21432Re: [PrimeNumbers] Strong pseudosermon (was 62-digit IsPrime)
- Apr 26, 2010Alan you will be happy to see this article on the BBC, UK :)
Web security attack 'makes silicon chips more reliable'
An attack on a widely used web security system could soon help make silicon chips more powerful and reliable.
Many websites use cryptographic systems to scramble key data, such as credit card numbers, when customers pay.
Scientists have found that by varying the voltage to key parts of a computer's processor, the ability to keep this data secret is compromised.
The researchers also discovered that a method that helps chips beat the attack could also make them more reliable.
Many modern security systems, such as the ones websites use to encrypt the credit card numbers of their customers, are based around a system known as public key cryptography.
This uses two keys, one public and one private, to scramble data. One of the most widely used implementations of this is known as RSA authentication.
"If data is locked with a public key, it can only be unlocked with the corresponding private key," said Professor Todd Austin, from the electrical engineering and computer science department at the University of Michigan who helped conduct the research.
Within 10 years a chip will have transistor failures every day
Professor Valeria Bertacco
"It's the kind of algorithm you use when you go to a website and you see the little padlock in the lower right hand corner to indicate a secure connection," he said.
The keys take the form of large numbers more than 1,000 digits long. Security is ensured because trying to guess a private number by trying all possible combinations would take longer than the age of the universe, using current computer technology.
Professor Austin, working with Andrea Pellegrini and Professor Valeria Bertacco, found a much quicker route to guessing the keys by varying the voltage to a processor.
"You need to be able to control the voltage to the power source to the device," said Professor Bertacco. "By putting the voltage just below where it should be means the device makes computational mistakes - it suffers temporary transistor failure."
The voltage was varied when a target machine was communicating with another machine via the web and the data flying between the two was encrypted using the public key system.
"It makes one mistake every now and again," she said. "But we need just a few mistakes."
During their test, the three researchers collected 8800 corrupted signatures in 10 hours and then analysed them using software that could call on 81 separate machines to boost its number crunching power.
The end result of the research was an attack method that could extract all the parts of a 1024 bit key in about 100 hours.
Initially, said Professor Bertacco, the work will lead to improvements in the way the public key security system works to make it less susceptible to such an attack. Future versions of the system will be "salted" with fake values to confuse any attempt to reconstruct a private key.
"It's part of the ongoing process of hardening RSA," said Professor Austin.
The implications of the research do not stop at security. It is also helping to produce error correction systems that spot when transistors fail and ensure that data is not corrupted as a result.
Professor Bertacco said the research would be useful when chips are made of even smaller components than those in use today. The widely-known Moore's Law predicts that the number of transistors on a given size of silicon wafer doubles roughly every 18 months.
Often that doubling is due to the transistors on the chip getting smaller. The transistors on Intel's most up to date desktop computers are about 32 nanometres in size.
Intel has said that it expects to soon start producing chips with components 22 and 16nm wide. A nanometre is a billionth of a metre.
However, as components get smaller they can get less reliable and need error checking and correction software to help cope with any errors that get introduced.
"Our mainstream research in this area is to make microchips operate correctly even in the face of transistor failure," she said. "Within 10 years a chip will have transistor failures every day. As transistors get smaller so they are more prone to failure."
God > infinity
From: Alan Eliasen <eliasen@...>
To: Ali Adams <alipoland@...>; Prime Number <email@example.com>
Sent: Sun, April 25, 2010 3:59:17 PM
Subject: [PrimeNumbers] Strong pseudosermon (was 62-digit IsPrime)
On 04/21/2010 09:01 PM, Ali Adams wrote:
> Greetings all,
> Can someone please help with validating if the following 62-digit
> number is a prime or not:
> 1110819595668080516 5653650502135350 6057690906175754 64617311659
> I am aware of IsProbablePrime but need a definite primality test.
(in another posting, asked)
> What about this 309-digit number?
>259336006801222696 0141827989906545 7702032918541745 1253947966996203 3747780569585929 9412847062270835 2120230964321433 3705453431960894 5822253023887835 9827583627468563 4622332103985890 9085250794700726 5127498998595582 3067653695374111 7527587085881465 1979141558307396 3161542913121427 0318567530145291 6463755740936626 397
(Which is not even probable-prime, as one single strong pseudoprime
I'm about to sermonize, so be prepared, my brothers and sisters.
Whenever someone says they "need" a definite primality test, they, by
their actions, usually prove that they don't understand that a
primality-proving test, by itself, is not an infallible proof of
primality (in the real world) and they rarely if ever do the correct
thing to decrease the probability of error, giving them a result that's
effectively no more certain than a working pseudoprime test (and, by
probabilities and failure modes that I'll cite here, probably quite a
bit weaker. And by "quite a bit" I mean 20 orders of magnitude, easy.)
Note: Keep in mind that all of this analysis applies in the *real
world*, not to some perfect, fictitious mathematical abstraction where
programmers don't make errors and hardware never fails and you can
ignore most algorithms and divide out big pesky constants because the
ideal theoretical asymptotic performance is always what you get when you
It is simple to implement and perform a strong pseudoprime test in
which the probability that a randomly-chosen composite number is
mistakenly stated to be prime is so low that it would never happen in
your lifetime. Not only that, you can trivially make the probability
of this mistake so low that you could test trillions of numbers every
second for the lifetime of the universe, and the probability of *any* of
those tests failing are still astronomically low, and that's even taking
the old ultra-conservative bound that as many as 1/4 of strong
pseudoprime tests can fail. See citations in the link below for numbers
about how conservative that estimate really is.
Many people have pointed out over the years that the probability that
your *hardware* or *software* fails (say, due to a high-energy particle
passing through your processor, or random thermal drift of electrons ,
or a bad transistor) during your primality test or primality "proof"
becomes rapidly much, much higher than the probability that, say, a
Rabin-Miller test incorrectly declares a composite to be prime.
Depending on the size of the number, this is true *even if you only do
one pass of a Rabin-Miller test*, and the probability of error in the
algorithm is far less than the probability of hardware failure, possibly
by hundreds or thousands of tens of thousands of *orders of magnitude*!
It may also be far, far more probable that the number you meant to
test or the reply (with certificate) was corrupted during communication
with someone else. For example, TCP only has a 16-bit checksum, and can
miss, say, two single-bit errors separated by 16 bits, for a possible
undetected error rate of 2^-16, or 1/65536 in a noisy channel. (Other
error-correcting algorithms apply to communication across the internet,
and most channels have fairly good s/n ratios so the end-to-end
probability of error is luckily usually lower than this.)
There's also the probability that Yahoo does something stupid with
the long line and cuts off a digit or something. (Some would say the
probability of Yahoo's software doing something stupid when adding their
cruft to a posting is 1, but that's being mean. Their software isn't
really that bad. The probability is actually 1-epsilon.)
To see approximate probabilities of the Rabin-Miller test failing
for certain number sizes, even with a *single* round of tests, see:
http://primes. utm.edu/notes/ prp_prob. html
Those probabilities of failure, especially for large numbers, are
mind-bogglingly small! Far smaller than the probability of some other
source of error.
Note that for the 309-digit number posted above, the probability of a
strong pseudoprime test mistakenly returning "prime" for a composite
number is less than 5.8*10^-29. Is that probability higher than other
probabilities we've already listed? (And note that since a strong
pseudoprime test easily catches that the number is actually composite.
No matter what base you choose. I'm sure that many here wouldn't
hesistate to give a cash reward for anyone who can find *any* prime base
smaller than the number for which a strong pseudoprime test fails. I
will initially offer all the money in my wallet. And I haven't even
tried to look for one. I'm that confident in the probabilities, or my
wallet is at its usual sad level.)
It's hard to approximate the probability that a particular piece of
software or hardware or communication will fail, but you can never
expect your primality "proof" to be any stronger than the most likely
error in the chain.
If you really *need* a prime number, you *must* then be sure to take
the certificate produced by one program's prime number proof *and verify
this certificate* , presumably with different software on a different
machine and hardware architecture! (I feel I should repeat this
sentence many times!) And then verify it again with another piece of
software! If you *don't* do this, the probability that the primality
"proof" is in error is approximately the probability of the thing most
likely to go wrong in the entire chain. (Again, maybe cosmic rays,
software bugs, Pentiums, race conditions in software, power glitches,
probability of human error, etc.) The primality certificate gives you a
way of verifying (without performing the whole proof again) that the
proof is indeed valid for that number.
However, there's nothing that prevents even the *verification* of the
certificate from having a similar unlikely failure! (Or similar
implementation bugs.) Assuming the implementations and hardware are
completely independent, the best you can do is to multiply the
probability of failure in both systems. Thus, if the probability of
failure in each system independently is 10^-32, then the probability
that *both* fail completely independently but in a way that gives the
same flawed result is at best 10^-64. Look at the URL cited above and
compare that to the probability of a probable-prime test failing. Is it
larger or smaller? If the probability is larger, the likelihood of your
"proof" being better than a pseudoprime test is purely illusory.
I'm not sure what the exact failure probability for a single
instruction is in modern hardware, but it's almost certainly *not*
better than 10^-32. (Multiply this by the number of instructions
required to perform the calculation. )
One of the best sources of information I've seen about actual failure
rates in installed hardware was done by the SETI@Home team, (which had
the world's largest distributed supercomputer at some points) which
cited detected error rates in returned work units (each work unit was
given to at least two people for validation, so errors could be
detected.) I don't remember the exact numbers, but the actual failure
rate appeared to be many, many orders of magnitude lower than 10^-32. I
don't seem to be able to find these stats at the moment. Does anyone
have the link?
If you haven't verified a primality certificate independently to
decrease your probability of hardware or software error, (and there is
*always* a non-zero probability of error on real physical hardware and
where humans are involved, even in a primality "proof") you're likely
not doing any better than a probabilistic prime test, in which the
unreliability of hardware, software, communications, and humans, rapidly
become the limiting factors.
You're also much more likely that some random mischiefmaker will say,
"sure it's prime" when it's not because they're annoyed with you for not
doing these simple tests yourself, and for refusing to take advice on
tools that will do the work for you. But if people didn't refuse to
listen, I wouldn't get a chance to gratuitously sermonize.
If you didn't verify the primality certificate as many ways as
possible, you clearly can't claim you understand that you "need" a prime
number, and don't really understand the probability of all the different
ways that failure could have occurred. Do you even know that the
primality certificate that was posted here was valid for the number you
gave, or did someone maybe miss the last digit digit during
cut-and-paste? Do you even know that the certificate was anything but a
cat walking across a keyboard? The probability that *you*
cut-and-pasted the number incorrectly is orders of magnitude higher than
probability of failure of a pseudoprime algorithm.
Homework: 1.) Can anyone else estimate probabilities of certain
types of errors? What do you think are the probabilities of various
failure modes in posting "prove this number prime for me" to a public
group? Do any of those probabilities exceed that of failure of a
probabilistic test? (Hint: The answer is yes.)
2.) Even if you get a provable prime number handed down impeachably
from the ghost of Fermat, what's the probability that when you're
*using* this prime number that something goes wrong in those
calculations? (Hint: Weakest link in the chain.)
 I intentionally state "randomly-chosen" because there are ways
to generate a very sparse set of numbers that can fool one of these
tests if you know in advance the bases it's going to test, which is why
most Rabin-Miller tests choose some bases randomly if there is the
potential for adversarial assault. See, for example:
François Arnault, Constructing Carmichael numbers which are strong
pseudoprimes to several bases, Journal of Symbolic Computation, v.20
n.2, p.151-161, Aug. 1995
 There's a certain nonzero probability that an electron will go
"the wrong way" and potentially tunnel "backwards" through a potential
barrier. As voltages used in processors get lower, and as gates get
smaller and the number of electrons required to switch a gate get fewer,
this becomes increasingly more probable. (I don't have my "Feynman
Lectures on Computation" at hand or I'd post some equations for this
probability. Very highly recommended!
http://tinyurl. com/9q4p8o )
 Multiple programmers creating similar software bugs is more
common than you might think. Sun's Java implementation had a famous bug
in their calculation of the Jacobi symbol, which caused
primality-testing routines to fail:
http://bugs. sun.com/bugdatab ase/view_ bug.do?bug_ id=4624738
When I implemented a function to calculate the Jacobi symbol in my
Frink programming language ( http://futureboy. us/frinkdocs/ ) I
initially created a similar bug, because my algorithm didn't work for
some negative numbers either. I didn't use Sun's algorithm, but I can
see how they went wrong. The algorithm often cited on the web, in
several number theory books, etc. needs preconditioning to handle
negative numbers correctly. Even more insanely, when evaluating
negative numbers, Java uses different sign conventions for the %
(modulus) operator (used for "int" values) and the BigInteger.mod( n)
function! It's easy to see how Sun even confuses their own programmers.
This bug in Sun's implementation caused some significant pain. Their
primality testing was originally a probabilistic Rabin-Miller test (with
probabilities that can easily be set that it won't return a wrong result
during the lifetime of the universe, and many users set them *way*
higher than that to be very safe) yet that bug caused failures much more
often, and introduced a method of failure that Rabin-Miller *can't*
produce, (Rabin-Miller can't ever declare a prime to be composite, but
it can, very rarely, declare a composite to be probably prime,) which is
why this failure mode was particularly unexpected to a lot of people,
and caused numbers above a certain size to fail mysteriously and
Note: When I was informed of this bug in my implementation of my
JacobiSymbol[ x] function, (it didn't affect my primality testing, which
has always worked properly,) I was ashamed that I wasn't able to release
a fix until I got home from work later that day. Sun, on the other
hand, took *3 years* to release a fix. (This bug was present in Java
versions 1.3 through 1.4.2.)
 Of course, some of their "failures" were due to intentional
attempts of people to corrupt their results, so exact probabilities of
failure are impossible to come by. There were interesting stories,
though, of *all* or *almost all* of the work units returned by several
well-meaning participants returning incorrect results. The SETI team
contacted these people and were able to verify that their computers'
floating-point units were indeed failing. (Sometimes subtly enough to
not make everything crash, but enough to make all extended calculations
wrong.) This is interesting and probably indicates that the frequency
of subtly or explicitly broken processors in the world is far less than
10^-9, setting a bound for what reliability we might expect for
 Don't laugh. This happened on this very list. A few years ago,
an intrepid researcher stated that they had been running a computer for
3+ years to find the factors of one of the RSA factoring challenge
numbers. Awesome persistence! And then one day it beeped! (I can't
imagine the excitement!) He announced to this list that he had
submitted his solution to RSA and was awaiting confirmation of the
factors, and he wasn't sure if the numbers were factors. I asked him
the obvious question, "did you multiply the two factors together and did
they come up with the original number?" The next day, with a leaden
heart, he responded to me and indicated that he had accidentally cut off
a digit when pasting in the original RSA number to his factoring
program. It still hurts me to think about it.
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