update 2-adic generator to 0x64fdd1a46201e246 (#1579)

* update 2-adic generator to `0x64fdd1a46201e246`

this necesstitates that we also change the various lifts of the two-adic generator; i.e. the lifts to mult. generators of the _entire_ field, as well as the lifts to 2-adic generators of the various extension fields.

* cargo fmt

---------

Co-authored-by: Benjamin Diamond <bdiamond@ulvetanna.io>

field/src/goldilocks_extensions.rs

@@ -21,10 +21,9 @@ impl Extendable<2> for GoldilocksField {
     // DTH_ROOT = W^((ORDER - 1)/2)
     const DTH_ROOT: Self = Self(18446744069414584320);
 
-    const EXT_MULTIPLICATIVE_GROUP_GENERATOR: [Self; 2] =
-        [Self(18081566051660590251), Self(16121475356294670766)];
+    const EXT_MULTIPLICATIVE_GROUP_GENERATOR: [Self; 2] = [Self(0), Self(11713931119993638672)];
 
-    const EXT_POWER_OF_TWO_GENERATOR: [Self; 2] = [Self(0), Self(15659105665374529263)];
+    const EXT_POWER_OF_TWO_GENERATOR: [Self; 2] = [Self(0), Self(7226896044987257365)];
 }
 
 impl Mul for QuadraticExtension<GoldilocksField> {
@@ -45,15 +44,11 @@ impl Extendable<4> for GoldilocksField {
     // DTH_ROOT = W^((ORDER - 1)/4)
     const DTH_ROOT: Self = Self(281474976710656);
 
-    const EXT_MULTIPLICATIVE_GROUP_GENERATOR: [Self; 4] = [
-        Self(5024755240244648895),
-        Self(13227474371289740625),
-        Self(3912887029498544536),
-        Self(3900057112666848848),
-    ];
+    const EXT_MULTIPLICATIVE_GROUP_GENERATOR: [Self; 4] =
+        [Self(0), Self(8295451483910296135), Self(0), Self(0)];
 
     const EXT_POWER_OF_TWO_GENERATOR: [Self; 4] =
-        [Self(0), Self(0), Self(0), Self(12587610116473453104)];
+        [Self(0), Self(0), Self(0), Self(17216955519093520442)];
 }
 
 impl Mul for QuarticExtension<GoldilocksField> {
@@ -75,11 +70,11 @@ impl Extendable<5> for GoldilocksField {
     const DTH_ROOT: Self = Self(1041288259238279555);
 
     const EXT_MULTIPLICATIVE_GROUP_GENERATOR: [Self; 5] = [
-        Self(2899034827742553394),
-        Self(13012057356839176729),
-        Self(14593811582388663055),
-        Self(7722900811313895436),
-        Self(4557222484695340057),
+        Self(4624713872807171977),
+        Self(381988216716071028),
+        Self(14499722700050429911),
+        Self(4870631734967222356),
+        Self(4518902370426242880),
     ];
 
     const EXT_POWER_OF_TWO_GENERATOR: [Self; 5] = [

field/src/goldilocks_field.rs

@@ -77,14 +77,14 @@ impl Field for GoldilocksField {
     const CHARACTERISTIC_TWO_ADICITY: usize = Self::TWO_ADICITY;
 
     // Sage: `g = GF(p).multiplicative_generator()`
-    const MULTIPLICATIVE_GROUP_GENERATOR: Self = Self(7);
+    const MULTIPLICATIVE_GROUP_GENERATOR: Self = Self(14293326489335486720);
 
     // Sage:
     // ```
     // g_2 = g^((p - 1) / 2^32)
     // g_2.multiplicative_order().factor()
     // ```
-    const POWER_OF_TWO_GENERATOR: Self = Self(1753635133440165772);
+    const POWER_OF_TWO_GENERATOR: Self = Self(7277203076849721926);
 
     const BITS: usize = 64;