From 550ff72727a1829bb72f5c40cffb96f2225fae84 Mon Sep 17 00:00:00 2001
From: tomsmeding <tom.smeding@gmail.com>
Date: Tue, 4 Oct 2016 11:07:14 +0200
Subject: More primes and primality testing

---
 bigint.cpp  |  3 ++-
 bigint.h    |  3 ++-
 main.cpp    | 14 ++++++++++++-
 numalgo.cpp | 20 ++++++++++++++++++
 numalgo.h   |  2 ++
 primes.cpp  | 70 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++--
 primes.h    | 10 +++++++--
 7 files changed, 115 insertions(+), 7 deletions(-)

diff --git a/bigint.cpp b/bigint.cpp
index 644ffb3..21c4ab2 100644
--- a/bigint.cpp
+++ b/bigint.cpp
@@ -7,9 +7,10 @@
 
 using namespace std;
 
+Bigint Bigint::mone(-1);
 Bigint Bigint::zero(0);
 Bigint Bigint::one(1);
-Bigint Bigint::mone(-1);
+Bigint Bigint::two(2);
 
 Bigint::Bigint()
 	:sign(1){}
diff --git a/bigint.h b/bigint.h
index 213bf5c..76f5415 100644
--- a/bigint.h
+++ b/bigint.h
@@ -89,9 +89,10 @@ public:
 	digit_t _digit(int idx) const;
 
 
+	static Bigint mone;
 	static Bigint zero;
 	static Bigint one;
-	static Bigint mone;
+	static Bigint two;
 };
 
 std::istream& operator>>(std::istream&,Bigint&);
diff --git a/main.cpp b/main.cpp
index 2d652b4..db846a9 100644
--- a/main.cpp
+++ b/main.cpp
@@ -144,10 +144,22 @@ void performrsa(){
 	cout<<"msg = "<<msg2<<endl;
 }
 
+void fermatpseudo(){
+	fillsmallprimes();
+	for(int i=2;i<65000;i++){
+		if(!strongPseudoPrime2(Bigint(i)))continue;
+		if(!binary_search(smallprimes.begin(),smallprimes.end(),i))cout<<i<<' ';
+	}
+	cout<<endl;
+}
+
 int main(int argc,char **argv){
+	(void)argc;
+	(void)argv;
 	// biginttest();
 	// repl(argc,argv);
 	// performrsa();
 	// testisqrt(argc,argv);
-	fillsmallprimes();
+	// randprime(Bigint(20),Bigint(42));
+	// fermatpseudo();
 }
diff --git a/numalgo.cpp b/numalgo.cpp
index f2ebac5..9b1681c 100644
--- a/numalgo.cpp
+++ b/numalgo.cpp
@@ -43,6 +43,26 @@ Bigint expmod(const Bigint &b,const Bigint &e,const Bigint &m){
 	return res;
 }
 
+int jacobiSymbol(Bigint a,Bigint n){
+	//https://en.wikipedia.org/wiki/Jacobi_symbol#Calculating_the_Jacobi_symbol
+	assert(n>0&&n.odd());
+	int res=1;
+	while(true){
+		a=a.divmod(n).second;
+		if(a<0)a+=n;
+		int nmod8=n.divmod(Bigint(8)).second.lowdigits();
+		int two_over_n=nmod8==1||nmod8==7?1:-1;
+		while(a.even()){
+			res*=two_over_n;
+			a>>=1;
+		}
+		if(a==1)return res;
+		if(gcd(a,n)!=1)return 0;
+		if(a.divmod(Bigint(4)).second.lowdigits()==3&&nmod8%4==3)res*=-1;
+		swap(a,n);
+	}
+}
+
 Bigint isqrt(const Bigint &n){
 	assert(n>=0);
 	if(n<=1)return n;
diff --git a/numalgo.h b/numalgo.h
index 997a67c..2304e94 100644
--- a/numalgo.h
+++ b/numalgo.h
@@ -8,6 +8,8 @@ Bigint egcd(const Bigint &a,const Bigint &b,Bigint &x,Bigint &y);
 
 Bigint expmod(const Bigint &base,const Bigint &exponent,const Bigint &modulus);
 
+int jacobiSymbol(Bigint a,Bigint n);
+
 // Returns sqrt(n), rounded down if necessary
 Bigint isqrt(const Bigint &n);
 
diff --git a/primes.cpp b/primes.cpp
index d144e5e..135e7bb 100644
--- a/primes.cpp
+++ b/primes.cpp
@@ -1,3 +1,4 @@
+#include <stdexcept>
 #include <cstring>
 #include <cmath>
 #include <cassert>
@@ -10,6 +11,7 @@ vector<int> smallprimes;
 bool smallprimes_inited=false;
 
 void fillsmallprimes(){
+	// cerr<<"Filling small primes..."<<endl;
 	smallprimes_inited=true;
 	//TODO: reserve expected amount of space in smallprimes
 	smallprimes.push_back(2);
@@ -44,6 +46,7 @@ pair<Bigint,Bigint> genprimepair(int nbits){
 }
 
 Bigint randprime(const Bigint &biglow,const Bigint &bighigh){
+	//https://en.wikipedia.org/wiki/Generating_primes#Large_primes
 	if(!smallprimes_inited)fillsmallprimes();
 
 	assert(bighigh>biglow);
@@ -53,30 +56,93 @@ Bigint randprime(const Bigint &biglow,const Bigint &bighigh){
 	if(diff<=maxrangesize){
 		low=biglow;
 		high=bighigh;
+		// cerr<<"low=biglow="<<low<<" high=bighigh="<<high<<endl;
 	} else {
 		high=low=cryptrandom_big(diff-maxrangesize);
 		high+=maxrangesize;
+		// cerr<<"low="<<low<<" high="<<high<<endl;
 	}
 	if(low.even())low+=1;
 	if(high.even())high-=1;
+	// cerr<<"low="<<low<<" high="<<high<<endl;
 	Bigint sizeb(high-low+1);
 	assert(sizeb>=0&&sizeb<=maxrangesize);
 	int nnums=(sizeb.lowdigits()+1)/2;
+	// cerr<<"nnums="<<nnums<<endl;
+
+	int nleft=nnums;
 
 	bool composite[nnums]; //low, low+2, low+4, ..., high (all odd numbers)
 	memset(composite,0,nnums*sizeof(bool));
 	int nsmallprimes=smallprimes.size();
 	for(int i=1;i<nsmallprimes;i++){
 		int pr=smallprimes[i];
+		if(pr>=2*nnums)break;
 		int lowoffset=low.divmod(Bigint(pr)).second.lowdigits();
 		int startat;
 		if(lowoffset==0)startat=0;
 		else if((pr-lowoffset)%2==0)startat=(pr-lowoffset)/2;
-		else startat=pr-lowoffset;
+		else startat=(pr-lowoffset+pr)/2;
+		// cerr<<"pr="<<pr<<" lowoffset="<<lowoffset<<" startat="<<startat<<endl;
 		for(int i=startat;i<nnums;i+=pr){ //skips ahead `2*pr` each time (so `pr` array elements)
+			// cerr<<"sieving composite["<<i<<"]: "<<low+2*i<<endl;
+			nleft-=!composite[i];
 			composite[i]=true;
 		}
 	}
+	vector<int> maybeprimes;
+	maybeprimes.reserve(nleft);
+	// cerr<<"Left ("<<nleft<<"):";
+	for(int i=0;i<nnums;i++){
+		if(!composite[i]){
+			// cerr<<' '<<low+2*i;
+			maybeprimes.push_back(i);
+		}
+	}
+	// cerr<<endl;
+
+	while(maybeprimes.size()){
+		int idx=arc4random_uniform(maybeprimes.size());
+		int i=maybeprimes[idx];
+		Bigint bi(low+2*i);
+		if(bailliePSW(bi))return bi;
+		maybeprimes.erase(maybeprimes.begin()+idx);
+	}
+	throw range_error("No primes");
 }
 
-bool bailliePSW(const Bigint&);
+bool strongPseudoPrime2(const Bigint &n){
+	//https://en.wikipedia.org/wiki/Strong_pseudoprime#Formal_definition
+	if(n<2||n.even())return false;
+	Bigint nm1(n);
+	nm1-=1;
+	Bigint d(nm1);
+	while(d.even()){ //TODO: optimise using __builtin_ctz
+		d>>=1;
+		if(expmod(Bigint::two,d,n)==nm1)return true;
+	}
+	if(expmod(Bigint::two,d,n)==1)return true;
+	return false;
+}
+
+bool bailliePSW(const Bigint &n){
+	//https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test#The_test
+	if(!strongPseudoPrime2(n))return false;
+	int D;
+	if     ((D=  5),jacobiSymbol(Bigint(D),n)==-1);
+	else if((D= -7),jacobiSymbol(Bigint(D),n)==-1);
+	else if((D=  9),jacobiSymbol(Bigint(D),n)==-1);
+	else if((D=-11),jacobiSymbol(Bigint(D),n)==-1);
+	else {
+		Bigint root(isqrt(n));
+		if(root*root==n)return false; //perfect square
+		for(int i=6;;i++){
+			D=2*i+1;
+			if(i%2==1)D=-D;
+			if(jacobiSymbol(Bigint(D),n)==-1)break;
+		}
+	}
+	//now we have a D
+	int P=1,Q=(1-D)/4;
+	return strongLucasPrime(n,D,P,Q);
+}
diff --git a/primes.h b/primes.h
index 61e5fc7..744fa3e 100644
--- a/primes.h
+++ b/primes.h
@@ -11,9 +11,15 @@ void fillsmallprimes();
 //for use in RSA (pass target number of bits of N)
 std::pair<Bigint,Bigint> genprimepair(int nbits);
 
-//finds random in range [low,high]; throws domain_error if no prime found
+//finds random in range [low,high]; throws range_error("No primes") if no prime found
 //Will call fillsmallprimes() if not yet done
 Bigint randprime(const Bigint &low,const Bigint &high);
 
+//checks Fermat [pseudo- or actual] primality
+bool strongPseudoPrime2(const Bigint &n);
+
+//checks Lucas [pseudo- or actual] primality
+bool strongLucasPrime(const Bigint &n,int D,int P,int Q);
+
 //checks primality
-bool bailliePSW(const Bigint&);
+bool bailliePSW(const Bigint &n);
-- 
cgit v1.2.3-70-g09d2