diff --git a/README.md b/README.md
index b5bfb57..78606cc 100644
--- a/README.md
+++ b/README.md
@@ -8,3 +8,46 @@ Implementations from scratch done while studying some FHE papers; do not use in
 - `tfhe`: https://eprint.iacr.org/2018/421.pdf scheme implementation
 
 `cargo test --release`
+
+
+## Example of usage
+> the repo is a work in progress, interfaces will change.
+
+This example shows usage of TFHE, but the idea is that the same interface would
+work for using CKKS & BFV, the only thing to be changed would be the parameters
+and the line `type S = TWLE<K>` to use `CKKS<Q, N>` or `BFV<Q, N, T>`.
+
+```rust
+const T: u64 = 128; // msg space (msg modulus)
+const K: usize = 16;
+type S = TLWE<K>;
+
+let mut rng = rand::thread_rng();
+let msg_dist = Uniform::new(0_u64, T);
+
+let (sk, pk) = S::new_key(&mut rng)?;
+
+// get two random msgs in Z_t
+let m1 = Rq::<T, 1>::rand_u64(&mut rng, msg_dist)?;
+let m2 = Rq::<T, 1>::rand_u64(&mut rng, msg_dist)?;
+let m3 = Rq::<T, 1>::rand_u64(&mut rng, msg_dist)?;
+
+// encode the msgs into the plaintext space
+let p1: Tn<1> = S::encode::<T>(&m1); // plaintext
+let p2: Tn<1> = S::encode::<T>(&m2); // plaintext
+let c3_const: Tn<1> = Tn(array::from_fn(|i| T64(m3.coeffs()[i].0))); // encode it as constant value
+
+let c1 = S::encrypt(&mut rng, &pk, &p1)?;
+let c2 = S::encrypt(&mut rng, &pk, &p2)?;
+
+// now we can do encrypted operations (notice that we do them using simple
+// operations by operator overloading):
+let c3 = c1 + c2;
+let c4 = c2 * c3_const;
+
+// decrypt & decode
+let p4_recovered = c4.decrypt(&sk);
+let m4 = S::decode::<T>(&p4_recovered);
+
+// m4 is equal to (m1+m2)*m3
+```
diff --git a/arith/src/ring_torus.rs b/arith/src/ring_torus.rs
index 9da3656..6cc1abc 100644
--- a/arith/src/ring_torus.rs
+++ b/arith/src/ring_torus.rs
@@ -167,15 +167,6 @@ fn naive_poly_mul<const N: usize>(poly1: &Tn<N>, poly2: &Tn<N>) -> Tn<N> {
     // apply mod (X^N + 1))
     modulus_u128::<N>(&mut result);
 
-    // sanity check: check that there are no coeffs > i64_max
-    assert_eq!(
-        result,
-        Tn::<N>(array::from_fn(|i| T64(result[i] as u64)))
-            .coeffs()
-            .iter()
-            .map(|c| c.0 as u128)
-            .collect::<Vec<_>>()
-    );
     Tn(array::from_fn(|i| T64(result[i] as u64)))
 }
 fn modulus_u128<const N: usize>(p: &mut Vec<u128>) {
diff --git a/arith/src/torus.rs b/arith/src/torus.rs
index 0c6abb2..dc85eb6 100644
--- a/arith/src/torus.rs
+++ b/arith/src/torus.rs
@@ -44,7 +44,7 @@ impl Add<T64> for T64 {
 }
 impl AddAssign for T64 {
     fn add_assign(&mut self, rhs: Self) {
-        self.0 += rhs.0;
+        self.0 = self.0.wrapping_add(rhs.0)
     }
 }
 
@@ -57,7 +57,7 @@ impl Sub<T64> for T64 {
 }
 impl SubAssign for T64 {
     fn sub_assign(&mut self, rhs: Self) {
-        self.0 -= rhs.0;
+        self.0 = self.0.wrapping_sub(rhs.0)
     }
 }
 
@@ -87,7 +87,7 @@ impl Mul<T64> for T64 {
 }
 impl MulAssign for T64 {
     fn mul_assign(&mut self, rhs: Self) {
-        self.0 *= rhs.0;
+        self.0 = self.0.wrapping_mul(rhs.0)
     }
 }
 
@@ -96,14 +96,14 @@ impl Mul<u64> for T64 {
     type Output = Self;
 
     fn mul(self, s: u64) -> Self {
-        Self(self.0 * s)
+        Self(self.0.wrapping_mul(s))
     }
 }
 impl Mul<&u64> for &T64 {
     type Output = T64;
 
     fn mul(self, s: &u64) -> Self::Output {
-        T64(self.0 * s)
+        T64(self.0.wrapping_mul(*s))
     }
 }
 
diff --git a/arith/src/tuple_ring.rs b/arith/src/tuple_ring.rs
index af18e76..f60b75b 100644
--- a/arith/src/tuple_ring.rs
+++ b/arith/src/tuple_ring.rs
@@ -55,7 +55,7 @@ impl<R: Ring, const K: usize> Sub<TR<R, K>> for TR<R, K> {
     }
 }
 
-/// for (TR,TR), the Mul operation is defined as:
+/// for (TR,TR), the Mul operation is defined as the dot product:
 /// for A, B \in R^k, result = Σ A_i * B_i \in R
 impl<R: Ring, const K: usize> Mul<TR<R, K>> for TR<R, K> {
     type Output = R;
diff --git a/tfhe/src/tlwe.rs b/tfhe/src/tlwe.rs
index a5669a4..68fb56a 100644
--- a/tfhe/src/tlwe.rs
+++ b/tfhe/src/tlwe.rs
@@ -133,6 +133,15 @@ impl<const K: usize> Sub<Tn<1>> for TLWE<K> {
         Self(a, b)
     }
 }
+// plaintext multiplication
+impl<const K: usize> Mul<Tn<1>> for TLWE<K> {
+    type Output = Self;
+    fn mul(self, plaintext: Tn<1>) -> Self {
+        let a: TR<Tn<1>, K> = TR(self.0 .0.iter().map(|r_i| *r_i * plaintext).collect());
+        let b: Tn<1> = self.1 * plaintext;
+        Self(a, b)
+    }
+}
 
 #[cfg(test)]
 mod tests {
@@ -235,4 +244,34 @@ mod tests {
 
         Ok(())
     }
+
+    #[test]
+    fn test_mul_plaintext() -> Result<()> {
+        const T: u64 = 128;
+        const K: usize = 16;
+        type S = TLWE<K>;
+
+        let mut rng = rand::thread_rng();
+
+        for _ in 0..200 {
+            let (sk, pk) = S::new_key(&mut rng)?;
+
+            let msg_dist = Uniform::new(0_u64, T);
+            let m1 = Rq::<T, 1>::rand_u64(&mut rng, msg_dist)?;
+            let m2 = Rq::<T, 1>::rand_u64(&mut rng, msg_dist)?;
+            let p1: Tn<1> = S::encode::<T>(&m1);
+            // don't scale up p2, set it directly from m2
+            let p2: Tn<1> = Tn(array::from_fn(|i| T64(m2.coeffs()[i].0)));
+
+            let c1 = S::encrypt(&mut rng, &pk, &p1)?;
+
+            let c3 = c1 * p2;
+
+            let p3_recovered: Tn<1> = c3.decrypt(&sk);
+            let m3_recovered = S::decode::<T>(&p3_recovered);
+            assert_eq!((m1.to_r() * m2.to_r()).to_rq::<T>(), m3_recovered);
+        }
+
+        Ok(())
+    }
 }