diff --git a/src/lib/gadgets/sha256.ts b/src/lib/gadgets/sha256.ts
index c85dc47ff3..d513b610f4 100644
--- a/src/lib/gadgets/sha256.ts
+++ b/src/lib/gadgets/sha256.ts
@@ -1,7 +1,8 @@
 // https://csrc.nist.gov/pubs/fips/180-4/upd1/final
-
 import { Field } from '../field.js';
 import { UInt32 } from '../int.js';
+import { TupleN } from '../util/types.js';
+import { assert, bitSlice, exists } from './common.js';
 import { Gadgets } from './gadgets.js';
 
 export { SHA256 };
@@ -149,36 +150,21 @@ function Maj(x: UInt32, y: UInt32, z: UInt32) {
 }
 
 function SigmaZero(x: UInt32) {
-  let rotr2 = ROTR(2, x);
-  let rotr13 = ROTR(13, x);
-  let rotr22 = ROTR(22, x);
-
-  return rotr2.xor(rotr13).xor(rotr22);
+  return sigma(x, [2, 13, 22]);
 }
 
 function SigmaOne(x: UInt32) {
-  let rotr6 = ROTR(6, x);
-  let rotr11 = ROTR(11, x);
-  let rotr25 = ROTR(25, x);
-
-  return rotr6.xor(rotr11).xor(rotr25);
+  return sigma(x, [6, 11, 25]);
 }
 
 // lowercase sigma = delta to avoid confusing function names
 
 function DeltaZero(x: UInt32) {
-  let rotr7 = ROTR(7, x);
-  let rotr18 = ROTR(18, x);
-  let shr3 = SHR(3, x);
-
-  return rotr7.xor(rotr18).xor(shr3);
+  return sigma(x, [3, 7, 18], true);
 }
 
 function DeltaOne(x: UInt32) {
-  let rotr17 = ROTR(17, x);
-  let rotr19 = ROTR(19, x);
-  let shr10 = SHR(10, x);
-  return rotr17.xor(rotr19).xor(shr10);
+  return sigma(x, [10, 17, 19], true);
 }
 
 function ROTR(n: number, x: UInt32) {
@@ -189,3 +175,82 @@ function SHR(n: number, x: UInt32) {
   let val = x.rightShift(n);
   return val;
 }
+
+function sigmaSimple(u: UInt32, bits: TupleN<number, 3>, firstShifted = false) {
+  let [r0, r1, r2] = bits;
+  let rot0 = firstShifted ? SHR(r0, u) : ROTR(r0, u);
+  let rot1 = ROTR(r1, u);
+  let rot2 = ROTR(r2, u);
+  return rot0.xor(rot1).xor(rot2);
+}
+
+function sigma(u: UInt32, bits: TupleN<number, 3>, firstShifted = false) {
+  if (u.isConstant()) return sigmaSimple(u, bits, firstShifted);
+
+  let [r0, r1, r2] = bits; // TODO assert bits are sorted
+  let x = u.value;
+
+  let d0 = r0;
+  let d1 = r1 - r0;
+  let d2 = r2 - r1;
+  let d3 = 32 - r2;
+
+  // decompose x into 4 chunks of size d0, d1, d2, d3
+  let [x0, x1, x2, x3] = exists(4, () => {
+    let xx = x.toBigInt();
+    return [
+      bitSlice(xx, 0, d0),
+      bitSlice(xx, r0, d1),
+      bitSlice(xx, r1, d2),
+      bitSlice(xx, r2, d3),
+    ];
+  });
+
+  // range check each chunk
+  // we only need to range check to 16 bits relying on the requirement that
+  // the rotated values are range-checked to 32 bits later; see comments below
+  rangeCheck16(x0);
+  rangeCheck16(x1);
+  rangeCheck16(x2);
+  rangeCheck16(x3);
+
+  // prove x decomposition
+
+  // x === x0 + x1*2^d0 + x2*2^(d0+d1) + x3*2^(d0+d1+d2)
+  let x23 = x2.add(x3.mul(1 << d2)).seal();
+  let x123 = x1.add(x23.mul(1 << d1)).seal();
+  x0.add(x123.mul(1 << d0)).assertEquals(x);
+  // ^ proves that 2^(32-d3)*x3 < x < 2^32 => x3 < 2^d3
+
+  // reassemble chunks into rotated values
+
+  let xRotR0: Field;
+
+  if (!firstShifted) {
+    // rotr(x, r0) = x1 + x2*2^d1 + x3*2^(d1+d2) + x0*2^(d1+d2+d3)
+    xRotR0 = x123.add(x0.mul(1 << (d1 + d2 + d3))).seal();
+    // ^ proves that 2^(32-d0)*x0 < xRotR0 => x0 < 2^d0 if we check xRotR0 < 2^32 later
+  } else {
+    // shr(x, r0) = x1 + x2*2^d1 + x3*2^(d1+d2)
+    xRotR0 = x123;
+
+    // finish x0 < 2^d0 proof:
+    rangeCheck16(x0.mul(1 << (16 - d0)).seal());
+  }
+
+  // rotr(x, r1) = x2 + x3*2^d2 + x0*2^(d2+d3) + x1*2^(d2+d3+d0)
+  let x01 = x0.add(x1.mul(1 << d0)).seal();
+  let xRotR1 = x23.add(x01.mul(1 << (d2 + d3))).seal();
+  // ^ proves that 2^(32-d1)*x1 < xRotR1 => x1 < 2^d1 if we check xRotR1 < 2^32 later
+
+  // rotr(x, r2) = x3 + x0*2^d3 + x1*2^(d3+d0) + x2*2^(d3+d0+d1)
+  let x012 = x01.add(x2.mul(1 << (d0 + d1))).seal();
+  let xRotR2 = x3.add(x012.mul(1 << d3)).seal();
+  // ^ proves that 2^(32-d2)*x2 < xRotR2 => x2 < 2^d2 if we check xRotR2 < 2^32 later
+
+  return UInt32.from(xRotR0).xor(xRotR1).xor(xRotR2);
+}
+
+function rangeCheck16(x: Field) {
+  x.rangeCheckHelper(16).assertEquals(x);
+}
diff --git a/src/lib/int.ts b/src/lib/int.ts
index ac59f5c78e..f161e83187 100644
--- a/src/lib/int.ts
+++ b/src/lib/int.ts
@@ -735,8 +735,10 @@ class UInt32 extends CircuitValue {
    * c.assertEquals(0b0110);
    * ```
    */
-  xor(x: UInt32) {
-    return UInt32.from(Gadgets.xor(this.value, x.value, UInt32.NUM_BITS));
+  xor(x: UInt32 | Field) {
+    return UInt32.from(
+      Gadgets.xor(this.value, UInt32.from(x).value, UInt32.NUM_BITS)
+    );
   }
 
   /**