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#include <stdexcept>
#include <cstring>
#include <cmath>
#include <cassert>
#include "numalgo.h"
#include "primes.h"
using namespace std;
vector<int> smallprimes;
bool smallprimes_inited=false;
void fillsmallprimes(){
smallprimes_inited=true;
//TODO: reserve expected amount of space in smallprimes
smallprimes.push_back(2);
const int highbound=65000;
bool composite[highbound/2]; //entries for 3, 5, 7, 9, etc.
memset(composite,0,highbound/2*sizeof(bool));
int roothighbound=sqrt(highbound);
for(int i=3;i<=highbound;i+=2){
if(composite[i/2-1])continue;
smallprimes.push_back(i);
if(i>roothighbound)continue;
for(int j=i*i;j<=highbound;j+=2*i){
composite[j/2-1]=true;
}
}
}
pair<Bigint,Bigint> genprimepair(Rng &rng,int nbits){
// for x = nbits/2:
// (2^x)^2 = 2^(2x)
// (2^x + 2^(x-2))^2 = 2^(2x) + 2^(2x-1) + 2^(2x-4)
// ergo: (2^x + lambda*2^(x-2))^2 \in [2^(2x), 2^(2x+1)), for lambda \in [0,1]
// To make sure the primes "differ in length by a few digits" [RSA78], we use x1=x-2 in the first
// prime and x2-x+2 in the second
int x1=nbits/2-2,x2=(nbits+1)/2+2;
assert(x1+x2==nbits);
return make_pair(
randprime(rng,Bigint::one<<x1,(Bigint::one<<x1)+(Bigint::one<<(x1-2))),
randprime(rng,Bigint::one<<x2,(Bigint::one<<x2)+(Bigint::one<<(x2-2))));
}
Bigint randprime(Rng &rng,const Bigint &biglow,const Bigint &bighigh){
//https://en.wikipedia.org/wiki/Generating_primes#Large_primes
if(!smallprimes_inited)fillsmallprimes();
assert(bighigh>biglow);
static const int maxrangesize=100001;
Bigint diff(bighigh-biglow);
Bigint low,high; //inclusive
if(diff<=maxrangesize){
low=biglow;
high=bighigh;
} else {
high=low=bigrandom(rng,diff-maxrangesize);
high+=maxrangesize;
}
if(low.even())low+=1;
if(high.even())high-=1;
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+pr)/2;
for(int i=startat;i<nnums;i+=pr){ //skips ahead `2*pr` each time (so `pr` array elements)
nleft-=!composite[i];
composite[i]=true;
}
}
vector<int> maybeprimes;
maybeprimes.reserve(nleft);
for(int i=0;i<nnums;i++){
if(!composite[i]){
maybeprimes.push_back(i);
}
}
while(maybeprimes.size()){
int idx=rng.get_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 strongPseudoPrime2(const Bigint &n){
//https://en.wikipedia.org/wiki/Strong_pseudoprime#Formal_definition
if(n<2)return false;
if(n==2)return true;
if(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 strongLucasPrime(const Bigint &n){
//https://en.wikipedia.org/wiki/Lucas_pseudoprime#Implementing_a_Lucas_probable_prime_test
//TODO: use d and s already calculated in strongPseudoPrime2
if(n<2)return false;
if(n==2)return true;
if(n.even())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,iQ=(1-D)/4;
if(gcd(n,Bigint(P)).compareAbs(1)!=0||gcd(n,Bigint(iQ)).compareAbs(1)!=0)return false; //would be too easy to ignore
//now begin the actual sequence algorithm
int s=0;
Bigint d(n);
d+=1;
while(d.even()){
d>>=1;
s++;
}
vector<bool> dbits=d.bits();
assert(dbits.size()>0);
assert(dbits[dbits.size()-1]==true);
Bigint U(1),V(P);
Bigint Qk(iQ);
for(int i=dbits.size()-2;i>=0;i--){
U*=V;
V*=V;
V-=Qk<<1;
Qk*=Qk;
if(dbits[i]){
Bigint unext(U*P);
unext+=V;
if(unext.odd())unext+=n;
assert(unext.even());
unext>>=1;
Bigint vnext(U*D);
vnext+=V*P;
if(vnext.odd())vnext+=n;
assert(vnext.even());
vnext>>=1;
U=unext.divmod(n).second;
V=vnext.divmod(n).second;
Qk=(Qk*iQ).divmod(n).second;
} else {
U=U.divmod(n).second;
V=V.divmod(n).second;
Qk=Qk.divmod(n).second;
}
}
if(U==0)return true;
if(V==0)return true; //r=0 check
for(int r=1;r<s;r++){
V*=V;
V-=Qk<<1;
V=V.divmod(n).second;
Qk=(Qk*Qk).divmod(n).second;
if(V==0)return true;
}
return false;
}
bool bailliePSW(const Bigint &n){
//https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test#The_test
return strongPseudoPrime2(n)&&strongLucasPrime(n);
}
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