So imagine a blindfolded jeweler. OK. And they're
tasked with cutting this priceless, incredibly
fragile diamond. But to keep the gem secure,
it's been locked inside a thick, opaque titanium
box. Right. So they can't see it at all. Exactly.
The jeweler has to use these remote -controlled
mechanical arms to reach inside and cut the gem.
They can feel their way through the delicate
movements and they execute the work perfectly,
but they never actually see the jewel itself.
They just hand the lock box back to the owner
when they're finished. That is a really vivid
way to look at it. Yeah. And welcome to today's
deep dive. Our mission today is to explore a
concept that cryptographers consider like the
holy grail of their field. It's called homomorphic
encryption. The big one. The really big one.
And we're pulling from a comprehensive set of
source materials today, mostly this really extensive
Wikipedia overview on the subject, just to understand
how we can process and analyze data while it
remains entirely encrypted. Which is huge for,
you know, safely outsourcing our digital lives
to the cloud. Right. Because right now you have
to trust the cloud provider. But if this works,
maybe you don't and I want to make sure you the
listener have a really clear Jargon -free shortcut
to understanding this tech because it's probably
going to be protecting your personal data very
soon So, okay, let's unpack this How close is
that blindfolded jeweler analogy to the actual
mechanics of homomorphic encryption? I mean,
the analogy actually holds up incredibly well
to the underlying mechanics. The jeweler essentially
represents a computer server in the cloud, right?
OK. And the mechanical arms, those are the mathematical
operations being performed. And the diamond is
your sensitive data. But the critical part of
your analogy is that the jeweler never sees the
diamond. Because the box is locked. Exactly.
In conventional computing, if you want a server
to, say, search a database or organize files
or run analytics, you have to give that server
the decryption key. Right. The data has to be
translated from encrypted gibberish back into
plain text before the computer can even do anything
with it. Yeah, and the moment it's in plain text,
it's completely exposed. That creates a massive
opening for what we call a privilege escalation
attack. Which means if a hacker gets in, while
the data is being processed. They can just walk
away with the plain text. Yeah. But homomorphic
encryption, or HE, aims to close that window
of vulnerability permanently. The server performs
computations directly on the ciphertext, which
is the encrypted gibberish. So the output of
the computation is just... more encrypted gibberish.
Exactly. It's only when you download that result
back to your own device and apply your private
key that the gibberish suddenly resolves into
the correct unencrypted answer. That's wild.
And the source material gives a couple of really
practical examples of this in action, which I
think highlights why you should care about this.
Think about predictive analytics in health care.
That's a perfect example. A hospital has mountains
of patient data that could be used to train AI
models to predict disease outbreaks or patient
outcomes. But because of privacy laws, they obviously
can't just hand patient records over to a third
-party tech company. Right, high PK and all that.
Exactly. But with homomorphic encryption, the
hospital could basically lock the records in
that opaque box, send the boxes to the cloud,
and the cloud runs its machine learning models
without ever seeing the files. Then it just sends
back an encrypted prediction. The cloud provider
learns absolutely nothing about the patients.
You see similar applications in image processing,
too. Like, you could send an encrypted photograph
to a lookup service to scan it for points of
interest. Like trying to identify a landmark
or something? Yeah, exactly. The service runs
its algorithms over the encrypted data, finds
the landmark, and sends you the answer all without
ever actually viewing your photo. Though I will
say, the sources provide a very necessary warning
right up front about side channels. Ah, yes.
the metadata issue. Right. Because the math itself
might be perfectly secure, meaning the box cannot
be broken into. But observing the behavior of
the system still leaks metadata. Like, an observer
might not see the photo of the landmark, but
they can see that your device sent a massive
packet of data to a lookup service at, you know,
three in the afternoon. It's a really vital carryout.
Cryptography secures the contents, but traffic
analysis reveals context. The metadata is still
visible. But to understand how the locked box
computation actually works without decrypting
the contents, we really have to look at the different
flavors of homomorphic encryption. Right, yeah,
they're categorized by the type and complexity
of the mathematical operations, the circuits
that they can handle. Right, and they range from
basic arithmetic to incredibly complex operations.
Like first, there's partially homomorphic encryption,
which means the system supports only one type
of mathematical gate. Just one. Right, so you
can do addition or you can do multiplication,
but you definitely cannot do both. The source
notes that the famous RSA cryptosystem is actually
partially homomorphic. It allows you to do an
unbounded number of multiplications on encrypted
data. And then there's the Pellier cryptosystem,
which allows for unbounded additions. But building
on that, if you want more flexibility, you move
up to somewhat homomorphic encryption. This one
allows for two types of gates. both addition
and multiplication, but only for a limited subset
of circuits. Meaning it has a ceiling on how
much math you can do, right? Exactly. It hits
a limit. From there, we reach leveled fully homomorphic
encryption, which supports arbitrary surface
with multiple types of gates, but only up to
a predetermined bounded depth. OK. And finally,
the ultimate goal, fully homomorphic encryption,
or FHE. This allows for the evaluation of arbitrary
circuits of unbounded depth, meaning you can
run literally any computer program of any length
or complexity on the encrypted data. Okay, I
want to push back on that bounded depth concept
for a second, because this is where the physical
reality of the math seems to get really weird.
It does get weird. Yeah. The sources focus heavily
on this idea of a multiplicative ceiling. Let's
say... Partially homomorphic encryption is a
calculator with only an addition button. And
somewhat homomorphic encryption gives you a calculator
with a multiplication button, but the battery
magically dies after five uses. That's a fun
way to picture it. Thanks. But why does doing
too much math break the encryption? Like, what
is actually happening inside the lockbox that
causes it to fail? Well, what's fascinating here
is that To make homomorphic encryption function
at all, the ciphertext must be inherently malleable.
Malleable how? The server has to be able to manipulate
the encrypted gibberish in a way that fundamentally
changes the hidden plaintext inside. But to prevent
attackers from just easily reverse engineering
that malleability, cryptographers have to inject
a small amount of mathematical noise or intentional
error into the encryption. So the noise is like
static on a radio transmission. That is a highly
useful way to visualize it. Yes, every time the
server performs an addition on the encrypted
data, that static grows just a tiny bit. OK.
But every time the server performs a multiplication,
the static compounds exponentially. It multiplies
the noise. So if the mathematical circuit is
too deep, meaning you perform too many multiplications,
the static eventually overpowers the signal.
Oh, man. So when you finally download the result
and use your key to open the box. The original
data is completely drowned out, the decryption
fails, and the data is lost forever. That is
just a wild concept. The very feature that keeps
the data secure, the noise, becomes the exact
thing that destroys the data if you manipulate
it too much. It's a huge paradox. Yeah. And the
problem of how to stop that noise from destroying
the data went unsolved for decades. I mean, the
source notes the concept of homomorphic encryption
was first proposed back in 1978, barely a year
after RSA was published. Right. And for over
30 years, the cryptography community didn't even
know if a solution was mathematically possible,
right? Yeah, they had no idea. Researchers built
systems that allowed unlimited additions, but
maybe only a single multiplication. But the holy
grail of fully homomorphic encryption remained
purely theoretical. It was this massive bottleneck
in computer science. Well, until 2009, when a
researcher named Craig Gentry figured it out.
using lattice -based cryptography, creating the
first plausible fully homomorphic encryption
scheme. Yes, Gentry's breakthrough. And his workaround
for the noise problem was a mechanism he called
bootstrapping. Bootstrapping is honestly one
of the most elegant concepts in modern cryptography.
Gentry realized that it was impossible to stop
the noise from growing during multiplication.
The static was essentially a law of physics for
this kind of math. So he couldn't stop it. Right.
So if he couldn't stop it, he had to find a way
to clean it up mid -computation before it destroyed
the signal. Here's where it gets really interesting.
Let me see if I have this right. It's like having
a photocopy of a document. Every time you copy
the photocopy, it degrades and gets grainier.
Exactly. That grain is the noise. So boot scrapping
sounds like having a machine that perfectly redraws
the faded text onto a fresh sheet of paper. But
the machine itself is entirely blind and never
actually reads the words. It just resets the
noise back to zero. Give you a fresh, clean ciphertext.
That captures the outcome perfectly. But to explain
the mechanics of how the blind machine does that,
we have to look at recursive self -embedding.
Recursive self -embedding. That sounds intense.
It is a bit. Gentry designed a scheme that was
capable of homomorphically evaluating its own
decryption circuit. Wait, evaluating its own
decryption circuit? How does that even work?
Okay, let's go back to your locked box. You have
a titanium box, and the static inside is getting
dangerously loud. Before it ruins the diamond,
the server wraps your box inside a second, slightly
larger titanium box. Okay, so now you have a
box inside a box. Right. Then you, the user,
provide the server with an encrypted version
of your key. The server uses its mechanical arms
to take that encrypted key, reach inside the
outer box, and unlock the inner box. Oh, so the
inner box is open? but the diamond is still safely
locked inside the outer box. Precisely. The data
is never exposed to the open air. But because
the inner box was unlocked, the accumulated noise
from the previous computations is stripped away.
That is brilliant. It really is. You're left
with a brand new, fresh ciphertext with baseline
noise holding the exact same value. And because
of this, you can string together an infinite
number of additions and multiplications. You
just pause and bootstrap whenever the noise gets
too high. Gentry proved that any scheme capable
of this self -evaluation could be converted into
a fully homomorphic scheme. Okay, but there is
a massive catch in the source material, like
a really big one. The processing time. Yes. The
original implementation by Gentry and Halevi
took about 30 minutes just to process a single
basic bit operation. And a bit is literally just
a one or a zero. It was very slow. If it takes
half an hour to do one microscopic calculation,
aren't we talking about years? just to load a
basic web page or process a simple text document.
How could anyone consider this practical? Well,
you have to view that 30 -minute metric through
the lens of scientific progress. At that moment
in 2009, speed was completely irrelevant. It
was entirely a proof of concept. Oh, so just
proving it could be done at all. Exactly. The
value was in proving that the underlying cryptographic
assumptions like problems over ideal lattices
could be used to achieve the impossible. Gentry
broke a 30 -year stalemate. And once a community
knew it was possible, other pathways opened up
almost immediately. Like that? Well, by 2010,
Van Dyke and a team of researchers presented
an alternative scheme using simple integers instead
of complex ideal lattices. The floodgates were
open. Okay, so the academic world knew the lockbox
worked. The blindfolded jeweler could reset their
tools without destroying the diamond. But to
make this viable for commercial cloud computing,
where servers crunch petabytes of data, they
needed to speed it up by several orders of magnitude.
Which brings us to the race for efficiency, Generations
2 and 3. Yeah, the subsequent generations of
FHE schemes focused entirely on optimization.
In the years following Gentry's breakthrough,
roughly 2011 to 2012, Researchers developed schemes
like BGV and BFE. And what did those do? They
drastically slowed the growth of the noise during
computations. The static simply didn't build
up as fast, meaning you didn't have to perform
that slow bootstrapping process nearly as often.
Though the source did mention some missteps,
right? Like NTRU -based schemes, I think. They
were called LTV and BLLN. Ah, yes. Those failed
because they relied on an overstretched variant
that ended up being vulnerable to subfield lattice
attacks. It was a dead end. But the successful
Gen 2 schemes thrived. The source mentions they
also achieved nearly optimal asymptotic complexity
using a technique involving SIMD single instruction,
multiple data. They were packing multiple plaintext
values into a single Cypher text to compute on
all of them simultaneously. Yes, Cindy packing
was a game changer. I want to try an analogy
for this. If we're packing multiple plaintexts
into one Cypher text, is it like stuffing hundreds
of passengers onto a massive train so you only
have to make one trip instead of driving them
all individually in cars? That's actually a great
way to put it. But wait, it definitely saves
time, but does that limit the route the train
can take? Like, do we lose computational flexibility
just to gain speed? If we connect this to the
bigger picture, you do face routing constraints
with SIMD packing. To use your train analogy,
you cannot tell passenger A to get off in New
York and passenger B to get off in Chicago. The
entire train goes to one destination. Right.
In computational terms, operations must be applied
uniformly across all the packed data slots at
the exact same time. You apply the exact same
mathematical instruction to the entire vector
of data. Which sounds pretty limiting, to be
honest. It does, until you look at how cloud
analytics actually work. Think about it. If you're
searching a massive encrypted database or performing
statistical analysis on health care records,
you usually want to apply the exact same comparative
operation to millions of data points anyway.
Oh, I see. So the constraint doesn't really matter
for those use cases. Exactly. The vector constraint
doesn't hinder the goal, and the speed game is
astronomical. In fact, these optimizations made
many applications so efficient they could run
in what's called leveled FHE mode. That's the
one where you know the depth of the circuit ahead
of time, right? Correct. If a software engineer
knows exactly how many multiplications a specific
program requires and the noise grows slowly enough,
the program might never even hit the noise ceiling.
So no bootstrapping at all? None. The server
can run the entire operation from start to finish
without ever needing to pause for the slow, expensive
bootstrapping process. That's a huge leap in
practicality. But the cryptography community
didn't stop there. The source highlights a massive
paradigm shift around 2013 and 2014, the third
generation. Researchers proposed new techniques,
I think GSW has mentioned, that avoided expensive
relinearization steps. Yes. And that led to schemes
like FHUW and TFHE, where they made a really
bold choice. They decided to refresh the ciphertext
after every single operation. Which seems completely
counterintuitive at first glance. Like, if bootstrapping
is the slowest part of the entire process, why
would doing it more often make the system faster?
That sounds like a terrible idea. Right. It seems
backward. was in how they simplified the logic.
By restricting the computations to very basic
Boolean gates, the bootstrapping procedure itself
became incredibly lightweight. So it wasn't this
massive heavy process anymore. Exactly. They
weren't running a massive heavy decryption circuit.
They were running a highly optimized tiny one.
So by 2016, these schemes dropped the bootstrapping
time from Gentry's original 30 minutes down to
a fraction of a second. We're talking less than
0 .1 seconds per operation. Wow. From taking
a coffee break to the bleak of an eye. So the
community conquered basic, exact computations,
making them lightning fast. But the tech landscape
in 2016 was undergoing a seismic shift, wasn't
it? It absolutely was. Yeah. The biggest demand
for cloud computing was no longer running exact
database searches. It was machine learning. was
AI. And machine learning doesn't really rely
on perfect exact integer math. It relies on messy
continuous approximate math. That's a challenge.
Machine learning models utilize weights and biases
that are often tiny decimal fractions. The FHE
schemes we've discussed so far, they handled
integers perfectly. Well, not fractions. Right.
If you try to multiply complex fractions in those
older schemes, the scale of the numbers blows
up and you have to use heavy bootstrapping just
to manage the sheer size of the data. To make
privacy -preserving machine learning work, the
industry needed an entirely different approach.
points to a specific scheme introduced in 2016
called CKKS. This scheme supports fixed point
arithmetic. which is commonly referred to as
block floating point arithmetic, meaning it's
designed specifically to handle those complex
decimals. Crucially, CKKS includes an efficient
rescaling operation. Instead of the scale of
the numbers blowing up after a multiplication,
CKKS allows the server to simply scale down the
encrypted message. Oh, that makes sense. Yeah.
This makes evaluating the polynomial approximations
used in machine learning incredibly fast and
efficient. It basically became the gold standard
for running AI models on encrypted data. But,
and this is a massive but in the source material,
CKKS is an approximate homomorphic encryption
scheme. It introduces non -deterministic and
deterministic approximation errors. It's like
slightly floppy by design. Yes, it trades exactness
for speed. So what does this all mean? If CKKS
uses approximate math, kind of like a busy accountant
just rounding numbers to the nearest thousand
to save time, the source notes this sloppiness
caused a major vulnerability. It did, the 2020
paper. Right. In 2020, researchers Lian Miciancio
published a passive attack showing that if the
results of these approximate decryptions are
shared, the secret key itself could actually
be recovered. I mean, doesn't that defeat the
entire purpose of the lockbox? Are we sacrificing
our perfect security just to get the speed we
need for AI? Well, this raises an important question
about how we evaluate cryptographic models. Let's
look objectively at the mechanics of what Lye
and Miciancio discovered. In exact FHE schemes,
when you use your private key to open the box,
the noise is removed entirely. The final answer
you read is mathematically perfect. OK. But in
CKKS, because the math is approximate, a tiny
fraction of that noise remains in the final decrypted
result. Like the accountant's rounding error.
Exactly. And what the researchers realize is
that this leftover error isn't just random static.
That error is mathematically tied to the secret
key. It's an algebraic artifact. Wait, really?
So the noise is essentially a fingerprint of
the key. It is. If a user decrypts the result
and then shares that slightly noisy data publicly
or, say, feeds it back to the server, an adversary
can compare the noisy result to the true mathematical
answer. Oh, wow. Yeah. By isolating the error
across multiple queries, the adversary can reverse
-engineer the secret key. The attacker doesn't
even have to break into the system. They just
have to observe the shared results. That sounds
completely catastrophic for privacy. It required
a fundamental rething, certainly, but it doesn't
mean the technology is broken. It means new technologies
that solve old problems inevitably create new
threat vectors. The standard security model cryptographers
use, which is known as IND -CPA, assumed that
decrypted results would remain completely private.
But people share AI outputs all the time. Exactly.
CCATAS challenged that assumption because in
modern machine learning, outputs are frequently
shared or fed back into public models. The standard
simply wasn't built for that use case. And the
tech industry had to scramble to fix it. The
source mentions that major tech companies maintaining
homomorphic encryption libraries like Heon, Seal,
Healib, Palisade, They all had to implement mitigations.
Right. They patched the vulnerabilities through
responsible disclosure before the paper even
went public. That's a relief. It really highlights
how rapidly the ecosystem is maturing. We're
no longer relying on isolated academic papers
here. There are standard setting bodies now.
The Homomorphic Encryption Standardization Consortium
was formed back in 2017 to maintain community
security standards across these different schemes.
With companies like IBM, Microsoft, and Intel
involved, right? Yes. And the technology is robust
enough now that it's even expanding into decentralized
applications. There are blockchain projects like
Phoenix leveraging FHE for privacy preserving
financial use cases. We have massive open source
frameworks pushing this complex math directly
into the hands of everyday software developers.
Which brings us right back to you the listener.
Why should you care about the mechanics of lattice
based cryptography or how a server manages floating
point arithmetic? Because you live your life
in the cloud. We all do. Every time you use a
cloud -based AI tool to draft a sensitive email,
or use a financial app to track your personal
budget, or a digital healthcare platform to message
your doctor, your privacy is increasingly going
to depend on these invisible, mathematical locked
boxes. As computing power scales up, homomorphic
encryption ensure your data doesn't have to be
exposed just to be useful. It represents a fundamental
decoupling of data utility from data privacy.
For the entire history of computing, we've had
to choose between keeping our data secret and
gaining the benefits of analytics. That compromise
is finally disappearing. But there is one final
thought I want to leave you with, building on
the warning from the source material we mentioned
earlier. Even with perfect homomorphic encryption,
side channels can still reveal metadata. Right,
the context. They can reveal that an encrypted
photograph was sent to a lookup service, even
if the photograph itself is perfectly secure.
So, here's something to ponder as we wrap up.
If homomorphic encryption eventually make the
actual contents of our data mathematically unbreakable,
will the future of cyber espionage and hacking
completely abandon trying to read our messages?
Will they stop trying to see the diamond and
instead focus entirely on the metadata building
complete intimate profiles of our lives based
solely on when, where, and how often we send
our locked boxes? The context speaks volumes.
The jeweler might be blindfolded, but everyone
can still see how often they show up to work.
Thank you for joining us on this deep dive into
homomorphic encryption. Keep questioning the
systems around you, and keep exploring the digital
world. We'll see you next time.