CBMC proofs for type-safety of reduce.c (#232)

* Add CBMC proof for `montgomery_reduce()`

Signed-off-by: Rod Chapman <[email protected]>

* Add proof of fqmul()

In particular, this proof shows that for any parameters a and b,
the pre-condition of montgomery_reduce() is satisfied.

Signed-off-by: Rod Chapman <[email protected]>

* Add CBMC proof for `barrett_reduce()`

Signed-off-by: Rod Chapman <[email protected]>

---------

Signed-off-by: Rod Chapman <[email protected]>

cbmc/proofs/barrett_reduce/Makefile

@@ -0,0 +1,54 @@
+# SPDX-License-Identifier: Apache-2.0
+
+include ../Makefile_params.common
+
+HARNESS_ENTRY = harness
+HARNESS_FILE = barrett_reduce_harness
+
+# This should be a unique identifier for this proof, and will appear on the
+# Litani dashboard. It can be human-readable and contain spaces if you wish.
+PROOF_UID = barrett_reduce
+
+DEFINES +=
+INCLUDES +=
+
+REMOVE_FUNCTION_BODY +=
+UNWINDSET +=
+
+PROOF_SOURCES += $(PROOFDIR)/$(HARNESS_FILE).c
+PROJECT_SOURCES += $(SRCDIR)/mlkem/reduce.c
+
+CHECK_FUNCTION_CONTRACTS=$(KYBER_NAMESPACE)barrett_reduce
+USE_FUNCTION_CONTRACTS=
+APPLY_LOOP_CONTRACTS=on
+USE_DYNAMIC_FRAMES=1
+
+# Disable any setting of EXTERNAL_SAT_SOLVER, and choose SMT backend instead
+EXTERNAL_SAT_SOLVER=
+CBMCFLAGS=--smt2
+
+FUNCTION_NAME = $(KYBER_NAMESPACE)barrett_reduce
+
+# If this proof is found to consume huge amounts of RAM, you can set the
+# EXPENSIVE variable. With new enough versions of the proof tools, this will
+# restrict the number of EXPENSIVE CBMC jobs running at once. See the
+# documentation in Makefile.common under the "Job Pools" heading for details.
+# EXPENSIVE = true
+
+# This function is large enough to need...
+CBMC_OBJECT_BITS = 8
+
+# If you require access to a file-local ("static") function or object to conduct
+# your proof, set the following (and do not include the original source file
+# ("mlkem/poly.c") in PROJECT_SOURCES).
+# REWRITTEN_SOURCES = $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i
+# include ../Makefile.common
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_SOURCE = $(SRCDIR)/mlkem/poly.c
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_FUNCTIONS = foo bar
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_OBJECTS = baz
+# Care is required with variables on the left-hand side: REWRITTEN_SOURCES must
+# be set before including Makefile.common, but any use of variables on the
+# left-hand side requires those variables to be defined. Hence, _SOURCE,
+# _FUNCTIONS, _OBJECTS is set after including Makefile.common.
+
+include ../Makefile.common

cbmc/proofs/barrett_reduce/barrett_reduce_harness.c

@@ -0,0 +1,29 @@
+// Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved.
+// SPDX-License-Identifier: MIT-0 AND Apache-2.0
+
+/*
+ * Insert copyright notice
+ */
+
+/**
+ * @file barrett_reduce_harness.c
+ * @brief Implements the proof harness for barrett_reduce function.
+ */
+#include "reduce.h"
+
+/*
+ * Insert project header files that
+ *   - include the declaration of the function
+ *   - include the types needed to declare function arguments
+ */
+
+/**
+ * @brief Starting point for formal analysis
+ *
+ */
+void harness(void) {
+  int16_t a;
+  int16_t r;
+
+  r = barrett_reduce(a);
+}

cbmc/proofs/barrett_reduce/cbmc-proof.txt

@@ -0,0 +1,3 @@
+# SPDX-License-Identifier: Apache-2.0
+
+# This file marks this directory as containing a CBMC proof.

cbmc/proofs/fqmul/Makefile

@@ -0,0 +1,54 @@
+# SPDX-License-Identifier: Apache-2.0
+
+include ../Makefile_params.common
+
+HARNESS_ENTRY = harness
+HARNESS_FILE = fqmul_harness
+
+# This should be a unique identifier for this proof, and will appear on the
+# Litani dashboard. It can be human-readable and contain spaces if you wish.
+PROOF_UID = fqmul
+
+DEFINES +=
+INCLUDES +=
+
+REMOVE_FUNCTION_BODY +=
+UNWINDSET +=
+
+PROOF_SOURCES += $(PROOFDIR)/$(HARNESS_FILE).c
+PROJECT_SOURCES += $(SRCDIR)/mlkem/reduce.c
+
+CHECK_FUNCTION_CONTRACTS=fqmul
+USE_FUNCTION_CONTRACTS=$(KYBER_NAMESPACE)montgomery_reduce
+APPLY_LOOP_CONTRACTS=on
+USE_DYNAMIC_FRAMES=1
+
+# Disable any setting of EXTERNAL_SAT_SOLVER, and choose SMT backend instead
+EXTERNAL_SAT_SOLVER=
+CBMCFLAGS=--smt2
+
+FUNCTION_NAME = fqmul
+
+# If this proof is found to consume huge amounts of RAM, you can set the
+# EXPENSIVE variable. With new enough versions of the proof tools, this will
+# restrict the number of EXPENSIVE CBMC jobs running at once. See the
+# documentation in Makefile.common under the "Job Pools" heading for details.
+# EXPENSIVE = true
+
+# This function is large enough to need...
+CBMC_OBJECT_BITS = 8
+
+# If you require access to a file-local ("static") function or object to conduct
+# your proof, set the following (and do not include the original source file
+# ("mlkem/poly.c") in PROJECT_SOURCES).
+# REWRITTEN_SOURCES = $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i
+# include ../Makefile.common
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_SOURCE = $(SRCDIR)/mlkem/poly.c
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_FUNCTIONS = foo bar
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_OBJECTS = baz
+# Care is required with variables on the left-hand side: REWRITTEN_SOURCES must
+# be set before including Makefile.common, but any use of variables on the
+# left-hand side requires those variables to be defined. Hence, _SOURCE,
+# _FUNCTIONS, _OBJECTS is set after including Makefile.common.
+
+include ../Makefile.common

cbmc/proofs/fqmul/cbmc-proof.txt

@@ -0,0 +1,3 @@
+# SPDX-License-Identifier: Apache-2.0
+
+# This file marks this directory as containing a CBMC proof.

cbmc/proofs/fqmul/fqmul_harness.c

@@ -0,0 +1,28 @@
+// Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved.
+// SPDX-License-Identifier: MIT-0 AND Apache-2.0
+
+/*
+ * Insert copyright notice
+ */
+
+/**
+ * @file montgomery_reduce_harness.c
+ * @brief Implements the proof harness for montgomery_reduce function.
+ */
+#include "reduce.h"
+
+/*
+ * Insert project header files that
+ *   - include the declaration of the function
+ *   - include the types needed to declare function arguments
+ */
+
+/**
+ * @brief Starting point for formal analysis
+ *
+ */
+void harness(void) {
+  int16_t a, b, r;
+
+  r = fqmul(a, b);
+}

cbmc/proofs/montgomery_reduce/Makefile

@@ -0,0 +1,54 @@
+# SPDX-License-Identifier: Apache-2.0
+
+include ../Makefile_params.common
+
+HARNESS_ENTRY = harness
+HARNESS_FILE = montgomery_reduce_harness
+
+# This should be a unique identifier for this proof, and will appear on the
+# Litani dashboard. It can be human-readable and contain spaces if you wish.
+PROOF_UID = montgomery_reduce
+
+DEFINES +=
+INCLUDES +=
+
+REMOVE_FUNCTION_BODY +=
+UNWINDSET +=
+
+PROOF_SOURCES += $(PROOFDIR)/$(HARNESS_FILE).c
+PROJECT_SOURCES += $(SRCDIR)/mlkem/reduce.c
+
+CHECK_FUNCTION_CONTRACTS=$(KYBER_NAMESPACE)montgomery_reduce
+USE_FUNCTION_CONTRACTS=
+APPLY_LOOP_CONTRACTS=on
+USE_DYNAMIC_FRAMES=1
+
+# Disable any setting of EXTERNAL_SAT_SOLVER, and choose SMT backend instead
+EXTERNAL_SAT_SOLVER=
+CBMCFLAGS=--smt2
+
+FUNCTION_NAME = $(KYBER_NAMESPACE)montgomery_reduce
+
+# If this proof is found to consume huge amounts of RAM, you can set the
+# EXPENSIVE variable. With new enough versions of the proof tools, this will
+# restrict the number of EXPENSIVE CBMC jobs running at once. See the
+# documentation in Makefile.common under the "Job Pools" heading for details.
+# EXPENSIVE = true
+
+# This function is large enough to need...
+CBMC_OBJECT_BITS = 8
+
+# If you require access to a file-local ("static") function or object to conduct
+# your proof, set the following (and do not include the original source file
+# ("mlkem/poly.c") in PROJECT_SOURCES).
+# REWRITTEN_SOURCES = $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i
+# include ../Makefile.common
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_SOURCE = $(SRCDIR)/mlkem/poly.c
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_FUNCTIONS = foo bar
+# $(PROOFDIR)/<__SOURCE_FILE_BASENAME__>.i_OBJECTS = baz
+# Care is required with variables on the left-hand side: REWRITTEN_SOURCES must
+# be set before including Makefile.common, but any use of variables on the
+# left-hand side requires those variables to be defined. Hence, _SOURCE,
+# _FUNCTIONS, _OBJECTS is set after including Makefile.common.
+
+include ../Makefile.common

cbmc/proofs/montgomery_reduce/cbmc-proof.txt

@@ -0,0 +1,3 @@
+# SPDX-License-Identifier: Apache-2.0
+
+# This file marks this directory as containing a CBMC proof.

cbmc/proofs/montgomery_reduce/montgomery_reduce_harness.c

@@ -0,0 +1,29 @@
+// Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved.
+// SPDX-License-Identifier: MIT-0 AND Apache-2.0
+
+/*
+ * Insert copyright notice
+ */
+
+/**
+ * @file montgomery_reduce_harness.c
+ * @brief Implements the proof harness for montgomery_reduce function.
+ */
+#include "reduce.h"
+
+/*
+ * Insert project header files that
+ *   - include the declaration of the function
+ *   - include the types needed to declare function arguments
+ */
+
+/**
+ * @brief Starting point for formal analysis
+ *
+ */
+void harness(void) {
+  int32_t a;
+  int16_t r;
+
+  r = montgomery_reduce(a);
+}

mlkem/reduce.c

@@ -3,6 +3,35 @@
 #include <stdint.h>
 #include "params.h"
 
+// QINV == -3327 converted to uint16_t == -3327 + 65536 == 62209
+static const uint32_t QINV = 62209;  // q^-1 mod 2^16
+
+/*************************************************
+ * Name:        cast_uint16_to_int16
+ *
+ * Description: Cast uint16 value to int16
+ *
+ * Returns:
+ *   input x in     0 .. 32767: returns value unchanged
+ *   input x in 32768 .. 65535: returns (x - 65536)
+ **************************************************/
+#ifdef CBMC
+#pragma CPROVER check push
+#pragma CPROVER check disable "conversion"
+#endif
+static inline int16_t cast_uint16_to_int16(uint16_t x) {
+  // PORTABILITY: This relies on uint16_t -> int16_t
+  // being implemented as the inverse of int16_t -> uint16_t,
+  // which is implementation-defined (C99 6.3.1.3 (3))
+  //
+  // CBMC (correctly) fails to prove this conversion is OK,
+  // so we have to suppress that check here
+  return (int16_t)x;
+}
+#ifdef CBMC
+#pragma CPROVER check pop
+#endif
+
 /*************************************************
  * Name:        montgomery_reduce
  *
@@ -41,36 +70,50 @@ int16_t montgomery_reduce(int32_t a) {
   // Replace C -> C * q in the above and estimate
   // q / 2^17 < 0.0254.
 
-  uint16_t u;
-  int16_t t;
-  // Compute a*q^{-1} mod 2^16 in unsigned representatives
-  u = (uint16_t)a * QINV;
+  //  Compute a*q^{-1} mod 2^16 in unsigned representatives
+  const uint16_t a_reduced = a & UINT16_MAX;
+  const uint16_t a_inverted = (a_reduced * QINV) & UINT16_MAX;
+
   // Lift to signed canonical representative mod 2^16.
-  // PORTABILITY: This relies on uint16_t -> int16_t
-  // being implemented as the inverse of int16_t -> uint16_t,
-  // which is not mandated by the standard.
-  t = (int16_t)u;
-  // By construction, the LHS is divisible by 2^16
-  t = (a - (int32_t)t * KYBER_Q) >> 16;
-  return t;
+  const int16_t t = cast_uint16_to_int16(a_inverted);
+
+  int32_t r = a - ((int32_t)t * KYBER_Q);
+
+  // PORTABILITY: Right-shift on a signed integer is, strictly-speaking,
+  // implementation-defined for negative left argument. Here,
+  // we assume it's sign-preserving "arithmetic" shift right. (C99 6.5.7 (5))
+  r = r >> 16;
+
+  return (int16_t)r;
 }
 
+// To divide by KYBER_Q using Barrett multiplication, the "magic number"
+// multiplier is round_to_nearest(2**26/KYBER_Q)
+#define BPOWER 26
+static const int32_t barrett_multiplier =
+    ((1 << BPOWER) + KYBER_Q / 2) / KYBER_Q;
+
 /*************************************************
  * Name:        barrett_reduce
  *
  * Description: Barrett reduction; given a 16-bit integer a, computes
  *              centered representative congruent to a mod q in
- *{-(q-1)/2,...,(q-1)/2}
+ *              {-(q-1)/2,...,(q-1)/2}
  *
  * Arguments:   - int16_t a: input integer to be reduced
  *
  * Returns:     integer in {-(q-1)/2,...,(q-1)/2} congruent to a modulo q.
  **************************************************/
 int16_t barrett_reduce(int16_t a) {
-  int16_t t;
-  const int16_t v = ((1 << 26) + KYBER_Q / 2) / KYBER_Q;
+  // Compute round_to_nearest(a/KYBER_Q) using the multiplier
+  // above and shift by BPOWER places.
+  //
+  // PORTABILITY: Right-shift on a signed integer is, strictly-speaking,
+  // implementation-defined for negative left argument. Here,
+  // we assume it's sign-preserving "arithmetic" shift right. (C99 6.5.7 (5))
+  const int32_t t = (barrett_multiplier * a + (1 << (BPOWER - 1))) >> BPOWER;
 
-  t = ((int32_t)v * a + (1 << 25)) >> 26;
-  t *= KYBER_Q;
-  return a - t;
+  // t is in -10 .. +10, so we need 32-bit math to
+  // evaluate t * KYBER_Q and the subsequent subtraction
+  return (int16_t)(a - t * KYBER_Q);
 }