diff --git a/README.md b/README.md
index b33a537..a3356dd 100644
--- a/README.md
+++ b/README.md
@@ -201,8 +201,6 @@ export type CurveFn = {
   ExtendedPoint: ExtendedPointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint, b?: bigint) => bigint;
-    invert: (number: bigint, modulo?: bigint) => bigint;
     randomPrivateKey: () => Uint8Array;
     getExtendedPublicKey: (key: PrivKey) => {
       head: Uint8Array;
@@ -306,6 +304,7 @@ export type CurveFn = {
   getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
   getSharedSecret: (privateA: PrivKey, publicB: PubKey, isCompressed?: boolean) => Uint8Array;
   sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => SignatureType;
+  signUnhashed: (msg: Uint8Array, privKey: PrivKey, opts?: SignOpts) => SignatureType;
   verify: (
     signature: Hex | SignatureType,
     msgHash: Hex,
@@ -316,8 +315,6 @@ export type CurveFn = {
   ProjectivePoint: ProjectivePointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint) => bigint;
-    invert: (number: bigint) => bigint;
     isValidPrivateKey(privateKey: PrivKey): boolean;
     hashToPrivateKey: (hash: Hex) => Uint8Array;
     randomPrivateKey: () => Uint8Array;
diff --git a/src/abstract/bls.ts b/src/abstract/bls.ts
index 76fe16c..e20c0d2 100644
--- a/src/abstract/bls.ts
+++ b/src/abstract/bls.ts
@@ -93,12 +93,9 @@ export type CurveFn<Fp, Fp2, Fp6, Fp12> = {
     publicKeys: (Hex | PointType<Fp>)[]
   ) => boolean;
   utils: {
-    bytesToHex: typeof ut.bytesToHex;
-    hexToBytes: typeof ut.hexToBytes;
     stringToBytes: typeof stringToBytes;
     hashToField: typeof hashToField;
     expandMessageXMD: typeof expandMessageXMD;
-    mod: typeof mod.mod;
     getDSTLabel: () => string;
     setDSTLabel(newLabel: string): void;
   };
@@ -177,7 +174,6 @@ export function bls<Fp2, Fp6, Fp12>(
   const utils = {
     hexToBytes: ut.hexToBytes,
     bytesToHex: ut.bytesToHex,
-    mod: mod.mod,
     stringToBytes: stringToBytes,
     // TODO: do we need to export it here?
     hashToField: (
diff --git a/src/abstract/edwards.ts b/src/abstract/edwards.ts
index d36a3de..a135f17 100644
--- a/src/abstract/edwards.ts
+++ b/src/abstract/edwards.ts
@@ -130,8 +130,6 @@ export type CurveFn = {
   ExtendedPoint: ExtendedPointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint) => bigint;
-    invert: (number: bigint) => bigint;
     randomPrivateKey: () => Uint8Array;
     getExtendedPublicKey: (key: PrivKey) => {
       head: Uint8Array;
@@ -146,7 +144,7 @@ export type CurveFn = {
 // NOTE: it is not generic twisted curve for now, but ed25519/ed448 generic implementation
 export function twistedEdwards(curveDef: CurveType): CurveFn {
   const CURVE = validateOpts(curveDef) as ReturnType<typeof validateOpts>;
-  const Fp = CURVE.Fp as mod.Field<bigint>;
+  const Fp = CURVE.Fp;
   const CURVE_ORDER = CURVE.n;
   const maxGroupElement = _2n ** BigInt(CURVE.nByteLength * 8);
 
@@ -662,9 +660,6 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
 
   const utils = {
     getExtendedPublicKey,
-    mod: modP,
-    invert: Fp.invert,
-
     /**
      * Not needed for ed25519 private keys. Needed if you use scalars directly (rare).
      */
diff --git a/src/abstract/weierstrass.ts b/src/abstract/weierstrass.ts
index 2e291b7..bdda6f1 100644
--- a/src/abstract/weierstrass.ts
+++ b/src/abstract/weierstrass.ts
@@ -608,13 +608,9 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       const { x, y } = this;
       // Check if x, y are valid field elements
       if (!Fp.isValid(x) || !Fp.isValid(y)) throw new Error(msg);
-      const left = Fp.square(y);
-      const right = weierstrassEquation(x);
-      // We subtract instead of comparing: it's safer
-      // (y²) - (x³ + ax + b) == 0
-      if (!Fp.isZero(Fp.sub(left, right))) throw new Error(msg);
-      // if (!Fp.equals(left, right))
-      // TODO: flag to disable this?
+      const left = Fp.square(y); // y²
+      const right = weierstrassEquation(x); // x³ + ax + b
+      if (!Fp.equals(left, right)) throw new Error(msg);
       if (!this.isTorsionFree()) throw new Error('Point must be of prime-order subgroup');
     }
 
@@ -771,8 +767,6 @@ export type CurveFn = {
   ProjectivePoint: ProjectiveConstructor<bigint>;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint, b?: bigint) => bigint;
-    invert: (number: bigint, modulo?: bigint) => bigint;
     _bigintToBytes: (num: bigint) => Uint8Array;
     _bigintToString: (num: bigint) => string;
     _normalizePrivateKey: (key: PrivKey) => bigint;
@@ -831,9 +825,7 @@ class HmacDrbg {
     }
     return ut.concatBytes(...out);
   }
-  // There is no need in clean() method
-  // It's useless, there are no guarantees with JS GC
-  // whether bigints are removed even if you clean Uint8Arrays.
+  // There are no guarantees with JS GC whether bigints are removed even if you clean Uint8Arrays.
 }
 
 export function weierstrass(curveDef: CurveType): CurveFn {
@@ -1049,8 +1041,6 @@ export function weierstrass(curveDef: CurveType): CurveFn {
   }
 
   const utils = {
-    mod: (n: bigint, modulo = Fp.ORDER) => mod.mod(n, modulo),
-    invert: Fp.invert,
     isValidPrivateKey(privateKey: PrivKey) {
       try {
         normalizePrivateKey(privateKey);
diff --git a/src/bls12-381.ts b/src/bls12-381.ts
index 8e51317..bdc4e09 100644
--- a/src/bls12-381.ts
+++ b/src/bls12-381.ts
@@ -135,6 +135,7 @@ const Fp2: mod.Field<Fp2> & Fp2Utils = {
     return { c0: Fp.mul(factor, Fp.create(a)), c1: Fp.mul(factor, Fp.create(-b)) };
   },
   sqrt: (num) => {
+    if (Fp2.equals(num, Fp2.ZERO)) return Fp2.ZERO; // Algo doesn't handles this case
     // TODO: Optimize this line. It's extremely slow.
     // Speeding this up would boost aggregateSignatures.
     // https://eprint.iacr.org/2012/685.pdf applicable?
diff --git a/src/ed25519.ts b/src/ed25519.ts
index 19df0ea..26cd6b2 100644
--- a/src/ed25519.ts
+++ b/src/ed25519.ts
@@ -260,7 +260,7 @@ const invertSqrt = (number: bigint) => uvRatio(_1n, number);
 
 const MAX_255B = BigInt('0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
 const bytes255ToNumberLE = (bytes: Uint8Array) =>
-  ed25519.utils.mod(bytesToNumberLE(bytes) & MAX_255B);
+  ed25519.CURVE.Fp.create(bytesToNumberLE(bytes) & MAX_255B);
 
 type ExtendedPoint = ExtendedPointType;
 
@@ -269,7 +269,7 @@ type ExtendedPoint = ExtendedPointType;
 function calcElligatorRistrettoMap(r0: bigint): ExtendedPoint {
   const { d } = ed25519.CURVE;
   const P = ed25519.CURVE.Fp.ORDER;
-  const { mod } = ed25519.utils;
+  const mod = ed25519.CURVE.Fp.create;
   const r = mod(SQRT_M1 * r0 * r0); // 1
   const Ns = mod((r + _1n) * ONE_MINUS_D_SQ); // 2
   let c = BigInt(-1); // 3
@@ -327,7 +327,7 @@ export class RistrettoPoint {
     hex = ensureBytes(hex, 32);
     const { a, d } = ed25519.CURVE;
     const P = ed25519.CURVE.Fp.ORDER;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     const emsg = 'RistrettoPoint.fromHex: the hex is not valid encoding of RistrettoPoint';
     const s = bytes255ToNumberLE(hex);
     // 1. Check that s_bytes is the canonical encoding of a field element, or else abort.
@@ -357,7 +357,7 @@ export class RistrettoPoint {
   toRawBytes(): Uint8Array {
     let { x, y, z, t } = this.ep;
     const P = ed25519.CURVE.Fp.ORDER;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     const u1 = mod(mod(z + y) * mod(z - y)); // 1
     const u2 = mod(x * y); // 2
     // Square root always exists
@@ -395,7 +395,7 @@ export class RistrettoPoint {
     assertRstPoint(other);
     const a = this.ep;
     const b = other.ep;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     // (x1 * y2 == y1 * x2) | (y1 * y2 == x1 * x2)
     const one = mod(a.x * b.y) === mod(a.y * b.x);
     const two = mod(a.y * b.y) === mod(a.x * b.x);
diff --git a/src/secp256k1.ts b/src/secp256k1.ts
index 383ba12..7edeb8f 100644
--- a/src/secp256k1.ts
+++ b/src/secp256k1.ts
@@ -127,7 +127,7 @@ export const secp256k1 = createCurve(
         const b1 = -_1n * BigInt('0xe4437ed6010e88286f547fa90abfe4c3');
         const a2 = BigInt('0x114ca50f7a8e2f3f657c1108d9d44cfd8');
         const b2 = a1;
-        const POW_2_128 = BigInt('0x100000000000000000000000000000000');
+        const POW_2_128 = BigInt('0x100000000000000000000000000000000'); // (2n**128n).toString(16)
 
         const c1 = divNearest(b2 * k, n);
         const c2 = divNearest(-b1 * k, n);
@@ -173,20 +173,17 @@ function normalizePublicKey(publicKey: Hex | PointType<bigint>): PointType<bigin
   } else {
     const bytes = ensureBytes(publicKey);
     // Schnorr is 32 bytes
-    if (bytes.length === 32) {
-      const x = bytesToNumberBE(bytes);
-      if (!isValidFieldElement(x)) throw new Error('Point is not on curve');
-      const y2 = secp256k1.utils._weierstrassEquation(x); // y² = x³ + ax + b
-      let y = sqrtMod(y2); // y = y² ^ (p+1)/4
-      const isYOdd = (y & _1n) === _1n;
-      // Schnorr
-      if (isYOdd) y = secp256k1.CURVE.Fp.negate(y);
-      const point = new secp256k1.Point(x, y);
-      point.assertValidity();
-      return point;
-    }
-    // Do we need that in schnorr at all?
-    return secp256k1.Point.fromHex(publicKey);
+    if (bytes.length !== 32) throw new Error('Schnorr pubkeys must be 32 bytes');
+    const x = bytesToNumberBE(bytes);
+    if (!isValidFieldElement(x)) throw new Error('Point is not on curve');
+    const y2 = secp256k1.utils._weierstrassEquation(x); // y² = x³ + ax + b
+    let y = sqrtMod(y2); // y = y² ^ (p+1)/4
+    const isYOdd = (y & _1n) === _1n;
+    // Schnorr
+    if (isYOdd) y = secp256k1.CURVE.Fp.negate(y);
+    const point = new secp256k1.Point(x, y);
+    point.assertValidity();
+    return point;
   }
 }
 
@@ -225,10 +222,13 @@ class SchnorrSignature {
   }
   static fromHex(hex: Hex) {
     const bytes = ensureBytes(hex);
-    if (bytes.length !== 64)
-      throw new TypeError(`SchnorrSignature.fromHex: expected 64 bytes, not ${bytes.length}`);
-    const r = bytesToNumberBE(bytes.subarray(0, 32));
-    const s = bytesToNumberBE(bytes.subarray(32, 64));
+    const len = 32; // group length
+    if (bytes.length !== 2 * len)
+      throw new TypeError(
+        `SchnorrSignature.fromHex: expected ${2 * len} bytes, not ${bytes.length}`
+      );
+    const r = bytesToNumberBE(bytes.subarray(0, len));
+    const s = bytesToNumberBE(bytes.subarray(len, 2 * len));
     return new SchnorrSignature(r, s);
   }
   assertValidity() {
diff --git a/src/stark.ts b/src/stark.ts
index 3ebcb58..b71895a 100644
--- a/src/stark.ts
+++ b/src/stark.ts
@@ -138,11 +138,12 @@ function hashKeyWithIndex(key: Uint8Array, index: number) {
 export function grindKey(seed: Hex) {
   const _seed = ensureBytes0x(seed);
   const sha256mask = 2n ** 256n;
-  const limit = sha256mask - starkCurve.utils.mod(sha256mask, starkCurve.CURVE.n);
+  const Fn = Fp(CURVE.n);
+  const limit = sha256mask - Fn.create(sha256mask);
   for (let i = 0; ; i++) {
     const key = hashKeyWithIndex(_seed, i);
     // key should be in [0, limit)
-    if (key < limit) return starkCurve.utils.mod(key, starkCurve.CURVE.n).toString(16);
+    if (key < limit) return Fn.create(key).toString(16);
   }
 }
 
diff --git a/test/ed25519.test.js b/test/ed25519.test.js
index 915f808..42ae307 100644
--- a/test/ed25519.test.js
+++ b/test/ed25519.test.js
@@ -646,7 +646,8 @@ for (let i = 0; i < VECTORS_RFC8032_PH.length; i++) {
 
 should('X25519 base point', () => {
   const { y } = ed25519.Point.BASE;
-  const u = ed25519.utils.mod((y + 1n) * ed25519.utils.invert(1n - y, ed25519.CURVE.P));
+  const { Fp } = ed25519.CURVE;
+  const u = Fp.create((y + 1n) * Fp.invert(1n - y));
   deepStrictEqual(hex(numberToBytesLE(u, 32)), x25519.Gu);
 });
 
diff --git a/test/ed448.test.js b/test/ed448.test.js
index c19c823..b9c5218 100644
--- a/test/ed448.test.js
+++ b/test/ed448.test.js
@@ -651,9 +651,10 @@ for (let i = 0; i < VECTORS_RFC8032_PH.length; i++) {
 
 should('X448 base point', () => {
   const { x, y } = ed448.Point.BASE;
-  const { P } = ed448.CURVE;
-  const invX = ed448.utils.invert(x * x, P); // x²
-  const u = ed448.utils.mod(y * y * invX, P); // (y²/x²)
+  const { Fp } = ed448.CURVE;
+  // const invX = Fp.invert(x * x); // x²
+  const u = Fp.div(Fp.create(y * y), Fp.create(x * x)); // (y²/x²)
+  // const u = Fp.create(y * y * invX);
   deepStrictEqual(hex(numberToBytesLE(u, 56)), x448.Gu);
 });
 
diff --git a/test/secp256k1.test.js b/test/secp256k1.test.js
index 13d38fd..7427399 100644
--- a/test/secp256k1.test.js
+++ b/test/secp256k1.test.js
@@ -1,5 +1,6 @@
 import * as fc from 'fast-check';
 import { secp256k1, schnorr } from '../lib/esm/secp256k1.js';
+import { Fp } from '../lib/esm/abstract/modular.js';
 import { readFileSync } from 'fs';
 import { default as ecdsa } from './vectors/ecdsa.json' assert { type: 'json' };
 import { default as ecdh } from './vectors/ecdh.json' assert { type: 'json' };
@@ -16,7 +17,6 @@ const privatesTxt = readFileSync('./test/vectors/privates-2.txt', 'utf-8');
 const schCsv = readFileSync('./test/vectors/schnorr.csv', 'utf-8');
 
 const FC_BIGINT = fc.bigInt(1n + 1n, secp.CURVE.n - 1n);
-const P = secp.CURVE.Fp.ORDER;
 // prettier-ignore
 const INVALID_ITEMS = ['deadbeef', Math.pow(2, 53), [1], 'xyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxy', secp.CURVE.n + 2n];
 
@@ -50,9 +50,9 @@ should('secp256k1.getPublicKey()', () => {
   }
 });
 should('secp256k1.getPublicKey() rejects invalid keys', () => {
-  // for (const item of INVALID_ITEMS) {
-  //   throws(() => secp.getPublicKey(item));
-  // }
+  for (const item of INVALID_ITEMS) {
+    throws(() => secp.getPublicKey(item));
+  }
 });
 should('secp256k1.precompute', () => {
   secp.utils.precompute(4);
@@ -434,17 +434,26 @@ should('secp256k1.utils.isValidPrivateKey()', () => {
     deepStrictEqual(secp.utils.isValidPrivateKey(d), expected);
   }
 });
+should('have proper curve equation in assertValidity()', () => {
+  throws(() => {
+    const { Fp } = secp.CURVE;
+    let point = new secp.Point(Fp.create(-2n), Fp.create(-1n));
+    point.assertValidity();
+  });
+});
+
+const Fn = Fp(secp.CURVE.n);
 const normal = secp.utils._normalizePrivateKey;
 const tweakUtils = {
   privateAdd: (privateKey, tweak) => {
     const p = normal(privateKey);
     const t = normal(tweak);
-    return secp.utils._bigintToBytes(secp.utils.mod(p + t, secp.CURVE.n));
+    return secp.utils._bigintToBytes(Fn.create(p + t));
   },
 
   privateNegate: (privateKey) => {
     const p = normal(privateKey);
-    return secp.utils._bigintToBytes(secp.CURVE.n - p);
+    return secp.utils._bigintToBytes(Fn.negate(p));
   },
 
   pointAddScalar: (p, tweak, isCompressed) => {