diff --git a/benches/bigint.rs b/benches/bigint.rs
index e1ec45e5..591e9a99 100644
--- a/benches/bigint.rs
+++ b/benches/bigint.rs
@@ -18,7 +18,7 @@ fn get_rng() -> StdRng {
     SeedableRng::from_seed(seed)
 }
 
-fn multiply_bench(b: &mut Bencher, xbits: usize, ybits: usize) {
+fn multiply_bench(b: &mut Bencher, xbits: u64, ybits: u64) {
     let mut rng = get_rng();
     let x = rng.gen_bigint(xbits);
     let y = rng.gen_bigint(ybits);
@@ -26,7 +26,7 @@ fn multiply_bench(b: &mut Bencher, xbits: usize, ybits: usize) {
     b.iter(|| &x * &y);
 }
 
-fn divide_bench(b: &mut Bencher, xbits: usize, ybits: usize) {
+fn divide_bench(b: &mut Bencher, xbits: u64, ybits: u64) {
     let mut rng = get_rng();
     let x = rng.gen_bigint(xbits);
     let y = rng.gen_bigint(ybits);
@@ -34,7 +34,7 @@ fn divide_bench(b: &mut Bencher, xbits: usize, ybits: usize) {
     b.iter(|| &x / &y);
 }
 
-fn remainder_bench(b: &mut Bencher, xbits: usize, ybits: usize) {
+fn remainder_bench(b: &mut Bencher, xbits: u64, ybits: u64) {
     let mut rng = get_rng();
     let x = rng.gen_bigint(xbits);
     let y = rng.gen_bigint(ybits);
@@ -245,7 +245,7 @@ fn from_str_radix_36(b: &mut Bencher) {
     from_str_radix_bench(b, 36);
 }
 
-fn rand_bench(b: &mut Bencher, bits: usize) {
+fn rand_bench(b: &mut Bencher, bits: u64) {
     let mut rng = get_rng();
 
     b.iter(|| rng.gen_bigint(bits));
diff --git a/benches/gcd.rs b/benches/gcd.rs
index e6c54b5a..1a65654c 100644
--- a/benches/gcd.rs
+++ b/benches/gcd.rs
@@ -18,7 +18,7 @@ fn get_rng() -> StdRng {
     SeedableRng::from_seed(seed)
 }
 
-fn bench(b: &mut Bencher, bits: usize, gcd: fn(&BigUint, &BigUint) -> BigUint) {
+fn bench(b: &mut Bencher, bits: u64, gcd: fn(&BigUint, &BigUint) -> BigUint) {
     let mut rng = get_rng();
     let x = rng.gen_biguint(bits);
     let y = rng.gen_biguint(bits);
diff --git a/benches/roots.rs b/benches/roots.rs
index e6113136..2433cf0a 100644
--- a/benches/roots.rs
+++ b/benches/roots.rs
@@ -43,7 +43,7 @@ fn check(x: &BigUint, n: u32) {
     assert_eq!((&hi - 1u32).nth_root(n), root);
 }
 
-fn bench_sqrt(b: &mut Bencher, bits: usize) {
+fn bench_sqrt(b: &mut Bencher, bits: u64) {
     let x = get_rng().gen_biguint(bits);
     eprintln!("bench_sqrt({})", x);
 
@@ -71,7 +71,7 @@ fn big4k_sqrt(b: &mut Bencher) {
     bench_sqrt(b, 4096);
 }
 
-fn bench_cbrt(b: &mut Bencher, bits: usize) {
+fn bench_cbrt(b: &mut Bencher, bits: u64) {
     let x = get_rng().gen_biguint(bits);
     eprintln!("bench_cbrt({})", x);
 
@@ -99,7 +99,7 @@ fn big4k_cbrt(b: &mut Bencher) {
     bench_cbrt(b, 4096);
 }
 
-fn bench_nth_root(b: &mut Bencher, bits: usize, n: u32) {
+fn bench_nth_root(b: &mut Bencher, bits: u64, n: u32) {
     let x = get_rng().gen_biguint(bits);
     eprintln!("bench_{}th_root({})", n, x);
 
diff --git a/ci/big_rand/src/lib.rs b/ci/big_rand/src/lib.rs
index 2c22db0a..1bea5750 100644
--- a/ci/big_rand/src/lib.rs
+++ b/ci/big_rand/src/lib.rs
@@ -102,7 +102,7 @@ mod biguint {
         let mut rng = R::from_seed(seed);
         for (i, &s) in expected.iter().enumerate() {
             let n: BigUint = s.parse().unwrap();
-            let r = rng.gen_biguint((1 << i) + i);
+            let r = rng.gen_biguint((1 << i) + i as u64);
             assert_eq!(n, r);
         }
     }
@@ -302,7 +302,7 @@ mod bigint {
         let mut rng = R::from_seed(seed);
         for (i, &s) in expected.iter().enumerate() {
             let n: BigInt = s.parse().unwrap();
-            let r = rng.gen_bigint((1 << i) + i);
+            let r = rng.gen_bigint((1 << i) + i as u64);
             assert_eq!(n, r);
         }
     }
diff --git a/src/algorithms.rs b/src/algorithms.rs
index d6a5625a..7952f939 100644
--- a/src/algorithms.rs
+++ b/src/algorithms.rs
@@ -521,7 +521,7 @@ fn mac3(acc: &mut [BigDigit], b: &[BigDigit], c: &[BigDigit]) {
         // Recomposition. The coefficients of the polynomial are now known.
         //
         // Evaluate at w(t) where t is our given base to get the result.
-        let bits = big_digit::BITS * i;
+        let bits = u64::from(big_digit::BITS) * i as u64;
         let result = r0
             + (comp1 << bits)
             + (comp2 << (2 * bits))
@@ -720,11 +720,11 @@ fn div_rem_core(mut a: BigUint, b: &BigUint) -> (BigUint, BigUint) {
 
 /// Find last set bit
 /// fls(0) == 0, fls(u32::MAX) == 32
-pub(crate) fn fls<T: PrimInt>(v: T) -> usize {
-    mem::size_of::<T>() * 8 - v.leading_zeros() as usize
+pub(crate) fn fls<T: PrimInt>(v: T) -> u8 {
+    mem::size_of::<T>() as u8 * 8 - v.leading_zeros() as u8
 }
 
-pub(crate) fn ilog2<T: PrimInt>(v: T) -> usize {
+pub(crate) fn ilog2<T: PrimInt>(v: T) -> u8 {
     fls(v) - 1
 }
 
diff --git a/src/bigint.rs b/src/bigint.rs
index 3a5d226f..1e20f503 100644
--- a/src/bigint.rs
+++ b/src/bigint.rs
@@ -875,7 +875,7 @@ impl_shift! { i8, i16, i32, i64, i128, isize }
 fn shr_round_down<T: PrimInt>(i: &BigInt, shift: T) -> bool {
     if i.is_negative() {
         let zeros = i.trailing_zeros().expect("negative values are non-zero");
-        shift > T::zero() && shift.to_usize().map(|shift| zeros < shift).unwrap_or(true)
+        shift > T::zero() && shift.to_u64().map(|shift| zeros < shift).unwrap_or(true)
     } else {
         false
     }
@@ -3195,7 +3195,7 @@ impl BigInt {
     /// Determines the fewest bits necessary to express the `BigInt`,
     /// not including the sign.
     #[inline]
-    pub fn bits(&self) -> usize {
+    pub fn bits(&self) -> u64 {
         self.data.bits()
     }
 
@@ -3290,7 +3290,7 @@ impl BigInt {
 
     /// Returns the number of least-significant bits that are zero,
     /// or `None` if the entire number is zero.
-    pub fn trailing_zeros(&self) -> Option<usize> {
+    pub fn trailing_zeros(&self) -> Option<u64> {
         self.data.trailing_zeros()
     }
 }
diff --git a/src/bigrand.rs b/src/bigrand.rs
index 864f75d7..ae00b7e4 100644
--- a/src/bigrand.rs
+++ b/src/bigrand.rs
@@ -11,17 +11,17 @@ use crate::bigint::{into_magnitude, magnitude};
 use crate::biguint::biguint_from_vec;
 
 use num_integer::Integer;
-use num_traits::Zero;
+use num_traits::{ToPrimitive, Zero};
 
 /// A trait for sampling random big integers.
 ///
 /// The `rand` feature must be enabled to use this. See crate-level documentation for details.
 pub trait RandBigInt {
     /// Generate a random `BigUint` of the given bit size.
-    fn gen_biguint(&mut self, bit_size: usize) -> BigUint;
+    fn gen_biguint(&mut self, bit_size: u64) -> BigUint;
 
     /// Generate a random BigInt of the given bit size.
-    fn gen_bigint(&mut self, bit_size: usize) -> BigInt;
+    fn gen_bigint(&mut self, bit_size: u64) -> BigInt;
 
     /// Generate a random `BigUint` less than the given bound. Fails
     /// when the bound is zero.
@@ -38,7 +38,7 @@ pub trait RandBigInt {
     fn gen_bigint_range(&mut self, lbound: &BigInt, ubound: &BigInt) -> BigInt;
 }
 
-fn gen_bits<R: Rng + ?Sized>(rng: &mut R, data: &mut [u32], rem: usize) {
+fn gen_bits<R: Rng + ?Sized>(rng: &mut R, data: &mut [u32], rem: u64) {
     // `fill` is faster than many `gen::<u32>` calls
     rng.fill(data);
     if rem > 0 {
@@ -49,25 +49,31 @@ fn gen_bits<R: Rng + ?Sized>(rng: &mut R, data: &mut [u32], rem: usize) {
 
 impl<R: Rng + ?Sized> RandBigInt for R {
     #[cfg(not(u64_digit))]
-    fn gen_biguint(&mut self, bit_size: usize) -> BigUint {
+    fn gen_biguint(&mut self, bit_size: u64) -> BigUint {
         let (digits, rem) = bit_size.div_rem(&32);
-        let mut data = vec![0u32; digits + (rem > 0) as usize];
+        let len = (digits + (rem > 0) as u64)
+            .to_usize()
+            .expect("capacity overflow");
+        let mut data = vec![0u32; len];
         gen_bits(self, &mut data, rem);
         biguint_from_vec(data)
     }
 
     #[cfg(u64_digit)]
-    fn gen_biguint(&mut self, bit_size: usize) -> BigUint {
+    fn gen_biguint(&mut self, bit_size: u64) -> BigUint {
         use core::slice;
 
         let (digits, rem) = bit_size.div_rem(&32);
+        let len = (digits + (rem > 0) as u64)
+            .to_usize()
+            .expect("capacity overflow");
         let native_digits = bit_size.div_ceil(&64);
-        let mut data = vec![0u64; native_digits];
+        let native_len = native_digits.to_usize().expect("capacity overflow");
+        let mut data = vec![0u64; native_len];
         unsafe {
             // Generate bits in a `&mut [u32]` slice for value stability
             let ptr = data.as_mut_ptr() as *mut u32;
-            let len = digits + (rem > 0) as usize;
-            debug_assert!(native_digits * 2 >= len);
+            debug_assert!(native_len * 2 >= len);
             let data = slice::from_raw_parts_mut(ptr, len);
             gen_bits(self, data, rem);
         }
@@ -79,7 +85,7 @@ impl<R: Rng + ?Sized> RandBigInt for R {
         biguint_from_vec(data)
     }
 
-    fn gen_bigint(&mut self, bit_size: usize) -> BigInt {
+    fn gen_bigint(&mut self, bit_size: u64) -> BigInt {
         loop {
             // Generate a random BigUint...
             let biguint = self.gen_biguint(bit_size);
@@ -253,12 +259,12 @@ impl SampleUniform for BigInt {
 /// The `rand` feature must be enabled to use this. See crate-level documentation for details.
 #[derive(Clone, Copy, Debug)]
 pub struct RandomBits {
-    bits: usize,
+    bits: u64,
 }
 
 impl RandomBits {
     #[inline]
-    pub fn new(bits: usize) -> RandomBits {
+    pub fn new(bits: u64) -> RandomBits {
         RandomBits { bits }
     }
 }
diff --git a/src/biguint.rs b/src/biguint.rs
index 53684185..ff256e48 100644
--- a/src/biguint.rs
+++ b/src/biguint.rs
@@ -216,14 +216,14 @@ impl FromStr for BigUint {
 
 // Convert from a power of two radix (bits == ilog2(radix)) where bits evenly divides
 // BigDigit::BITS
-fn from_bitwise_digits_le(v: &[u8], bits: usize) -> BigUint {
+fn from_bitwise_digits_le(v: &[u8], bits: u8) -> BigUint {
     debug_assert!(!v.is_empty() && bits <= 8 && big_digit::BITS % bits == 0);
     debug_assert!(v.iter().all(|&c| BigDigit::from(c) < (1 << bits)));
 
     let digits_per_big_digit = big_digit::BITS / bits;
 
     let data = v
-        .chunks(digits_per_big_digit)
+        .chunks(digits_per_big_digit.into())
         .map(|chunk| {
             chunk
                 .iter()
@@ -237,11 +237,15 @@ fn from_bitwise_digits_le(v: &[u8], bits: usize) -> BigUint {
 
 // Convert from a power of two radix (bits == ilog2(radix)) where bits doesn't evenly divide
 // BigDigit::BITS
-fn from_inexact_bitwise_digits_le(v: &[u8], bits: usize) -> BigUint {
+fn from_inexact_bitwise_digits_le(v: &[u8], bits: u8) -> BigUint {
     debug_assert!(!v.is_empty() && bits <= 8 && big_digit::BITS % bits != 0);
     debug_assert!(v.iter().all(|&c| BigDigit::from(c) < (1 << bits)));
 
-    let big_digits = (v.len() * bits + big_digit::BITS - 1) / big_digit::BITS;
+    let big_digits = (v.len() as u64)
+        .saturating_mul(bits.into())
+        .div_ceil(&big_digit::BITS.into())
+        .to_usize()
+        .unwrap_or(core::usize::MAX);
     let mut data = Vec::with_capacity(big_digits);
 
     let mut d = 0;
@@ -1587,7 +1591,7 @@ impl Integer for BigUint {
     #[inline]
     fn gcd(&self, other: &Self) -> Self {
         #[inline]
-        fn twos(x: &BigUint) -> usize {
+        fn twos(x: &BigUint) -> u64 {
             x.trailing_zeros().unwrap_or(0)
         }
 
@@ -1688,7 +1692,7 @@ impl Integer for BigUint {
 }
 
 #[inline]
-fn fixpoint<F>(mut x: BigUint, max_bits: usize, f: F) -> BigUint
+fn fixpoint<F>(mut x: BigUint, max_bits: u64, f: F) -> BigUint
 where
     F: Fn(&BigUint) -> BigUint,
 {
@@ -1739,7 +1743,8 @@ impl Roots for BigUint {
 
         // The root of non-zero values less than 2ⁿ can only be 1.
         let bits = self.bits();
-        if bits <= n as usize {
+        let n64 = u64::from(n);
+        if bits <= n64 {
             return BigUint::one();
         }
 
@@ -1748,7 +1753,7 @@ impl Roots for BigUint {
             return x.nth_root(n).into();
         }
 
-        let max_bits = bits / n as usize + 1;
+        let max_bits = bits / n64 + 1;
 
         #[cfg(feature = "std")]
         let guess = if let Some(f) = self.to_f64() {
@@ -1757,11 +1762,10 @@ impl Roots for BigUint {
         } else {
             // Try to guess by scaling down such that it does fit in `f64`.
             // With some (x * 2ⁿᵏ), its nth root ≈ (ⁿ√x * 2ᵏ)
-            let nsz = n as usize;
-            let extra_bits = bits - (f64::MAX_EXP as usize - 1);
-            let root_scale = (extra_bits + (nsz - 1)) / nsz;
-            let scale = root_scale * nsz;
-            if scale < bits && bits - scale > nsz {
+            let extra_bits = bits - (f64::MAX_EXP as u64 - 1);
+            let root_scale = extra_bits.div_ceil(&n64);
+            let scale = root_scale * n64;
+            if scale < bits && bits - scale > n64 {
                 (self >> scale).nth_root(n) << root_scale
             } else {
                 BigUint::one() << max_bits
@@ -1792,7 +1796,7 @@ impl Roots for BigUint {
         }
 
         let bits = self.bits();
-        let max_bits = bits / 2 as usize + 1;
+        let max_bits = bits / 2 + 1;
 
         #[cfg(feature = "std")]
         let guess = if let Some(f) = self.to_f64() {
@@ -1801,7 +1805,7 @@ impl Roots for BigUint {
         } else {
             // Try to guess by scaling down such that it does fit in `f64`.
             // With some (x * 2²ᵏ), its sqrt ≈ (√x * 2ᵏ)
-            let extra_bits = bits - (f64::MAX_EXP as usize - 1);
+            let extra_bits = bits - (f64::MAX_EXP as u64 - 1);
             let root_scale = (extra_bits + 1) / 2;
             let scale = root_scale * 2;
             (self >> scale).sqrt() << root_scale
@@ -1828,7 +1832,7 @@ impl Roots for BigUint {
         }
 
         let bits = self.bits();
-        let max_bits = bits / 3 as usize + 1;
+        let max_bits = bits / 3 + 1;
 
         #[cfg(feature = "std")]
         let guess = if let Some(f) = self.to_f64() {
@@ -1837,7 +1841,7 @@ impl Roots for BigUint {
         } else {
             // Try to guess by scaling down such that it does fit in `f64`.
             // With some (x * 2³ᵏ), its cbrt ≈ (∛x * 2ᵏ)
-            let extra_bits = bits - (f64::MAX_EXP as usize - 1);
+            let extra_bits = bits - (f64::MAX_EXP as u64 - 1);
             let root_scale = (extra_bits + 2) / 3;
             let scale = root_scale * 3;
             (self >> scale).cbrt() << root_scale
@@ -1864,7 +1868,7 @@ fn high_bits_to_u64(v: &BigUint) -> u64 {
             let mut ret_bits = 0;
 
             for d in v.data.iter().rev() {
-                let digit_bits = (bits - 1) % big_digit::BITS + 1;
+                let digit_bits = (bits - 1) % u64::from(big_digit::BITS) + 1;
                 let bits_want = cmp::min(64 - ret_bits, digit_bits);
 
                 if bits_want != 64 {
@@ -1932,9 +1936,9 @@ impl ToPrimitive for BigUint {
     #[inline]
     fn to_f32(&self) -> Option<f32> {
         let mantissa = high_bits_to_u64(self);
-        let exponent = self.bits() - fls(mantissa);
+        let exponent = self.bits() - u64::from(fls(mantissa));
 
-        if exponent > f32::MAX_EXP as usize {
+        if exponent > f32::MAX_EXP as u64 {
             None
         } else {
             let ret = (mantissa as f32) * 2.0f32.powi(exponent as i32);
@@ -1949,9 +1953,9 @@ impl ToPrimitive for BigUint {
     #[inline]
     fn to_f64(&self) -> Option<f64> {
         let mantissa = high_bits_to_u64(self);
-        let exponent = self.bits() - fls(mantissa);
+        let exponent = self.bits() - u64::from(fls(mantissa));
 
-        if exponent > f64::MAX_EXP as usize {
+        if exponent > f64::MAX_EXP as u64 {
             None
         } else {
             let ret = (mantissa as f64) * 2.0f64.powi(exponent as i32);
@@ -2170,13 +2174,17 @@ impl_to_biguint!(f32, FromPrimitive::from_f32);
 impl_to_biguint!(f64, FromPrimitive::from_f64);
 
 // Extract bitwise digits that evenly divide BigDigit
-fn to_bitwise_digits_le(u: &BigUint, bits: usize) -> Vec<u8> {
+fn to_bitwise_digits_le(u: &BigUint, bits: u8) -> Vec<u8> {
     debug_assert!(!u.is_zero() && bits <= 8 && big_digit::BITS % bits == 0);
 
     let last_i = u.data.len() - 1;
     let mask: BigDigit = (1 << bits) - 1;
     let digits_per_big_digit = big_digit::BITS / bits;
-    let digits = (u.bits() + bits - 1) / bits;
+    let digits = u
+        .bits()
+        .div_ceil(&u64::from(bits))
+        .to_usize()
+        .unwrap_or(core::usize::MAX);
     let mut res = Vec::with_capacity(digits);
 
     for mut r in u.data[..last_i].iter().cloned() {
@@ -2196,11 +2204,15 @@ fn to_bitwise_digits_le(u: &BigUint, bits: usize) -> Vec<u8> {
 }
 
 // Extract bitwise digits that don't evenly divide BigDigit
-fn to_inexact_bitwise_digits_le(u: &BigUint, bits: usize) -> Vec<u8> {
+fn to_inexact_bitwise_digits_le(u: &BigUint, bits: u8) -> Vec<u8> {
     debug_assert!(!u.is_zero() && bits <= 8 && big_digit::BITS % bits != 0);
 
     let mask: BigDigit = (1 << bits) - 1;
-    let digits = (u.bits() + bits - 1) / bits;
+    let digits = u
+        .bits()
+        .div_ceil(&u64::from(bits))
+        .to_usize()
+        .unwrap_or(core::usize::MAX);
     let mut res = Vec::with_capacity(digits);
 
     let mut r = 0;
@@ -2659,12 +2671,12 @@ impl BigUint {
 
     /// Determines the fewest bits necessary to express the `BigUint`.
     #[inline]
-    pub fn bits(&self) -> usize {
+    pub fn bits(&self) -> u64 {
         if self.is_zero() {
             return 0;
         }
-        let zeros = self.data.last().unwrap().leading_zeros();
-        self.data.len() * big_digit::BITS - zeros as usize
+        let zeros: u64 = self.data.last().unwrap().leading_zeros().into();
+        self.data.len() as u64 * u64::from(big_digit::BITS) - zeros
     }
 
     /// Strips off trailing zero bigdigits - comparisons require the last element in the vector to
@@ -2723,9 +2735,10 @@ impl BigUint {
 
     /// Returns the number of least-significant bits that are zero,
     /// or `None` if the entire number is zero.
-    pub fn trailing_zeros(&self) -> Option<usize> {
+    pub fn trailing_zeros(&self) -> Option<u64> {
         let i = self.data.iter().position(|&digit| digit != 0)?;
-        Some(i * big_digit::BITS + self.data[i].trailing_zeros() as usize)
+        let zeros: u64 = self.data[i].trailing_zeros().into();
+        Some(i as u64 * u64::from(big_digit::BITS) + zeros)
     }
 }
 
@@ -2745,7 +2758,7 @@ fn plain_modpow(base: &BigUint, exp_data: &[BigDigit], modulus: &BigUint) -> Big
     }
 
     let mut r = exp_data[i];
-    let mut b = 0usize;
+    let mut b = 0u8;
     while r.is_even() {
         base = &base * &base % modulus;
         r >>= 1;
@@ -2984,7 +2997,7 @@ impl<'de> serde::Deserialize<'de> for BigUint {
 
 /// Returns the greatest power of the radix for the given bit size
 #[inline]
-fn get_radix_base(radix: u32, bits: usize) -> (BigDigit, usize) {
+fn get_radix_base(radix: u32, bits: u8) -> (BigDigit, usize) {
     mod gen {
         include! { concat!(env!("OUT_DIR"), "/radix_bases.rs") }
     }
diff --git a/src/lib.rs b/src/lib.rs
index 7b757840..30c5abec 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -258,11 +258,11 @@ mod big_digit {
 
     // `DoubleBigDigit` size dependent
     #[cfg(not(u64_digit))]
-    pub(crate) const BITS: usize = 32;
+    pub(crate) const BITS: u8 = 32;
     #[cfg(u64_digit)]
-    pub(crate) const BITS: usize = 64;
+    pub(crate) const BITS: u8 = 64;
 
-    pub(crate) const HALF_BITS: usize = BITS / 2;
+    pub(crate) const HALF_BITS: u8 = BITS / 2;
     pub(crate) const HALF: BigDigit = (1 << HALF_BITS) - 1;
 
     const LO_MASK: DoubleBigDigit = (1 << BITS) - 1;
diff --git a/src/monty.rs b/src/monty.rs
index 0307ec82..b9a0bd06 100644
--- a/src/monty.rs
+++ b/src/monty.rs
@@ -150,7 +150,7 @@ pub(crate) fn monty_modpow(x: &BigUint, y: &BigUint, m: &BigUint) -> BigUint {
 
     // rr = 2**(2*_W*len(m)) mod m
     let mut rr = BigUint::one();
-    rr = (rr.shl(2 * num_words * big_digit::BITS)) % m;
+    rr = (rr.shl(2 * num_words as u64 * u64::from(big_digit::BITS))) % m;
     if rr.data.len() < num_words {
         rr.data.resize(num_words, 0);
     }