diff --git a/src/partially_blindrsa.ts b/src/partially_blindrsa.ts
index 91de3e8..81ccdb7 100644
--- a/src/partially_blindrsa.ts
+++ b/src/partially_blindrsa.ts
@@ -289,19 +289,40 @@ export class PartiallyBlindRSA {
         algorithm: Pick<RsaHashedKeyGenParams, 'modulusLength' | 'publicExponent' | 'hash'>,
         generateSafePrimeSync: (length: number) => sjcl.BigNumber | bigint = generateSafePrime,
     ): Promise<CryptoKeyPair> {
+        if (algorithm.modulusLength % 2 !== 0 || algorithm.modulusLength < 4) {
+            throw new Error('modulusLength must be an even number greater than or equal to 4');
+        }
+
         prepare_sjcl_random_generator();
 
         // 1. p = SafePrime(bits / 2)
         // 2. q = SafePrime(bits / 2)
         // 3. while p == q, go to step 2.
+        // Also retry if their product is one bit short of the requested modulus length.
         let p: sjcl.BigNumber;
         let q: sjcl.BigNumber;
+        let n: sjcl.BigNumber;
+        const primeBitLength = algorithm.modulusLength >> 1;
+        const MAX_NUM_TRIES = algorithm.modulusLength ** 2;
+        let validKeySize = false;
+        let i = 0;
         do {
-            const p_tmp = generateSafePrimeSync(algorithm.modulusLength >> 1);
-            const q_tmp = generateSafePrimeSync(algorithm.modulusLength >> 1);
-            p = typeof p_tmp === 'bigint' ? new sjcl.bn(p_tmp.toString(16)) : p_tmp;
-            q = typeof q_tmp === 'bigint' ? new sjcl.bn(q_tmp.toString(16)) : q_tmp;
-        } while (p.equals(q));
+            const p_tmp = generateSafePrimeSync(primeBitLength);
+            const q_tmp = generateSafePrimeSync(primeBitLength);
+            p = typeof p_tmp === 'bigint' ? new sjcl.bn('0x' + p_tmp.toString(16)) : p_tmp;
+            q = typeof q_tmp === 'bigint' ? new sjcl.bn('0x' + q_tmp.toString(16)) : q_tmp;
+            n = p.mul(q);
+            validKeySize =
+                !p.equals(q) &&
+                p.bitLength() === primeBitLength &&
+                q.bitLength() === primeBitLength &&
+                n.bitLength() === algorithm.modulusLength;
+            i++;
+        } while (!validKeySize && i < MAX_NUM_TRIES);
+
+        if (!validKeySize) {
+            throw new Error(`generateKey reached MAX_NUM_TRIES=${MAX_NUM_TRIES}`);
+        }
 
         // 4. phi = (p - 1) * (q - 1)
         const phi = p.sub(1).mul(q.sub(1));
@@ -318,9 +339,6 @@ export class PartiallyBlindRSA {
         // TODO: replace this applying Chinese Remainder Theorem.
         const d = inverseMod(e, phi);
 
-        // 7. n = p * q
-        const n = p.mul(q);
-
         // 7. sk = (n, p, q, phi, d)
         const sk: BigSecretKey = { n, p, q, d };
         // 8. pk = (n, e)
diff --git a/src/prime.ts b/src/prime.ts
index ba779f4..e76ed67 100644
--- a/src/prime.ts
+++ b/src/prime.ts
@@ -66,17 +66,20 @@ export function generatePrime(bitLength: number, NUM_TRIES_PRIMALITY = 20): sjcl
     const MAX_NUM_TRIES = NUM_TRIES_PRIMALITY * bitLength ** 4;
 
     // 2^b
-    const twoToN = new sjcl.bn(2);
+    const twoToN = new sjcl.bn(1);
     for (let i = 0; i < bitLength; i++) {
         twoToN.doubleM();
     }
     twoToN.normalize();
 
+    const twoToNMinusOne = new sjcl.bn(twoToN).halveM().normalize();
+
     let prime: sjcl.BigNumber;
     let i = 0;
 
     do {
-        prime = sjcl.bn.random(twoToN, SJCL_PARANOIA);
+        prime = sjcl.bn.random(twoToNMinusOne, SJCL_PARANOIA);
+        prime = prime.addM(twoToNMinusOne).normalize();
         if ((prime.getLimb(0) & 0x1) == 0) {
             prime = prime.addM(1).normalize();
         }
diff --git a/test/partially_blindrsa.test.ts b/test/partially_blindrsa.test.ts
index 44ce60a..e267f4d 100644
--- a/test/partially_blindrsa.test.ts
+++ b/test/partially_blindrsa.test.ts
@@ -78,6 +78,36 @@ test('Parameters', () => {
     }
 });
 
+test.each([2, 3, 513])('generateKey/rejects invalid modulus length/%d', async (modulusLength) => {
+    await expect(
+        PartiallyBlindRSA.generateKey({
+            modulusLength,
+            publicExponent: Uint8Array.of(0x01, 0x00, 0x01),
+            hash: 'SHA-384',
+        }),
+    ).rejects.toThrow('modulusLength must be an even number greater than or equal to 4');
+});
+
+test('generateKey/stops retrying invalid bigint safe-prime hook', async () => {
+    let primeCount = 0;
+
+    await expect(
+        PartiallyBlindRSA.generateKey(
+            {
+                modulusLength: 4,
+                publicExponent: Uint8Array.of(0x01, 0x00, 0x01),
+                hash: 'SHA-384',
+            },
+            () => {
+                primeCount++;
+                return 2n;
+            },
+        ),
+    ).rejects.toThrow('generateKey reached MAX_NUM_TRIES=16');
+
+    expect(primeCount).toBe(32);
+});
+
 describe.each(vectors)('Errors-vec%#', (v: Vector) => {
     test('non-extractable-keys', async () => {
         const { privateKey, publicKey } = await keysFromVector(v, false);
diff --git a/test/primes.test.ts b/test/primes.test.ts
index 6324749..a9ae1da 100644
--- a/test/primes.test.ts
+++ b/test/primes.test.ts
@@ -83,7 +83,7 @@ test.each([128, 256, 512, 1024])(
     (bitLength) => {
         const p = generatePrime(bitLength);
 
-        expect(p.bitLength()).toBeGreaterThanOrEqual(bitLength);
+        expect(p.bitLength()).toBe(bitLength);
         expect(isPrime(p)).toBe(true);
     },
     1_200_000,
@@ -94,7 +94,7 @@ test.each([128, 256])(
     (bitLength) => {
         const p = generateSafePrime(bitLength);
 
-        expect(p.bitLength()).toBeGreaterThanOrEqual(bitLength);
+        expect(p.bitLength()).toBe(bitLength);
         expect(isSafePrime(p)).toBe(true);
     },
     1_200_000,