1use crate::modulus::prime::Prime;
  2use crate::modulus::montgomery::{Montgomery, MontgomeryPrecomp};
  3use crate::modulus::shoup::{ShoupPrecomp};
  4use primality_test::is_prime;
  5use prime_factorization::Factorization;
  6
  7impl Prime<u64>{
  8
  9    /// Returns a new instance of Prime<u64>.
 10    /// Panics if q_base is not a prime > 2 and
 11    /// if q_base^q_power would overflow u64.
 12    pub fn new(q_base: u64, q_power: u64) -> Self{
 13		assert!(is_prime(q_base) && q_base > 2);
 14        Self::new_unchecked(q_base, q_power)
 15	}
 16
 17    /// Returns a new instance of Prime<u64>.
 18    /// Does not check if q_base is a prime > 2.
 19    /// Panics if q_base^q_power would overflow u64.
 20	pub fn new_unchecked(q_base: u64, q_power: u64) -> Self {
 21
 22        let mut q = q_base;
 23        for _i in 1..q_power{
 24            q *= q_base
 25        }
 26
 27        assert!(q.next_power_of_two().ilog2() <= 61);
 28
 29        let mut phi = q_base-1;
 30        for _i in 1..q_power{
 31            phi *= q_base
 32        }
 33
 34        Self {
 35            q:q,
 36            q_base:q_base,
 37            q_power:q_power,
 38            factors: Vec::new(),
 39            montgomery:MontgomeryPrecomp::new(q),
 40            shoup:ShoupPrecomp::new(q),
 41            phi:phi,
 42        }
 43    }
 44
 45    pub fn q(&self) -> u64{
 46        self.q
 47    }
 48
 49    pub fn q_base(&self) -> u64{
 50        self.q_base
 51    }
 52
 53    pub fn q_power(&self) -> u64{
 54        self.q_power
 55    }
 56
 57    /// Returns x^exponen mod q.
 58    #[inline(always)]
 59    pub fn pow(&self, x: u64, exponent: u64) -> u64{
 60        let mut y_mont: Montgomery<u64> = self.montgomery.one();
 61        let mut x_mont: Montgomery<u64> = self.montgomery.prepare(x);
 62        let mut i: u64 = exponent;
 63        while i > 0{
 64            if i & 1 == 1{
 65                self.montgomery.mul_internal_assign(x_mont, &mut y_mont);
 66            }
 67
 68            self.montgomery.mul_internal_assign(x_mont, &mut x_mont);
 69
 70            i >>= 1;
 71
 72        }
 73        
 74        self.montgomery.unprepare(y_mont)
 75    }
 76
 77    /// Returns x^-1 mod q.
 78    /// User must ensure that x is not divisible by q_base.
 79    #[inline(always)]
 80    pub fn inv(&self, x: u64) -> u64{
 81        self.pow(x, self.phi-1)
 82    }
 83}
 84
 85impl Prime<u64>{
 86    /// Returns the smallest nth primitive root of q_base.
 87    pub fn primitive_root(&mut self) -> u64{
 88
 89        self.check_factors();
 90
 91        let mut candidate: u64 = 1u64;
 92        let mut not_found: bool = true;
 93
 94        while not_found{
 95
 96            candidate += 1;
 97
 98            for &factor in &self.factors{
 99
100                if  Pow(candidate, (self.q_base-1)/factor, self.q_base) == 1{
101                    not_found = true;
102                    break
103                }
104                not_found = false;
105            }
106        }
107
108        if not_found{
109            panic!("failed to find a primitive root for q_base={}", self.q_base)
110        }
111
112        candidate
113    }
114
115    /// Returns an nth primitive root of q = q_base^q_power in Montgomery.
116    pub fn primitive_nth_root(&mut self, nth_root:u64) -> u64{
117
118        assert!(self.q & (nth_root-1) == 1, "invalid prime: q = {} % nth_root = {} = {} != 1", self.q, nth_root, self.q & (nth_root-1));
119
120        let psi: u64 = self.primitive_root();
121
122        // nth primitive root mod q_base: psi_nth^(prime.q_base-1)/nth_root mod q_base
123        let psi_nth_q_base: u64 = Pow(psi, (self.q_base-1)/nth_root, self.q_base);
124
125        // lifts nth primitive root mod q_base to q = q_base^q_power
126        let psi_nth_q: u64 = self.hensel_lift(psi_nth_q_base, nth_root);
127
128        assert!(self.pow(psi_nth_q, nth_root) == 1, "invalid nth primitive root: psi^nth_root != 1 mod q");
129        assert!(self.pow(psi_nth_q, nth_root>>1) == self.q-1, "invalid nth primitive root: psi^(nth_root/2) != -1 mod q");
130
131        psi_nth_q
132    }
133
134    /// Checks if the field self.factor is populated.
135    /// If not, factorize q_base-1 and populates self.factor.
136    /// If yes, checks that it contains the unique factors of q_base-1.
137    pub fn check_factors(&mut self){
138
139        if self.factors.len() == 0{
140
141            let factors = Factorization::run(self.q_base-1).prime_factor_repr();
142            let mut distincts_factors: Vec<u64> = Vec::with_capacity(factors.len());
143            for factor in factors.iter(){
144                distincts_factors.push(factor.0)
145            }
146            self.factors = distincts_factors
147            
148        }else{
149            let mut q_base: u64 = self.q_base;
150
151            for &factor in &self.factors{
152                if !is_prime(factor){
153                    panic!("invalid factor list: factor {} is not prime", factor)
154                }
155    
156                while q_base%factor != 0{
157                    q_base /= factor
158                }
159            }
160    
161            if q_base != 1{
162                panic!("invalid factor list: does not fully divide q_base: q_base % (all factors) = {}", q_base)
163            }
164        }
165
166        
167    }
168
169    /// Returns (psi + a * q_base)^{nth_root} = 1 mod q = q_base^q_power given psi^{nth_root} = 1 mod q_base.
170    /// Panics if psi^{nth_root} != 1 mod q_base.
171    fn hensel_lift(&self, psi: u64, nth_root: u64) -> u64{
172        assert!(Pow(psi, nth_root, self.q_base)==1, "invalid argument psi: psi^nth_root = {} != 1", Pow(psi, nth_root, self.q_base));
173
174        let mut psi_mont: Montgomery<u64> = self.montgomery.prepare(psi);
175        let nth_root_mont: Montgomery<u64> = self.montgomery.prepare(nth_root);
176
177        for _i in 1..self.q_power{
178
179            let psi_pow: Montgomery<u64> = self.montgomery.pow(psi_mont, nth_root-1);
180
181            let num: Montgomery<u64> = Montgomery(self.montgomery.one().value() + self.q - self.montgomery.mul_internal(psi_pow, psi_mont).value());
182
183            let mut den: Montgomery<u64> = self.montgomery.mul_internal(nth_root_mont, psi_pow);
184
185            den = self.montgomery.pow(den, self.phi-1);
186
187            psi_mont = self.montgomery.add_internal(psi_mont, self.montgomery.mul_internal(num, den));
188        }
189        
190        self.montgomery.unprepare(psi_mont)
191    }
192}
193
194/// Returns x^exponent mod q.
195/// This function internally instantiate a new MontgomeryPrecomp<u64>
196/// To be used when called only a few times and if there 
197/// is no Prime instantiated with q.
198pub fn Pow(x:u64, exponent:u64, q:u64) -> u64{
199    let montgomery: MontgomeryPrecomp<u64> = MontgomeryPrecomp::<u64>::new(q);
200    let mut y_mont: Montgomery<u64> = montgomery.one();
201    let mut x_mont: Montgomery<u64> = montgomery.prepare(x);
202    let mut i: u64 = exponent;
203    while i > 0{
204        if i & 1 == 1{
205            montgomery.mul_internal_assign(x_mont, &mut y_mont);
206        }
207
208        montgomery.mul_internal_assign(x_mont, &mut x_mont);
209
210        i >>= 1;
211    }
212    
213    montgomery.unprepare(y_mont)
214}