@@ -21,10 +21,34 @@ MontFp!("-1"),
     ];
 
     #[inline(always)]
-    fn mul_fp_by_nonresidue(fe: &Self::Fp) -> Self::Fp {
-        let original = fe;
-        let mut fe = -fe.double();
+    fn mul_fp_by_nonresidue_in_place(fe: &mut Self::Fp) -> &mut Self::Fp {
+        let original = *fe;
+        fe.neg_in_place();
         fe.double_in_place();
-        fe - original
+        fe.double_in_place();
+        *fe -= original;
+        fe
+    }
+
+    #[inline(always)]
+    fn sub_and_mul_fp_by_nonresidue(y: &mut Self::Fp, x: &Self::Fp) {
+        let mut original = *y;
+        original += x;
+        y.double_in_place().double_in_place();
+        *y += original;
+    }
+
+    #[inline(always)]
+    fn add_and_mul_fp_by_nonresidue_plus_one(y: &mut Self::Fp, x: &Self::Fp) {
+        y.double_in_place().double_in_place().neg_in_place();
+        *y += x;
+    }
+
+    fn add_and_mul_fp_by_nonresidue_in_place(y: &mut Self::Fp, x: &Self::Fp) {
+        let mut original = *y;
+        original.double_in_place().double_in_place();
+        original += &*y;
+        *y = *x;
+        *y -= original;
     }
 }

@@ -68,10 +68,12 @@ ),
     ];
 
     #[inline(always)]
-    fn mul_fp2_by_nonresidue(fe: &Fq2) -> Fq2 {
+    fn mul_fp2_by_nonresidue_in_place(fe: &mut Fq2) -> &mut Fq2 {
         // Karatsuba multiplication with constant other = u.
-        let c0 = Fq2Config::mul_fp_by_nonresidue(&fe.c1);
-        let c1 = fe.c0;
-        Fq2::new(c0, c1)
+        let old_c0 = fe.c0;
+        fe.c0 = fe.c1;
+        Fq2Config::mul_fp_by_nonresidue_in_place(&mut fe.c0);
+        fe.c1 = old_c0;
+        fe
     }
 }

@@ -136,7 +136,7 @@ assert_eq!(Zero, Fq2::zero().legendre());
     // i^2 = -1
     let mut m1 = -Fq2::one();
     assert_eq!(QuadraticResidue, m1.legendre());
-    m1 = Fq6Config::mul_fp2_by_nonresidue(&m1);
+    Fq6Config::mul_fp2_by_nonresidue_in_place(&mut m1);
     assert_eq!(QuadraticNonResidue, m1.legendre());
 }
 

@@ -21,7 +21,22 @@ MontFp!("-1"),
     ];
 
     #[inline(always)]
-    fn mul_fp_by_nonresidue(fp: &Self::Fp) -> Self::Fp {
-        -(*fp)
+    fn mul_fp_by_nonresidue_in_place(fp: &mut Self::Fp) -> &mut Self::Fp {
+        fp.neg_in_place()
+    }
+
+    #[inline(always)]
+    fn sub_and_mul_fp_by_nonresidue(y: &mut Self::Fp, x: &Self::Fp) {
+        *y += x;
+    }
+
+    #[inline(always)]
+    fn add_and_mul_fp_by_nonresidue_plus_one(y: &mut Self::Fp, x: &Self::Fp) {
+        *y = *x;
+    }
+
+    fn add_and_mul_fp_by_nonresidue_in_place(y: &mut Self::Fp, x: &Self::Fp) {
+        y.neg_in_place();
+        *y += x;
     }
 }

@@ -82,11 +82,10 @@ ];
 
     /// Multiply this element by the quadratic nonresidue 1 + u.
     /// Make this generic.
-    fn mul_fp2_by_nonresidue(fe: &Fq2) -> Fq2 {
-        let mut copy = *fe;
-        let t0 = copy.c0;
-        copy.c0 -= &fe.c1;
-        copy.c1 += &t0;
-        copy
+    fn mul_fp2_by_nonresidue_in_place(fe: &mut Fq2) -> &mut Fq2 {
+        let t0 = fe.c0;
+        fe.c0 -= &fe.c1;
+        fe.c1 += &t0;
+        fe
     }
 }

@@ -1718,7 +1718,7 @@ assert_eq!(Zero, Fq2::zero().legendre());
     // i^2 = -1
     let mut m1 = -Fq2::one();
     assert_eq!(QuadraticResidue, m1.legendre());
-    m1 = Fq6Config::mul_fp2_by_nonresidue(&m1);
+    Fq6Config::mul_fp2_by_nonresidue_in_place(&mut m1);
     assert_eq!(QuadraticNonResidue, m1.legendre());
 }
 
@@ -1731,7 +1731,7 @@
     for _ in 0..1000 {
         let mut a = Fq2::rand(&mut rng);
         let mut b = a;
-        a = Fq6Config::mul_fp2_by_nonresidue(&a);
+        Fq6Config::mul_fp2_by_nonresidue_in_place(&mut a);
         b.mul_assign(&nqr);
 
         assert_eq!(a, b);
@@ -1747,7 +1747,7 @@
     for _ in 0..1000 {
         let mut a = Fq6::rand(&mut rng);
         let mut b = a;
-        a = Fq12Config::mul_fp6_by_nonresidue(&a);
+        Fq12Config::mul_fp6_by_nonresidue_in_place(&mut a);
         b.mul_assign(&nqr);
 
         assert_eq!(a, b);

@@ -21,7 +21,7 @@ MontFp!("-1"),
     ];
 
     #[inline(always)]
-    fn mul_fp_by_nonresidue(fe: &Self::Fp) -> Self::Fp {
-        -(*fe)
+    fn mul_fp_by_nonresidue_in_place(fe: &mut Self::Fp) -> &mut Self::Fp {
+        fe.neg_in_place()
     }
 }

@@ -90,12 +90,16 @@ ),
     ];
 
     #[inline(always)]
-    fn mul_fp2_by_nonresidue(fe: &Fq2) -> Fq2 {
+    fn mul_fp2_by_nonresidue_in_place(fe: &mut Fq2) -> &mut Fq2 {
         // (c0+u*c1)*(9+u) = (9*c0-c1)+u*(9*c1+c0)
         let mut f = *fe;
         f.double_in_place().double_in_place().double_in_place();
-        let c0 = f.c0 + fe.c0 + Fq2Config::mul_fp_by_nonresidue(&fe.c1);
+        let mut c0 = fe.c1;
+        Fq2Config::mul_fp_by_nonresidue_in_place(&mut c0);
+        c0 += &f.c0;
+        c0 += &fe.c0;
         let c1 = f.c1 + fe.c1 + fe.c0;
-        Fq2::new(c0, c1)
+        *fe = Fq2::new(c0, c1);
+        fe
     }
 }

@@ -139,7 +139,7 @@ assert_eq!(Zero, Fq2::zero().legendre());
     // i^2 = -1
     let mut m1 = -Fq2::one();
     assert_eq!(QuadraticResidue, m1.legendre());
-    m1 = Fq6Config::mul_fp2_by_nonresidue(&m1);
+    Fq6Config::mul_fp2_by_nonresidue_in_place(&mut m1);
     assert_eq!(QuadraticNonResidue, m1.legendre());
 }
 

@@ -82,9 +82,7 @@ MontFp!("4922464560225523242118178942575080391082002530232324381063048548642823052024664478336818169867474395270858391911405337707247735739826664939444490469542109391530482826728203582549674992333383150446779312029624171857054392282775648"),
     ];
 
     #[inline(always)]
-    fn mul_fp_by_nonresidue(fe: &Self::Fp) -> Self::Fp {
-        let original = -(*fe);
-        let double = original + &original;
-        double + &double
+    fn mul_fp_by_nonresidue_in_place(fe: &mut Self::Fp) -> &mut Self::Fp {
+        fe.double_in_place().double_in_place()
     }
 }

@@ -77,11 +77,12 @@ MontFp!("2416169158604010336818399199316106389588878314690767988978701685873498866746813334102117883272276610365242925950967572554030909749205624998805208910209389668659757274773858916683688639755413288353778854399286396639505385648830027756861"),
     ];
 
     #[inline(always)]
-    fn mul_fp_by_nonresidue(fe: &Self::Fp) -> Self::Fp {
+    fn mul_fp_by_nonresidue_in_place(fe: &mut Self::Fp) -> &mut Self::Fp {
         let original = *fe;
-        let mut four_fe = fe.double();
-        four_fe.double_in_place();
-        let eight_fe = four_fe.double();
-        eight_fe + &four_fe + &original
+        fe.double_in_place();
+        *fe += original;
+        fe.double_in_place().double_in_place();
+        *fe += original;
+        fe
     }
 }

@@ -12,9 +12,9 @@
 #[test]
 fn test_fq3_more() {
     let mut rng = test_rng();
-    let a: Fq3 = rng.gen();
+    let mut a: Fq3 = rng.gen();
     assert_eq!(
         a * Fq6Config::NONRESIDUE,
-        <Fp6ConfigWrapper<Fq6Config>>::mul_base_field_by_nonresidue(&a)
+        *<Fp6ConfigWrapper<Fq6Config>>::mul_base_field_by_nonresidue_in_place(&mut a)
     );
 }

@@ -11,9 +11,9 @@
 #[test]
 fn test_fq3_more() {
     let mut rng = test_rng();
-    let a: Fq3 = rng.gen();
+    let mut a: Fq3 = rng.gen();
     assert_eq!(
         a * Fq6Config::NONRESIDUE,
-        <Fp6ConfigWrapper<Fq6Config>>::mul_base_field_by_nonresidue(&a)
+        *<Fp6ConfigWrapper<Fq6Config>>::mul_base_field_by_nonresidue_in_place(&mut a)
     );
 }