diff --git a/packages/bugc/CHANGELOG.md b/packages/bugc/CHANGELOG.md
index fb6e5e4ea..1b5cb6a56 100644
--- a/packages/bugc/CHANGELOG.md
+++ b/packages/bugc/CHANGELOG.md
@@ -65,6 +65,14 @@ support. Changes to the specification itself are tracked in the root
   the function has another `return;`. Before, return merging removed
   the call's continuation block, and code generation failed ("Jump
   target ... not found") ([#388]).
+- `==` and `!=` on `bytes` or `string` in memory compare the contents,
+  by the `keccak256` of each operand's data. Before, they compared the
+  two memory addresses, so equal contents in two places were unequal
+  ([#391]).
+- A hex literal longer than 32 bytes hashes and compares by its bytes
+  at every optimization level. Before, constant folding hashed its
+  text. A string literal that starts with "0x" is its text. Before,
+  code generation decoded it as hex ([#391]).
 - `push` of bytes in calldata onto an `array<bytes>` in storage now
   stores the bytes. Before, it stored the calldata reference's word
   (offset and length) ([#379]).
@@ -563,3 +571,4 @@ First publication.
 [#383]: https://github.com/ethdebug/format/pull/383
 [#384]: https://github.com/ethdebug/format/pull/384
 [#388]: https://github.com/ethdebug/format/pull/388
+[#391]: https://github.com/ethdebug/format/pull/391
diff --git a/packages/bugc/src/evmgen/__snapshots__/activation-pairing.test.ts.snap b/packages/bugc/src/evmgen/__snapshots__/activation-pairing.test.ts.snap
index 4a23c8dd1..8195b3707 100644
--- a/packages/bugc/src/evmgen/__snapshots__/activation-pairing.test.ts.snap
+++ b/packages/bugc/src/evmgen/__snapshots__/activation-pairing.test.ts.snap
@@ -46,10 +46,10 @@ exports[`the code does not depend on the markers > is the same for every example
   "advanced/nested-structs.bug at O1": "ae8772307e46e33f94b8e36017ef8a4c23743fbb26085693a495239d25da3008",
   "advanced/nested-structs.bug at O2": "d141d683602322f4195283df977a6c5618e72e995bfca7d8b285ae3a0f779bb4",
   "advanced/nested-structs.bug at O3": "d141d683602322f4195283df977a6c5618e72e995bfca7d8b285ae3a0f779bb4",
-  "advanced/token-registry.bug at O0": "430a947f02160f905a04efdeea0b5d69fb7aa6be6b2dc54734be26ec0511988a",
-  "advanced/token-registry.bug at O1": "2de1cfe1d642a35a409d46df00f9ebb08ce0c43736020a5aa06cae30983b72e6",
-  "advanced/token-registry.bug at O2": "2c0dcbce9d26b8182ce6607aeb480005fc485e7e21c44405be48c451c945c2e3",
-  "advanced/token-registry.bug at O3": "989c5871f2e3c40b13af199fcc441a7aee8427a7935243af8f152d6758124642",
+  "advanced/token-registry.bug at O0": "5e8c272398293d3005131bbb42f9ff447f7eeb34c37f8a0ace51189986d53815",
+  "advanced/token-registry.bug at O1": "e64bc9fa74ad41315751723cbe5f5efa67496523d60c4d41a23eb7274a1efd7d",
+  "advanced/token-registry.bug at O2": "f70c11b4702b7d5eab762a68809048055f512bbaad6fb8e584d62a678b325b37",
+  "advanced/token-registry.bug at O3": "72c79740aed4c6a0a2e79347b2652c96a6cbbcb260f0fd6c710809fcb08c144b",
   "advanced/voting-system.bug at O0": "e49f6349c9c8fc5d3e14cd737a860891139518b0e0ea73d3c01f314a60e80a30",
   "advanced/voting-system.bug at O1": "32f9c89a0890e1591b4e1aa29ec2b52d83a13c6ed06c4fc3391245002c505e38",
   "advanced/voting-system.bug at O2": "3479679f427cb43fcc7288d34666f865d94c20a05f216f719f41a218624823ec",
diff --git a/packages/bugc/src/evmgen/bytes-equality.test.ts b/packages/bugc/src/evmgen/bytes-equality.test.ts
new file mode 100644
index 000000000..71f4d033b
--- /dev/null
+++ b/packages/bugc/src/evmgen/bytes-equality.test.ts
@@ -0,0 +1,131 @@
+import { describe, it, expect } from "vitest";
+
+import { keccak256 } from "ethereum-cryptography/keccak";
+
+import { executeProgram } from "#test/evm/behavioral";
+
+const levels = [0, 1, 2, 3] as const;
+
+const word = (n: number) => n.toString(16).padStart(64, "0");
+const hex = (s: string) => Buffer.from(s).toString("hex");
+
+/** Calldata: the two lengths, then the two byte strings */
+const calldata = (a: string, b: string) =>
+  word(a.length) + word(b.length) + hex(a) + hex(b);
+
+/**
+ * Two byte strings from the calldata, each copied to its own place in
+ * memory, compared as `bytes` and as `string`, with a literal, and
+ * with a string in storage
+ */
+const source = `name Equality;
+storage {
+  [0] eq: bool; [1] ne: bool; [2] seq: bool; [3] sne: bool;
+  [4] lit: bool; [5] stored: string; [6] seqs: bool;
+}
+code {
+  let la = msg.data[0:32] as bytes32 as uint256;
+  let lb = msg.data[32:64] as bytes32 as uint256;
+  let a: bytes = msg.data[64:64 + la];
+  let b: bytes = msg.data[64 + la:64 + la + lb];
+  eq = a == b;
+  ne = a != b;
+  let x = a as string;
+  let y = b as string;
+  seq = x == y;
+  sne = x != y;
+  lit = x == "abc";
+  stored = x;
+  seqs = stored == y;
+}`;
+
+const cases: [string, string, string][] = [
+  ["equal", "abc", "abc"],
+  ["unequal, same length", "abc", "abd"],
+  ["different lengths", "abc", "abcd"],
+  ["empty", "", ""],
+  ["empty and not", "", "a"],
+  ["equal, 32 bytes", "a".repeat(32), "a".repeat(32)],
+  ["equal, 33 bytes", "a".repeat(33), "a".repeat(33)],
+  ["unequal past 32 bytes", "a".repeat(33), "a".repeat(32) + "b"],
+  ["32 and 33 bytes", "a".repeat(32), "a".repeat(33)],
+  ["unequal in the first byte, 64", "b" + "a".repeat(63), "a".repeat(64)],
+];
+
+describe("== and != on bytes and string compare contents", () => {
+  for (const level of levels) {
+    describe(`at O${level}`, () => {
+      for (const [name, a, b] of cases) {
+        it(name, async () => {
+          const result = await executeProgram(source, {
+            calldata: calldata(a, b),
+            optimizationLevel: level,
+          });
+          expect(result.callSuccess).toBe(true);
+          const equal = BigInt(a === b);
+          expect(await result.getStorage(0n)).toBe(equal);
+          expect(await result.getStorage(1n)).toBe(1n - equal);
+          expect(await result.getStorage(2n)).toBe(equal);
+          expect(await result.getStorage(3n)).toBe(1n - equal);
+          expect(await result.getStorage(4n)).toBe(BigInt(a === "abc"));
+          expect(await result.getStorage(6n)).toBe(equal);
+        });
+      }
+    });
+  }
+});
+
+/** A hex literal longer than 32 bytes, as dynamic `bytes` */
+const long = "11".repeat(32) + "22";
+
+/** A string literal that starts as a hex literal does */
+const hexLike = "0x12";
+
+/**
+ * Literals compared with copies from the calldata, and hashed: the
+ * same at every level, whether or not the optimizer folds them
+ */
+const literals = `name Literals;
+storage {
+  [0] longEq: bool; [1] longHash: bytes32;
+  [2] textEq: bool; [3] textHash: bytes32; [4] castHash: bytes32;
+}
+code {
+  let la = msg.data[0:32] as bytes32 as uint256;
+  let lb = msg.data[32:64] as bytes32 as uint256;
+  let a: bytes = msg.data[64:64 + la];
+  let b: bytes = msg.data[64 + la:64 + la + lb];
+  longEq = a == 0x${long};
+  longHash = keccak256(0x${long});
+  textEq = (b as string) == "${hexLike}";
+  textHash = keccak256("${hexLike}");
+  castHash = keccak256("${hexLike}" as bytes);
+}`;
+
+describe("bytes and string literals", () => {
+  const keccak = (data: Uint8Array) =>
+    BigInt("0x" + Buffer.from(keccak256(data)).toString("hex"));
+
+  for (const level of levels) {
+    it(`compare and hash by their bytes at O${level}`, async () => {
+      const a = Buffer.from(long, "hex");
+      const b = Buffer.from(hexLike);
+      const result = await executeProgram(literals, {
+        calldata:
+          word(a.length) +
+          word(b.length) +
+          a.toString("hex") +
+          b.toString("hex"),
+        optimizationLevel: level,
+      });
+      expect(result.callSuccess).toBe(true);
+      expect([
+        await result.getStorage(0n),
+        await result.getStorage(1n),
+        await result.getStorage(2n),
+        await result.getStorage(3n),
+        await result.getStorage(4n),
+      ]).toEqual([1n, keccak(a), 1n, keccak(b), keccak(b)]);
+    });
+  }
+});
diff --git a/packages/bugc/src/evmgen/generation/instructions/const.ts b/packages/bugc/src/evmgen/generation/instructions/const.ts
index 95821deb7..1828f5027 100644
--- a/packages/bugc/src/evmgen/generation/instructions/const.ts
+++ b/packages/bugc/src/evmgen/generation/instructions/const.ts
@@ -1,4 +1,4 @@
-import type * as Ir from "#ir";
+import * as Ir from "#ir";
 import type { Stack } from "#evm";
 
 import {
@@ -42,27 +42,8 @@ export function generateConst<S extends Stack>(
 
   // References need memory allocation for the data they point to
   if (inst.type.kind === "ref" && inst.type.location === "memory") {
-    let bytes: Uint8Array;
-    let byteLength: bigint;
-
-    // For memory references, we need to check the origin type to understand what data to store
-    // For now, we'll handle hex strings and regular strings
-    if (typeof inst.value === "string" && inst.value.startsWith("0x")) {
-      // Dynamic bytes from hex string - decode the hex
-      const hexStr = inst.value.slice(2); // Remove 0x prefix
-      const hexBytes = [];
-      for (let i = 0; i < hexStr.length; i += 2) {
-        hexBytes.push(parseInt(hexStr.substr(i, 2), 16));
-      }
-      bytes = new Uint8Array(hexBytes);
-      byteLength = BigInt(bytes.length);
-    } else {
-      // String or non-hex bytes - use UTF-8 encoding
-      const strValue = String(inst.value);
-      const encoder = new TextEncoder();
-      bytes = encoder.encode(strValue);
-      byteLength = BigInt(bytes.length);
-    }
+    const bytes = Ir.Utils.constantBytes(String(inst.value), inst.type);
+    const byteLength = BigInt(bytes.length);
 
     // Calculate memory needed: 32 bytes for length + actual data (padded to 32-byte words)
     const dataWords = (byteLength + 31n) / 32n;
diff --git a/packages/bugc/src/ir/utils/constant.ts b/packages/bugc/src/ir/utils/constant.ts
new file mode 100644
index 000000000..2df501f38
--- /dev/null
+++ b/packages/bugc/src/ir/utils/constant.ts
@@ -0,0 +1,27 @@
+import { Type as BugType } from "#types";
+
+import { Type } from "../spec/type.js";
+
+/**
+ * The bytes of a string constant of dynamic `bytes` or a `string`: a
+ * `string` is its text, in UTF-8, and `bytes` (from a hex literal too
+ * long for a word) are the bytes the "0x…" hex names. Code generation
+ * stores these bytes, and constant folding hashes them.
+ */
+export function constantBytes(value: string, type: Type): Uint8Array {
+  const isString =
+    Type.isRef(type) &&
+    type.origin !== "synthetic" &&
+    BugType.isElementary(type.origin) &&
+    BugType.Elementary.isString(type.origin);
+  if (isString || !value.startsWith("0x")) {
+    return new TextEncoder().encode(value);
+  }
+
+  const hex = value.slice(2);
+  const bytes = new Uint8Array(Math.ceil(hex.length / 2));
+  for (let i = 0; i < bytes.length; i++) {
+    bytes[i] = parseInt(hex.slice(2 * i, 2 * i + 2), 16);
+  }
+  return bytes;
+}
diff --git a/packages/bugc/src/ir/utils/index.ts b/packages/bugc/src/ir/utils/index.ts
index 9a9db3395..13c031aeb 100644
--- a/packages/bugc/src/ir/utils/index.ts
+++ b/packages/bugc/src/ir/utils/index.ts
@@ -3,3 +3,4 @@ export * from "./cast.js";
 export * from "./def-sites.js";
 export * from "./debug-info.js";
 export * from "./signed.js";
+export * from "./constant.js";
diff --git a/packages/bugc/src/irgen/generate/expressions/operator.ts b/packages/bugc/src/irgen/generate/expressions/operator.ts
index 10545359b..886397ec2 100644
--- a/packages/bugc/src/irgen/generate/expressions/operator.ts
+++ b/packages/bugc/src/irgen/generate/expressions/operator.ts
@@ -4,6 +4,7 @@ import { Severity } from "#result";
 
 import { Error as IrgenError, assertExhausted } from "#irgen/errors";
 import { fromBugType } from "#irgen/type";
+import { Type } from "#types";
 
 import { Process } from "../process.js";
 import type { Context } from "./context.js";
@@ -137,19 +138,31 @@ const makeBuildBinaryOperator = (
     const resultType = fromBugType(nodeType);
 
     // Evaluate operands
-    const leftVal = yield* buildExpression(expr.operands[0], {
+    let leftVal = yield* buildExpression(expr.operands[0], {
       kind: "rvalue",
     });
-    const rightVal = yield* buildExpression(expr.operands[1], {
+    let rightVal = yield* buildExpression(expr.operands[1], {
       kind: "rvalue",
     });
 
-    // Generate temp for result
-    const tempId = yield* Process.Variables.newTemp();
-
     const op = mapBinaryOp(expr.operator);
     const operationDebug = yield* Process.Debug.forAstNode(expr);
 
+    // `==` and `!=` on dynamic bytes or strings compare their
+    // contents, by comparing the hashes of the data. The operands need
+    // no copy to memory (`emitInMemory`): the type checker rejects
+    // bytes and strings in calldata here, so both are in memory.
+    if (
+      (op === "eq" || op === "ne") &&
+      isDynamicBytes(yield* Process.Types.nodeType(expr.operands[0]))
+    ) {
+      leftVal = yield* emitHash(leftVal, operationDebug);
+      rightVal = yield* emitHash(rightVal, operationDebug);
+    }
+
+    // Generate temp for result
+    const tempId = yield* Process.Variables.newTemp();
+
     // Emit binary operation
     yield* Process.Instructions.emit({
       kind: "binary",
@@ -165,6 +178,29 @@ const makeBuildBinaryOperator = (
       : Ir.Value.temp(tempId, resultType);
   };
 
+/** Whether a type is dynamic `bytes` or a `string` */
+function isDynamicBytes(type: Type | undefined): boolean {
+  if (!type || !Type.isElementary(type)) return false;
+  return Type.Elementary.isBytes(type)
+    ? Type.Elementary.Bytes.isDynamic(type)
+    : Type.Elementary.isString(type);
+}
+
+/** keccak256 of the data of dynamic bytes or a string in memory */
+function* emitHash(
+  value: Ir.Value,
+  operationDebug: Ir.Instruction.Debug,
+): Process<Ir.Value> {
+  const dest = yield* Process.Variables.newTemp();
+  yield* Process.Instructions.emit({
+    kind: "hash",
+    values: [value],
+    dest,
+    operationDebug,
+  } as Ir.Instruction.Hash);
+  return Ir.Value.temp(dest, Ir.Type.Scalar.bytes32);
+}
+
 /**
  * Whether an operation's result can fall outside its type. A quotient
  * or remainder of unsigned values is no greater than the dividend, and
diff --git a/packages/bugc/src/optimizer/steps/constant-folding.ts b/packages/bugc/src/optimizer/steps/constant-folding.ts
index 28e617cc9..5d0d09180 100644
--- a/packages/bugc/src/optimizer/steps/constant-folding.ts
+++ b/packages/bugc/src/optimizer/steps/constant-folding.ts
@@ -321,7 +321,7 @@ export class ConstantFoldingStep extends BaseOptimizationStep {
     );
 
     if (values.length === 1 && typeof values[0] === "string") {
-      return new TextEncoder().encode(values[0]);
+      return Ir.Utils.constantBytes(values[0], inst.values[0].type);
     }
 
     const words: Uint8Array[] = [];
diff --git a/packages/bugc/src/typechecker/checker.test.ts b/packages/bugc/src/typechecker/checker.test.ts
index 55dcd5707..92ad77a6b 100644
--- a/packages/bugc/src/typechecker/checker.test.ts
+++ b/packages/bugc/src/typechecker/checker.test.ts
@@ -148,6 +148,52 @@ describe("checkProgram", () => {
       expect(Result.hasMessages(result)).toBe(false);
     });
 
+    it("should type check == and != on bytes and strings", () => {
+      const result = check(`
+        name Test;
+        storage { [0] s: string; }
+        code {
+          let a: bytes = msg.data[0:4];
+          let b: bytes = msg.data[4:8];
+          let x = a as string;
+          let b1 = a == b;
+          let b2 = a != b;
+          let b3 = x == "abc";
+          let b4 = s != x;
+          let b5 = msg.data[0:4] as bytes4 == 0x12345678 as bytes4;
+          let c: bytes calldata = msg.data[0:4];
+          let b6 = (c as bytes) == a;
+        }
+      `);
+
+      expect(result.success).toBe(true);
+      expect(Result.hasMessages(result)).toBe(false);
+    });
+
+    it("should reject == between bytes, strings and bytesN", () => {
+      const result = check(`
+        name Test;
+        storage {}
+        code {
+          let a: bytes = msg.data[0:4];
+          let x = a as string;
+          let b1 = a == x;
+          let b2 = a == (msg.data[0:4] as bytes4);
+        }
+      `);
+
+      expect(result.success).toBe(false);
+      expect(Result.countErrors(result)).toBe(2);
+      expect(result).toHaveMessage({
+        severity: Severity.Error,
+        message: "Cannot compare bytes with string",
+      });
+      expect(result).toHaveMessage({
+        severity: Severity.Error,
+        message: "Cannot compare bytes with bytes4",
+      });
+    });
+
     it("should type check logical operators", () => {
       const result = check(`
         name Test;
diff --git a/packages/web/docs/implementation-guides/compiler/case-study-bug.mdx b/packages/web/docs/implementation-guides/compiler/case-study-bug.mdx
index 7cdd86d6a..c1bb09696 100644
--- a/packages/web/docs/implementation-guides/compiler/case-study-bug.mdx
+++ b/packages/web/docs/implementation-guides/compiler/case-study-bug.mdx
@@ -106,6 +106,24 @@ so `x + 1` with `x: uint8` is a `uint8` and wraps at 8 bits.
 -(-128 as int8);                 // -128
 ```
 
+### Equality
+
+`==` and `!=` on dynamic `bytes` or on `string` compare the contents,
+not the place in memory: the compiler compares the `keccak256` of the
+data of each operand. As for a mapping key, this assumes that two
+different contents do not have the same hash. Both operands must have
+the same type, in memory: bytes or a string in calldata do not
+compare, so cast or copy them to memory first. A `bytesN` is a value,
+and compares as a word.
+
+```bug
+let a: bytes = msg.data[0:4];
+let b: bytes = msg.data[0:4];    // a second copy
+a == b;                          // true
+(a as string) == "abcd";         // true if a is "abcd"
+msg.data[0:4] as bytes4 == 0xa9059cbb as bytes4;
+```
+
 ### Bounds
 
 An index out of bounds reverts, as in Solidity. `a[i]` on an array (in
@@ -174,7 +192,8 @@ motd = name;                     // stored as Solidity stores a string
 As a `string` in memory, a `string calldata` has no index or slice,
 and bytes do not assign to it without a cast. It too can only be the
 type of a `let`, a parameter or a cast. Bytes or a string in calldata
-cannot be compared with `==` or `!=`, or be a mapping key.
+cannot be compared with `==` or `!=`, or be a mapping key: cast or
+copy them to memory first, as in `(text as bytes) == other`.
 
 A parameter of bytes or a string in calldata takes its argument's word,
 with no copy, both in a real call and when the optimizer inlines the