Checksumming large swathes of prime numbers? (for verification) - primes

Question

Are there any clever algorithms for computing high-quality checksums on millions or billions of prime numbers? I.e. with maximum error-detection capability and perhaps segmentable?
Motivation:
Small primes - up to 64 bits in size - can be sieved on demand to the tune of millions per second, by using a small bitmap for sieving potential factors (up to 2^32-1) and a second bitmap for sieving the numbers in the target range.
Algorithm and implementation are reasonably simple and straightforward but the devil is in the details: values tend to push against - or exceed - the limits of builtin integral types everywhere, boundary cases abound (so to speak) and even differences in floating point strictness can cause breakage if programming is not suitably defensive. Not to mention the mayhem that an optimising compiler can wreak, even on already-compiled, already-tested code in a static lib (if link-time code generation is used). Not to mention that faster algorithms tend to be a lot more complicated and thus even more brittle.
This has two consequences: test results are basically meaningless unless the tests are performed using the final executable image, and it becomes highly desirable to verify proper operation at runtime, during normal use.
Checking against pre-computed values would give the highest degree of confidence but the required files are big and clunky. A text file with 10 million primes has on the order of 100 MB uncompressed and more than 10 MB compressed; storing byte-encoded differences requires one byte per prime and entropy coding can at best reduce the size to half (5 MB for 10 million primes). Hence even a file that covers only the small factors up to 2^32 would weigh in at about 100 MB, and the complexity of the decoder would exceed that of the windowed sieve itself.
This means that checking against files is not feasible except as a final release check for a newly-built executable. Not to mention that the trustworthy files are not easy to come by. The Prime Pages offer files for the first 50 million primes, and even the amazing primos.mat.br goes only up to 1,000,000,000,000. This is unfortunate since many of the boundary cases (== need for testing) occur between 2^62 and 2^64-1.
This leaves checksumming. That way the space requirements would be marginal, and only proportional to the number of test cases. I don't want to require that a decent checksum like MD5 or SHA-256 be available, and with the target numbers all being prime it should be possible to generate a high-quality, high-resolution checksum with some simple ops on the numbers themselves.
This is what I've come up with so far. The raw digest consists of four 64-bit numbers; at the end it can be folded down to the desired size.
for (unsigned i = 0; i < ELEMENTS(primes); ++i)
{
digest[0] *= primes[i]; // running product (must be initialised to 1)
digest[1] += digest[0]; // sum of sequence of running products
digest[2] += primes[i]; // running sum
digest[3] += digest[2] * primes[i]; // Hornerish sum
}
At two (non-dependent) muls per prime the speed is decent enough, and except for the simple sum each of the components has always uncovered all errors I tried to sneak past the digest. However, I'm not a mathematician, and empirical testing is not a guarantee of efficacy.
Are there some mathematical properties that can be exploited to design - rather than 'cook' as I did - a sensible, reliable checksum?
Is it possible to design the checksum in a way that makes it steppable, in the sense that subranges can be processed separately and then the results combined with a bit of arithmetic to give the same result as if the whole range had been checksummed in one go? Same thing as all advanced CRC implementations tend to have nowadays, to enable parallel processing.
EDIT The rationale for the current scheme is this: the count, the sum and the product do not depend on the order in which primes are added to the digest; they can be computed on separate blocks and then combined. The checksum does depend on the order; that's its raison d'être. However, it would be nice if the two checksums of two consecutive blocks could be combined somehow to give the checksum of the combined block.
The count and the sum can sometimes be verified against external sources, like certain sequences on oeis.org, or against sources like the batches of 10 million primes at primos.mat.br (the index gives first and last prime, the number == 10 million is implied). No such luck for product and checksum, though.
Before I throw major time and computing horsepower at the computation and verification of digests covering the whole range of small factors up to 2^64 I'd like to hear what the experts think about this...
The scheme I'm currently test-driving in 32-bit and 64-bit variants looks like this:
template<typename word_t>
struct digest_t
{
word_t count;
word_t sum;
word_t product;
word_t checksum;
// ...
void add_prime (word_t n)
{
count += 1;
sum += n;
product *= n;
checksum += n * sum + product;
}
};
This has the advantage that the 32-bit digest components are equal to the lower halves of the corresponding 64-bit values, meaning only 64-bit digests need to be computed stored even if fast 32-bit verification is desired. A 32-bit version of the digest can be found in this simple sieve test program # pastebin, for hands-on experimentation. The full Monty in a revised, templated version can be found in a newer paste for a sieve that works up to 2^64-1.

Answer 1

I've done a good bit of work parallelizing operations on Cell architectures. This has a similar feel.
In this case, I would use a hash function that's fast and possibly incremental (e.g. xxHash or MurmurHash3) and a hash list (which is a less flexible specialization of a Merkle Tree).
These hashes are extremely fast. It's going to be surprisingly hard to get better with some simple set of operations. The hash list affords parallelism -- different blocks of the list can be handled by different threads, and then you hash the hashes. You could also use a Merkle Tree, but I suspect that'd just be more complex without much benefit.
Virtually divide your range into aligned blocks -- we'll call these microblocks. (e.g. a microblock is a range such as [n<<15, (n+1)<<15) )
To handle a microblock, compute what you need to compute, add it to a buffer, hash the buffer. (An incremental hash function will afford a smaller buffer. The buffer doesn't have to be filled with the same length of data every time.)
Each microblock hash will be placed in a circular buffer.
Divide the circular buffer into hashable blocks ("macroblocks"). Incrementally hash these macroblocks in the proper order as they become available or if there's no more microblocks left.
The resulting hash is the one you want.
Some additional notes:
I recommend a design where threads reserve a range of pending microblocks that the circular buffer has space for, process them, dump the values in the circular buffer, and repeat.
This has the added benefit that you can decide how many threads you want to use on the fly. e.g. when requesting a new range of microblocks, each thread could detect if there's too many/little threads running and adjust.
I personally would have the thread adding the last microblock hash to a macroblock clean up that macroblock. Less parameters to tune this way.
Maintaining a circular buffer isn't as hard as it sounds -- the lowest order macroblock still unhandled defines what portion of the "macroblock space" the circular buffer represents. All you need is a simple counter that increments when appropriate to express this.
Another benefit is that since the threads go through a reserve/work/reserve/work cycle on a regular basis, a thread that is unexpectedly slow won't hinder the running time nearly as badly.
If you're looking to make something less robust but easier, you could forgo a good bit of the work by using a "striped" pattern -- decide on the max number of threads (N), and have each thread handle every N-th microblock (offset by its thread "ID") and hash the resulting macroblocks per thread instead. Then at the end, hash the macroblock hashes from the N threads. If you have less than N threads, you can divide the work up amongst the number of threads you do want. (e.g. 64 max threads, but three real threads, thread 0 handles 21 virtual threads, thread 1 handles 21 virtual threads, and thread 2 handles 22 virtual threads -- not ideal, but not terrible) This is essentially a shallow Merkel tree instead of a hash list.

Answer 2

Kaganar's excellent answer demonstrates how to make things work even if the digests for adjacent blocks cannot be combined mathematically to give the same result as if the combined block had been digested instead.
The only drawback of his solution is that the resulting block structure is by necessity rather rigid, rather like PKI with its official all-encompassing hierarchy of certifications vs. 'guerrilla style' PGP whose web of trust covers only the few subjects who are of interest. In other words, it requires devising a global addressing structure/hierarchy.
This is the digest in its current form; the change is that the order-dependent part has been simplified to its essential minimum:
void add_prime (word_t n)
{
count += 1;
sum += n;
product *= n;
checksum += n * count;
}
Here are the lessons learnt from practical work with that digest:
count, sum and product (i.e. partial primorial modulo word size) turned out to be exceedingly useful because of the fact that they relate to things also found elsewhere in the world, like certain lists at OEIS
count and sum were very useful because the first tends to be naturally available when manipulating (generating, using, comparing) batches of primes, and the sum is easily computed on the fly with zero effort; this allows partial verification against existing results without going the whole hog of instantiating and updating a digest, and without the overhead of two - comparatively slow - multiplications
count is also exceedingly useful as it must by necessity be part of any indexing superstructure built on systems of digests, and conversely it can guide the search straight to the block (range) containing the nth prime, or to the blocks overlapped by the nth through (n+k)th primes
the order dependency of the fourth component (checksum) turned out be less of a hindrance than anticipated, since small primes tend to 'occur' (be generated or used) in order, in situations where verification might be desired
the order dependency of the checksum - and lack of combinability - made it perfectly useless outside of the specific block for which it was generated
fixed-size auxiliary program structures - like the ubiquitous small factor bitmaps - are best verified as raw memory for startup self-checks, instead of running a primes digest on them; this drastically reduces complexity and speeds things up by several orders of magnitude
For many practical purposes the order-dependent checksum could simply be dropped, leaving you with a three-component digest that is trivially combinable for adjacent ranges.
For verification of fixed ranges (like in self-tests) the checksum component is still useful. Any other kind of checksum - the moral equivalent of a CRC - would be just as useful for that and probably faster. It would be even more useful if an order-independent (combinable) way of supplementing the resolution of the first three components could be found. Extending the resolution beyond the first three components is most relevant for bigger computing efforts, like sieving, verifying and digesting trillions of primes for posterity.
One such candidate for an order-independent, combinable fourth component is the sum of squares.
Overall the digest turned out to be quite useful as is, despite the drawbacks concerning the checksum component. The best way of looking at the digest is probably as consisting of a 'characteristic' part (the first three components, combinable) and a checksum part that is only relevant for the specific block. The latter could just as well be replaced with a hash of any desired resolution. Kaganar's solution indicates how this checksum/hash can be integrated into a system that extends beyond a single block, despite its inherent non-combinability.
The summary of prime number sources seems to have fallen by the wayside, so here it is:
up to 1,000,000,000,000 available as files from sites like primos.mat.br
up to 2^64-10*2^64 in super-fast bulk via the primesieve.org console program (pipe)
up to 2^64-1 - and beyond - via the gp/PARI program (pipe, about 1 million primes/minute)

Answer 3

I'm answering this question again in a second answer since this is a very different and hopefully better tack:
It occurred to me that what you're doing is basically looking for a checksum, not over a list of primes, but over a range of a bitfield where a number is prime (bit is set to 1) or it's not (bit is set to 0). You're going to have a lot more 0's than 1's for any interesting range, so you hopefully only have to do an operation for the 1's.
Typically the problem with using a trivial in-any-order hash is that they handle multiplicity poorly and are oblivious to order. But you don't care about either of these problems -- every bit can only be set or unset once.
From that point of view, a bitwise-exclusive-or or addition should be just fine if combined with a good hashing function of the index of the bit -- that is, the found prime. (If your primes are 64-bit you could go with some of the functions here.)
So, for the ultimate simplicity that will give you the same value for any set of ranges of inputs, yes, stick to hashing and combining it with a simple operation like you are. But change to a traditional hash function which appears "random" given its input -- hash64shift on the linked page is likely what you're looking for. The probability of a meaningful collision is remote. Most hash functions stink, however -- make sure you pick one that is known to have good properties. (Avalanches well, etc.) Thomas Wang's are usually not so bad. (Bob Jenkin's are fantastic, but he sticks mostly to 32 bit functions. Although his mix function on the linked page is very good, but probably overkill.)
Parallelizing the check is obviously trivial, the code size and effort is vastly reduced from my other answer, and there's much less synchronization and almost no buffering that needs to occur.