diff --git a/amozeika/comm/README.md b/amozeika/comm/README.md
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+This repository contains the python code which computes the number of committees, K, given the total number of nodes, N, (a detailed description of the algorithm is provided in https://www.overleaf.com/read/yqvczvrmsgts) and the simulation code which uses this K to compute the prob. that in at least one of these committees the number of adversarial nodes exceeds a fraction of the committee size.
diff --git a/amozeika/comm/comm.py b/amozeika/comm/comm.py
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+++ b/amozeika/comm/comm.py
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+import math
+from scipy.stats import binom
+def comm(N,delta,A,P):
+#N is the number of nodes, delta is the failure prob. which can be tolerated, A is the fraction of a committee (typical value is  1/3) and P  is the fraction of adversarial nodes (typical value is 1/4). 
+    K = 1
+    n = N
+    r = 0
+    Prob = 0.0
+    m = 0
+    while Prob < delta:
+        K_1 = K
+        n_1 = n
+        r_1 = r
+        Prob_1 = Prob
+        m = m +1
+        K = 2*m + 1
+        n = N // K
+        r = N % K
+        if 0 < r:
+            Prob_n = binom.cdf(math.floor(A * n), n, P)
+            Prob_n1 = binom.cdf(math.floor(A * (n+1)), n+1, P)
+            Prob = 1 - Prob_n ** (K - r) * Prob_n1 ** r
+        else:
+            Prob_n = binom.cdf(math.floor(A * n), n, P)
+            Prob = 1 - Prob_n ** K
+    #return number of committees, K_1, committee size, n_1, number of committees with size n_1+1, r_1 and prob. of failure, Prob_1. 
+    return K_1, n_1, r_1, Prob_1;
diff --git a/amozeika/comm/rand_part_3_v2.r b/amozeika/comm/rand_part_3_v2.r
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+#This code simulates the assignment  of N nodes  into the K committees.
+#A node  is adversarial with the probability  p1, i.e. the number of adversarial nodes is N x p1 on average.
+#The code computes the probability that in at least one of the committees the number of adversarial nodes  exceeds the A x the number of nodes  in the committee, where  A is the number between 0 and 1.
+#For details see https://www.overleaf.com/read/yqvczvrmsgts
+#To run code  launch R from the terminal and type  source('rand_part_3_v2.r'); The output is in rand_num* files.
+code_name<-'rand_part_3_v2.r';
+rand_num=9038;
+set.seed(rand_num);
+print(rand_num);
+##################parameters##############################################
+M=10^6; #number of samples
+N=773; #total number of nodes
+K=3; #number of committees
+p1=1/4; #fraction of advrsarial nodes
+A=1/3; #fraction of nodes  in a reservoir
+n=N%/%K; #compute quotient
+r=N%%K; #compute remainder
+###################generate array of committee sizes such that K-r committees are of size n and r committee are of size n+1########################
+comm=replicate(K,n);
+for (k in 1:r) comm[k]=comm[k]+1;
+###########################################################################
+in_file<-paste(rand_num,"_param.txt",sep="");
+out_file<-paste(rand_num,"_stats.csv", sep="");
+###########################################################################
+write(paste("r-code=", code_name,", rand_num=",rand_num, ", M=",M,", N=",N, ", K=", K, ", n=", n, ", r=", r, ", p1=",p1,", A=", A, sep=""), in_file, ncolumns =1, append = FALSE,  sep = "\n");
+write(paste("K", "Prob",  sep = "\t"), out_file, ncolumns =2, append = FALSE,  sep = "\t");
+start_time <- Sys.time();
+Prob=0;
+for (j in 1:M)
+{
+    s<-rbinom(N,1, p1); #generate 0/1
+    N0<-replicate(K,0);
+    N1<-replicate(K,0);
+    mu=1;
+    cntr=0;
+    for (i in 1:N)
+    {
+                    if (s[i]>0)
+                    {
+                        N1[mu]=N1[mu]+1;
+                    } else
+                        {
+                            N0[mu]=N0[mu]+1;
+                        }
+                    cntr=cntr+1;
+                    if (cntr==comm[mu])
+                    {
+                        cntr=0;
+                        mu=mu+1;
+                    }
+                    
+    }
+                SUM=0;
+                for (mu in 1:K)
+                {
+                    if (N1[mu]>=(floor((A*(N1[mu]+N0[mu])))+1))
+                    {
+                        SUM=SUM+1;
+                    }
+                }
+                if (SUM>0)
+                {
+                    Prob=Prob+1;
+                }
+}
+Prob=Prob/M;
+write(paste(K,Prob,sep = "\t"), out_file, ncolumns =1, append =TRUE,  sep = "\t");
+end_time <- Sys.time();
+print(end_time - start_time);
diff --git a/amozeika/comm/rand_part_4_v2.r b/amozeika/comm/rand_part_4_v2.r
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+#This code simulates random assignment  of N nodes  into the K committees.
+#The number of adversarial nodes is  N_r (N_r < N).
+#The code computes the probability that in at least one of the committees the number of adversarial nodes exceeds the A x the number of nodes  in the committee, where  A is the number between 0 and 1.
+#For details see https://www.overleaf.com/read/yqvczvrmsgts
+#To run code  launch R from the terminal and type  source('rand_part_4_v2.r'); The output is in rand_num* files.
+code_name<-'rand_part_4_v2.r';
+rand_num=7800; 
+set.seed(rand_num);
+print(rand_num);
+##################parameters##############################################
+M=10^6; #number of samples
+N=773; #total number of nodes
+K=3; #number of committees
+N_r=floor(N/4); #number of adversarial nodes
+A=1/3; #fraction of nodes  in a committee
+n=N%/%K; #compute quotient
+r=N%%K; #compute remainder
+###################generate array of committee sizes such that K-r committees are of size n and r committees are of size n+1########################
+comm=replicate(K,n);
+for (k in 1:r) comm[k]=comm[k]+1;
+###########################################################################
+in_file<-paste(rand_num,"_param.txt",sep="");
+out_file<-paste(rand_num,"_stats.csv", sep="");
+###########################################################################
+write(paste("r-code=", code_name,", rand_num=",rand_num, ", M=",M,", N=",N, ", K=",K,", n=",n, ", r=",r, ", N/N_r=",(N/N_r),", A=", A, sep=""), in_file, ncolumns =1, append = FALSE,  sep = "\n");
+write(paste("K", "Prob",  sep = "\t"), out_file, ncolumns =2, append = FALSE,  sep = "\t");
+start_time <- Sys.time();
+#generate binary sequence with exactly N_r of 1's
+s0<-replicate(N,0);
+for (i in 1:N_r)
+{
+    s0[i]=1;
+};
+Prob=0;
+for (j in 1:M)
+{
+    s<-sample(s0);
+    N0<-replicate(K,0);
+    N1<-replicate(K,0);
+    mu=1;
+    cntr=0;
+    for (i in 1:N)
+    {
+        if (s[i]>0)
+        {
+            N1[mu]=N1[mu]+1;
+        } else
+                {
+                    N0[mu]=N0[mu]+1;
+                }
+        cntr=cntr+1;
+        if (cntr==comm[mu])
+        {
+            cntr=0;
+            mu=mu+1;
+        }
+                        
+    }
+                    SUM=0;
+                    for (mu in 1:K)
+                    {
+                        if (N1[mu]>=(floor((A*(N1[mu]+N0[mu])))+1))
+                        {
+                            SUM=SUM+1;
+                        }
+                    }
+                    if (SUM>0)
+                    {
+                        Prob=Prob+1;
+                    }
+}
+Prob=Prob/M;
+write(paste(K,Prob,sep = "\t"), out_file, ncolumns =1, append =TRUE,  sep = "\t");
+end_time <- Sys.time();
+print(end_time - start_time);