\subsection{Practical Evaluation of Stopping Times}\label{sub:stop:exp}
+ 
+Let be given a function $f: \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$
+and an initial seed $x^0$.
+The pseudo code given in algorithm~\ref{algo:stop} returns the smallest 
+number of iterations such that all elements $\ell\in \llbracket 1,{\mathsf{N}} \rrbracket$ are fair. It allows to deduce an approximation of $E[\ts]$
+by calling this code many times with many instances of function and many 
+seeds.
+
+Practically speaking, for each number $\mathsf{N}$,$ 3 \le \mathsf{N} \le 16$, 
+10 functions have been generaed according to method presented in section~\ref{sec:hamilton}. For each of them, the calculus of the approximation of $E[\ts]$
+is executed 10000 times with a random seed. The table~\ref{table:stopping:moy}
+summarizes results. It can be observed that the approximation is largely
+smaller than the upper bound given in theorem~\ref{prop:stop}.
+
+\begin{algorithm}[ht]
+%\begin{scriptsize}
+\KwIn{a function $f$, an initial configuration $x^0$ ($\mathsf{N}$ bits)}
+\KwOut{a number of iterations $\textit{nbit}$}
+
+$\textit{nbit} \leftarrow 0$\;
+$x\leftarrow x^0$\;
+$\textit{visited}\leftarrow\emptyset$\;
+
+\While{$\left\vert{\textit{visited}}\right\vert < \mathsf{N} $}
+{
+        $ s \leftarrow \textit{Random}(n)$ \;
+        $\textit{image} \leftarrow f(x) $\;
+        \If{$x[s] \neq \textit{image}[s]$}{
+            $\textit{visited} \leftarrow \textit{visited} \cup \{s\}$
+          }
+        $x[s] \leftarrow \textit{image}[s]$\;
+        $\textit{nbit} \leftarrow \textit{nbit}+1$\;
+}
+\Return{$\textit{nbit}$}\;
+%\end{scriptsize}
+\caption{Pseudo Code of the stoping time calculus}
+\label{algo:stop}
+\end{algorithm}
+
+
+
+
+\begin{table}
+$$
+\begin{array}{|*{15}{l|}}
+\hline
+\mathsf{N}  & 3 & 4 & 5 & 6 & 7& 8 & 9 & 10& 11 & 12 & 13 & 14 & 15 & 16 \\
+\hline
+\mathsf{N}  & 3 & 10.9 & 5 & 17.7 & 7& 25 & 9 & 32.7& 11 & 40.8 & 13 & 49.2 & 15 & 16 \\
+\hline
+\end{array}
+$$
+\caption{Average Stopping Time}\label{table:stopping:moy}
+\end{table}
-- 
2.39.5