random: Account for entropy loss due to overwrites

From: "H. Peter Anvin" <[email protected]>

When we write entropy into a non-empty pool, we currently don't account at all for the fact that we will probabilistically overwrite some of the entropy in that pool.

Quoted 5 lines · H. Peter Anvin

From: "H. Peter Anvin" <[email protected]>

When we write entropy into a non-empty pool, we currently don't account at all for the fact that we will probabilistically overwrite some of the entropy in that pool.

Technically, no, nothing is overwritten. The key fact is that the mixing function is -reversible-. Thus, even if you mix in known data, you can't learn anything about the state and thus can't destroy any of the existing entropy.

But you are correct, mixing new actual entropy is not purely additive (with saturation). For that to happen, we'd need an input mixing function with perfect maximal cascading. Instead we effectively cascade across somewhere between 6 and 64 bits. So the truth lies somewhere between linear and your exponential estimate (which would be the case for mixing a single bit into the pool with XOR), but much closer to linear due to combinatoric expansion.

On the other hand, I don't think this sort of thing matters at all. There is so much more fundamentally wrong with even trying to do entropy accounting in the first place that these sorts of details don't even matter. Instead we should stop fooling ourselves and just drop the pretense of accounting entirely. Now that we've got a much richer set of inputs, I think the time is ripe... but of course, I'm no longer the maintainer.

From: "H. Peter Anvin" <[email protected]>

When we write entropy into a non-empty pool, we currently don't account at all for the fact that we will probabilistically overwrite some of the entropy in that pool. This means that unless the pool is fully empty, we are currently *guaranteed* to overestimate the amount of entropy in the pool!

Assuming Shannon entropy with zero correlations we end up with an exponentally decaying value of new entropy added:

	entropy <- entropy + (pool_size - entropy) *
		(1 - exp(-add_entropy/pool_size))

However, calculations involving fractional exponentials are not practical in the kernel, so apply a piecewise linearization:

For add_entropy <= pool_size then

(1 - exp(-add_entropy/pool_size)) >= (add_entropy/pool_size)*0.632...

... so we can approximate the exponential with add_entropy/(pool_size*2) and still be on the safe side by adding at most one pool_size at a time.

In order for the loop not to take arbitrary amounts of time if a bad ioctl is received, terminate if we are within one bit of full. This way the loop is guaranteed to terminate after no more than log2(poolsize) iterations, no matter what the input value is. The vast majority of the time the loop will be executed exactly once.

The piecewise linearization is very conservative, approaching 1/2 of the usable input value for small inputs, however, our entropy estimation is pretty weak at best, especially for small values; we have no handle on correlation; and the Shannon entropy measure (Rényi entropy of order 1) is not the correct one to use in the first place, but rather the correct entropy measure is the min-entropy, the Rényi entropy of infinite order.

As such, this conservatism seems more than justified. Note, however, that attempting to add one bit of entropy will never succeed; nor will two bits unless the pool is completely empty. These roundoff artifacts could be improved by using fixed-point arithmetic and adding some number of fractional entropy bits.

Signed-off-by H. Peter Anvin <[email protected]>

Cc DJ Johnston <[email protected]>

Cc <[email protected]>

---
  drivers/char/random.c | 104 ++++++++++++++++++++++++++++++++++----------------
  1 file changed, 72 insertions(+), 32 deletions(-)

drivers/char/random.c +72 -32

index b86eae9..5d870ad 100644
@@ -272,10 +272,12 @@
  /*
   * Configuration information
   */
-#define INPUT_POOL_WORDS 128
-#define OUTPUT_POOL_WORDS 32
-#define SEC_XFER_SIZE 512
-#define EXTRACT_SIZE 10
+#define INPUT_POOL_SHIFT	12
+#define INPUT_POOL_WORDS	(1 << (INPUT_POOL_SHIFT-5))
+#define OUTPUT_POOL_SHIFT	10
+#define OUTPUT_POOL_WORDS	(1 << (OUTPUT_POOL_SHIFT-5))
+#define SEC_XFER_SIZE		512
+#define EXTRACT_SIZE		10
 
  #define LONGS(x) (((x) + sizeof(unsigned long) - 1)/sizeof(unsigned long))
 

@@ -309,40 +311,41 @@ static DEFINE_PER_CPU(int, trickle_count);
   * scaled squared error sum) except for the last tap, which is 1 to
   * get the twisting happening as fast as possible.
   */
-static struct poolinfo {
+static const struct poolinfo {
+	int poolshift;		/* log2(POOLBITS) */
  	int poolwords;
  	int tap1, tap2, tap3, tap4, tap5;
  } poolinfo_table[] = {
-	/* x^128 + x^103 + x^76 + x^51 +x^25 + x + 1 -- 105 */
-	{ 128,	103,	76,	51,	25,	1 },
-	/* x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 -- 15 */
-	{ 32,	26,	20,	14,	7,	1 },
+	/*    x^128 + x^103 + x^76 + x^51 +x^25 + x + 1 -- 105 */
+	{ 12, 128,	103,	76,	51,	25,	1 },
+	/*    x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 -- 15 */
+	{ 10, 32,	26,	20,	14,	7,	1 },
  #if 0
-	/* x^2048 + x^1638 + x^1231 + x^819 + x^411 + x + 1  -- 115 */
-	{ 2048,	1638,	1231,	819,	411,	1 },
+	/*    x^2048 + x^1638 + x^1231 + x^819 + x^411 + x + 1  -- 115 */
+	{ 16, 2048,	1638,	1231,	819,	411,	1 },
 
-	/* x^1024 + x^817 + x^615 + x^412 + x^204 + x + 1 -- 290 */
-	{ 1024,	817,	615,	412,	204,	1 },
+	/*    x^1024 + x^817 + x^615 + x^412 + x^204 + x + 1 -- 290 */
+	{ 15, 1024,	817,	615,	412,	204,	1 },
 
-	/* x^1024 + x^819 + x^616 + x^410 + x^207 + x^2 + 1 -- 115 */
-	{ 1024,	819,	616,	410,	207,	2 },
+	/*    x^1024 + x^819 + x^616 + x^410 + x^207 + x^2 + 1 -- 115 */
+	{ 15, 1024,	819,	616,	410,	207,	2 },
 
-	/* x^512 + x^411 + x^308 + x^208 + x^104 + x + 1 -- 225 */
-	{ 512,	411,	308,	208,	104,	1 },
+	/*    x^512 + x^411 + x^308 + x^208 + x^104 + x + 1 -- 225 */
+	{ 14, 512,	411,	308,	208,	104,	1 },
 
-	/* x^512 + x^409 + x^307 + x^206 + x^102 + x^2 + 1 -- 95 */
-	{ 512,	409,	307,	206,	102,	2 },
-	/* x^512 + x^409 + x^309 + x^205 + x^103 + x^2 + 1 -- 95 */
-	{ 512,	409,	309,	205,	103,	2 },
+	/*    x^512 + x^409 + x^307 + x^206 + x^102 + x^2 + 1 -- 95 */
+	{ 14, 512,	409,	307,	206,	102,	2 },
+	/*    x^512 + x^409 + x^309 + x^205 + x^103 + x^2 + 1 -- 95 */
+	{ 14, 512,	409,	309,	205,	103,	2 },
 
-	/* x^256 + x^205 + x^155 + x^101 + x^52 + x + 1 -- 125 */
-	{ 256,	205,	155,	101,	52,	1 },
+	/*    x^256 + x^205 + x^155 + x^101 + x^52 + x + 1 -- 125 */
+	{ 13, 256,	205,	155,	101,	52,	1 },
 
-	/* x^128 + x^103 + x^78 + x^51 + x^27 + x^2 + 1 -- 70 */
-	{ 128,	103,	78,	51,	27,	2 },
+	/*    x^128 + x^103 + x^78 + x^51 + x^27 + x^2 + 1 -- 70 */
+	{ 12, 128,	103,	78,	51,	27,	2 },
 
-	/* x^64 + x^52 + x^39 + x^26 + x^14 + x + 1 -- 15 */
-	{ 64,	52,	39,	26,	14,	1 },
+	/*    x^64 + x^52 + x^39 + x^26 + x^14 + x + 1 -- 15 */
+	{ 11, 64,	52,	39,	26,	14,	1 },
  #endif
  };
 

@@ -424,7 +427,7 @@ module_param(debug, bool, 0644);
  struct entropy_store;
  struct entropy_store {
  	/* read-only data: */
-	struct poolinfo *poolinfo;
+	const struct poolinfo *poolinfo;
  	__u32 *pool;
  	const char *name;
  	struct entropy_store *pull;

@@ -585,11 +588,13 @@ static void fast_mix(struct fast_pool *f, const void *in, int nbytes)
  }
 
  /*
- * Credit (or debit) the entropy store with n bits of entropy
+ * Credit (or debit) the entropy store with n bits of entropy.
+ * The nbits value is given in units of 2^-16 bits, i.e. 0x10000 == 1 bit.
   */
  static void credit_entropy_bits(struct entropy_store *r, int nbits)
  {
  	int entropy_count, orig;
+	int pool_size = r->poolinfo->POOLBITS;
 
  	if (!nbits)
  		return;

@@ -597,13 +602,48 @@ static void credit_entropy_bits(struct entropy_store *r, int nbits)
  	DEBUG_ENT("added %d entropy credits to %s\n", nbits, r->name);
  retry:
  	entropy_count = orig = ACCESS_ONCE(r->entropy_count);
-	entropy_count += nbits;
+	if (nbits < 0) {
+		/* Debit. */
+		entropy_count += nbits;
+	} else {
+		/*
+		 * Credit: we have to account for the possibility of
+		 * overwriting already present entropy.  Even in the
+		 * ideal case of pure Shannon entropy, new contributions
+		 * approach the full value asymptotically:
+		 *
+		 * entropy <- entropy + (pool_size - entropy) *
+		 *	(1 - exp(-add_entropy/pool_size))
+		 *
+		 * For add_entropy <= pool_size then
+		 * (1 - exp(-add_entropy/pool_size)) >=
+		 *    (add_entropy/pool_size)*0.632...
+		 * so we can approximate the exponential with
+		 * add_entropy/(pool_size*2) and still be on the
+		 * safe side by adding at most one pool_size at a time.
+		 *
+		 * The use of pool_size-1 in the while statement is to
+		 * prevent rounding artifacts from making the loop
+		 * arbitrarily long; this limits the loop to poolshift
+		 * turns no matter how large nbits is.
+		 */
+		int pnbits = nbits;
+		int s      = r->poolinfo->poolshift + 1;
+
+		do {
+			int anbits = min(pnbits, pool_size);
+
+			entropy_count +=
+				((pool_size - entropy_count)*anbits) >> s;
+			pnbits -= anbits;
+		} while (unlikely(entropy_count < pool_size-1 && pnbits));
+	}
 
  	if (entropy_count < 0) {
  		DEBUG_ENT("negative entropy/overflow\n");
  		entropy_count = 0;
-	} else if (entropy_count > r->poolinfo->POOLBITS)
-		entropy_count = r->poolinfo->POOLBITS;
+	} else if (entropy_count > pool_size)
+		entropy_count = pool_size;
  	if (cmpxchg(&r->entropy_count, orig, entropy_count) != orig)
  		goto retry;
 

drivers/char/random.c static DEFINE_PER_CPU(int, trickle_count); · line 317

Quoted 5 lines

drivers/char/random.c +2 -1

-static struct poolinfo {
+static const struct poolinfo {
+	int poolshift;		/* log2(POOLBITS) */
 	int poolwords;
 	int tap1, tap2, tap3, tap4, tap5;

Quoted 5 lines · H. Peter Anvin

drivers/char/random.c static DEFINE_PER_CPU(int, trickle_count); · line 317

Quoted 5 lines

drivers/char/random.c +2 -1

-static struct poolinfo {
+static const struct poolinfo {
+	int poolshift;		/* log2(POOLBITS) */
 	int poolwords;
 	int tap1, tap2, tap3, tap4, tap5;

Poolshift is duplicated information; it's just log2(poolwords) + 5 (since POOLBITS is poolwords*32).

Granted you don't want to recalculate it every single time you need to use it, but perhaps it would be better to add poolshift to struct entropy_store, and set it in init_std_data()?

Or we could compute poolwords (and poolbits, and poolbytes) from it, since shifts generally are cheap. I don't strongly care, whatever your preference is.

... and it looks like include/linux/log2.h already has a definition for ilog2() which should definitely work for all versions of gcc, so we could do this instead:

#define shiftbits(w) (ilog2((w)) + 5)

	/* x^128 + x^103 + x^76 + x^51 +x^25 + x + 1 -- 105 */
	{ shiftbits(128), 103,	76,	51,	25,	1 },
	/* x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 -- 15 */
	{ shiftbits(32),  26,	20,	14,	7,	1 },