[An in-depth blog post by Daniel J. Bernstein, crossposted between the ECRYPT blog and the cr.yp.to blog.]

For many years NIST has officially claimed that
AES-128 has "comparable strength" to 256-bit ECC,
namely 128 "bits of security".
Ten years ago,
in a talk
"Is 2^{255}−19 big enough?",
I disputed this claim.
The underlying attack algorithms had already been known for years,
and it's not hard to see their impact on key-size selection;
but somehow NIST hadn't gotten the memo,
and still hasn't gotten it today.

Here are the relevant quotes from my slides, plus some explanations:

- "Given H(k) = AES
_{k}(0), find k using ≈2^{127}AES evaluations." This is simply brute-force search: at worst 2^{128}, maybe 2^{126}or smaller, on average 2^{127}. Actually, the inner loop of brute-force search is slightly cheaper than a normal AES evaluation, since there are many ways to merge work across the AES evaluations for multiple inputs; keep in mind this theme of merging work. - "Given H(k[1]),H(k[2]),...,H(k[2
^{40}]), find*all*k[i] using a*total*of ≈2^{127}AES evaluations." For example, sort the list of targets, and look up each of the 2^{128}possible H outputs in the sorted list. Accessing such a large volume of data (the sorted list of 2^{40}targets) might sound like a performance problem, but there's a fancier brute-force algorithm, using Rivest's idea of "distinguished points", that practically eliminates the communication costs; see, for example, my paper "Understanding brute force". This attack against a batch of targets is much, much, much faster than attacking each target separately; it's almost as fast as attacking a single key. - "Or find
*some*k[i] using ≈2^{87}AES evaluations." This is the same multi-target attack: within 2^{88}guesses it's already expected to find one key, within 2^{98}guesses it's expected to find about 1000 keys, etc. What's scary here is that an attack with cost many orders of magnitude below 2^{128}is accomplishing a useful result against somebody using AES-128. - "Given public key on 255-bit elliptic curve E, find secret key using ≈2
^{127}additions on E." This is making the 255-bit ECC situation sound a lot like the AES-128 situation at first, supporting NIST's claim. But what happens when the attacker has multiple targets? - "Given 2
^{40}public keys, find all secret keys using ≈2^{147}additions on E." This is a million times faster than finding each key separately; on the other hand, it's still much slower than finding a single targeted key. The basic idea of this attack algorithm was first published in a 1999 paper by Escott–Sager–Selkirk–Tsapakidis, and was improved in several subsequent papers; see my paper "Computing small discrete logarithms faster" with Lange for the history and the latest speedups. - "Finding
*some*key is as hard as finding first key: ≈2^{127}additions." This is what makes this multiple-target ECC attack much less scary than the multiple-target AES attack. If the attacker's resources are actually capable of carrying out 2^{100}computations then the 2^{40}-target AES attack will break*thousands of AES keys*while the ECC attack, unless it's astonishingly lucky, will break*zero ECC keys*. -
"Bottom line: 128-bit AES keys are not comparable in security
to 255-bit elliptic-curve keys.
Is 2
^{255}−19 big enough? Yes. Is 128-bit AES safe? Unclear."

What the attacker hopes to find inside the AES attack is a **key collision**.
This means that a key guessed by the attack matches a key chosen by a user.
Any particular guessed key has chance only 1/2^{128}
of matching any particular user key,
but the attack ends up merging costs across a batch of 2^{40} user keys,
amplifying the effectiveness of each guess by a factor 2^{40}.
A paper by
Chatterjee–Menezes–Sarkar
presents many more attacks along these lines,
including a quite serious 80-bit collision attack on
a version of HMAC that's fully compliant with the NIST HMAC standard:

After querying 2^{20}users for the MAC of some fixed message m, the adversary would be able to determine the secret key of one of the 2^{20}users after performing about 2^{60}MAC operations. Since the work can be easily and effectively parallelized, the attack should be considered feasible today.

The single-key ECC attack also looks for key collisions,
and on top of this exploits another attack tool
called a "worst-case-to-average-case reduction",
specifically a "random self-reduction".
This extra attack tool
expands the original user key (a curve point kP)
into a pool of many modified user keys (points kP+rP, where r is chosen randomly by the attacker).
Any collision between any of the guessed keys (points gP, where g is known to the attacker)
and any of the modified user keys
(i.e., any equation kP+rP = gP)
immediately reveals the original user's secret key.
Breaking 2^{40} user keys in the same way needs 2^{40} collisions,
and this takes only 2^{20} times as long as finding one collision.

The number of modified user keys is now chosen by the attacker,
rather than being naturally limited to the number of legitimate user keys.
This sounds much more destructive than the AES-128 attack,
and it *would* be much more destructive if we used 128-bit ECC:
the attacker would expand the original key into 2^{64} modified keys
and find a collision after just 2^{64} guessed keys,
for a total attack cost on the scale of 2^{64}.
Rivest's "distinguished points" idea again eliminates
issues regarding data volume, communication, parallelization, etc.;
the classic paper on this topic is
"Parallel collision search with cryptanalytic applications"
by van Oorschot and Wiener.

In 2007 NIST revised its
"Recommendation for Key Management"
to prohibit use of 160-bit ECC after 2010:
the minimum ECC key size allowed is 224 bits after 2010 (and 256 bits after 2030).
Evidently NIST understands that a 2^{80} attack breaking somebody's 160-bit ECC key
is enough of a problem to warrant moving to larger ECC keys.
But NIST is foolishly continuing to recommend AES-128 keys,
even though
a 2^{80} attack will break somebody's AES-128 key
out of a batch of 2^{48} keys.

The security section of my
Curve25519 paper
includes a subsection "Batch discrete logarithms"
that summarizes the cost of attacking multiple targets
and concludes that Curve25519 is big enough:
"The attacker is likely to wait for 2^{125} additions before finding the first key ...
Curve25519 is at a comfortable security level
where finding the first key is, conjecturally, far out of reach,
so the reduced cost of finding subsequent keys is not a threat.
The attacker can perform only 2^{125}ε additions for small ε,
so the attacker's chance of success—of finding *any* keys—is
only about ε^{2}."
The paper continues by comparing this to the cost of "brute-force search"
for finding "a batch of u keys" or "finding some key" in a secret-key system such as AES.
Ten years later this type of analysis is still missing from NIST's
latest key-size recommendations.

**Cutting things too close: the "attacker economist" philosophy.**
There are three common approaches to choosing cryptographic key sizes:

- The "attacker will probably fail" approach. People taking this approach say things like this: "We're designing this system so that the cost of breaking it is larger than the resources available to the attacker."
- The "attacker will definitely fail" approach, producing larger key sizes. I take this approach, and I say things like this: "We're designing this system so that the attacker has chance at most one in a billion of breaking it using the available resources."
- The "attacker economist" approach, producing smaller key sizes. People taking this approach say things like this: "We don't need attacks to be infeasible; that would be overkill. We're designing this system so that the attacker's expected cost of carrying out an attack exceeds the attacker's expected benefit from doing so."

For some attacks, notably the number-field sieve (NFS) for integer factorization, the success probability jumps very quickly from 0 to 1 as the cost increases past a particular threshold. It's then clear what the "attacker will probably fail" approach says, namely that the key size has to be chosen so that this threshold is beyond the attacker's resources.

However,
for many other attacks the success probability grows much more gradually,
and then it's not clear what the "attacker will probably fail" approach means.
If "the cost of breaking"
is defined as the median cost
then the "attacker will probably fail" approach
ends up saying that the attacker has chance below 50% of breaking the system;
but cryptographic users aren't happy to hear
that they've been given a cryptographic system
that the attacker can break with probability 40%.
NIST seems to focus on the *average* cost
("an algorithm that provides X bits of security would, on average,
take 2^{X-1} T units of time to attack"),
and with this definition
the attacker's success probability could be 60% or even larger.

The "attacker economist" approach provides a rationale for focusing on averages. It views the attacker as rationally deciding whether to carry out attacks, saying "I will carry out this attack because its average benefit to me is larger than its average cost" or "I will skip this attack because its average benefit to me is smaller than its average cost". Of course, if the attacker carries out an attack of the second type, the attacker might still end up luckily gaining something (maybe the benefit turns out to be unusually large or the cost turns out to be unusually small), but after a long series of such attacks the attacker expects to lose.

The attraction of the "attacker economist" approach
is that it produces smaller key sizes.
The attack cost is limited to min{B,C},
where B is the attacker's benefit and C is the attacker's resources;
for comparison,
in the "attacker will definitely fail" approach,
the attack cost is limited to C,
which is usually much larger than min{B,C}.
In other words,
the "attacker economist" approach
excludes not merely *infeasible* attacks,
but also *uneconomical* attacks.

There are, however, several problems with this approach. First, even if the attacker's expected cost obviously outweighs the attacker's expected benefit, sometimes the attacker goes ahead and carries out the attack anyway. The "attacker economist" approach assumes that rational attackers won't carry out uneconomical attacks, so users don't have to worry about such attacks; but attackers in the real world do not always act on the basis of rational economic analyses. The damage done by the attack, beyond its benefits to the attacker, is invisible to the "attacker economist" philosophy.

Second,
evaluating the attacker's benefit
requires understanding all the consequences of, e.g., successfully forging a signature.
How exactly is the signature being used?
What exactly does the attacker gain from the forgery?
What is the monetary value of this benefit?
The answers are complicated and constantly changing;
they have a dramatic dependence on the details
of how cryptography is integrated into real applications.
A single forgery might not sound so bad,
but it might also allow the attacker to take complete control over a critical computer.
Sure, we can tell cryptographic users
that they should evaluate the consequences of the cryptography failing,
and that they should limit the attacker's gain from a successful attack,
but the reality is that nobody gets this right:
people are constantly surprised by how much damage an attack can do.
For comparison,
there's at least some hope of accurately assessing the resources available to attackers
and of building cryptographic systems that are *infeasible* to break.

Third,
even in situations where it's clear how much damage is caused by each forgery,
"attacker economist" proponents
are constantly overestimating the attacker's *cost per forgery*.
The typical process here might sound reasonable at first:
take what everybody says is the state-of-the-art forgery algorithm,
and look at what everybody says is the cost of that algorithm.
The problem is that the algorithm designers are
practically always optimizing and reporting costs in a metric
that's different from, and much larger than, the cost per forgery.
Specifically,
the usual metric is the cost of forging *one* message under *one* key,
while the attacker is trying to forge *many* messages under *many* keys.

**Some examples of difficulties for the "attacker economist" philosophy.**
People typically say that NFS is the state-of-the-art algorithm for forging RSA signatures,
and that NFS has cost somewhere around 2^{80} for RSA-1024.
However, NFS produces much more than a single forgery:
it produces the RSA secret key.
The attacker then uses this key for any number of forgeries.
The cost per forgery is the cost of the usual signing algorithm
plus a small fraction of 2^{80}, depending on the number of forgeries.
The cost of one forgery thus wildly overestimates
the per-forgery cost of many forgeries under the same key.

As another example,
consider the
first draft
of the UMAC RFC (October 2000),
which claims that 64-bit UMAC tags are
"appropriate for most security architectures and applications".
This claim is backed by the following deceptive statement:
"A 1/2^{60} forging probability means that if an attacker could try out 2^{60} MACs,
then the attacker would probably get one right."
The correct statement is that an attacker trying out 2^{60} MACs
would probably get at least 2^{58} right.
I had already pointed out this mistake as part of a
paper
(see also the
latest version)
posted several months earlier:

Some writers claim that forgery probabilities around 1/2^{32}are adequate for most applications. The attacker's cost of 2^{32}forgery attempts, they say, is much larger than the attacker's benefit from forging a single message. Unfortunately, even if all attackers acted on the basis of rational economic analyses, this argument would be wrong, because it wildly underestimates the attacker's benefit. In a typical authentication system, as soon as the attacker is lucky enough to succeed at a few forgeries, he can immediately figure out enough secret information to let him forge messages of his choice. ... It is crucial for the forgery probability to be so small that attackers have no hope.

Several years later,
after I objected,
the UMAC authors tweaked their text as follows:
"If an attacker could have N forgery attempts,
then the attacker would have no more than N/2^{60} probability
of getting one or more of them right."
The actual situation, when there are successful forgeries,
is that the number of successful forgeries is very likely to be on the scale of N;
describing this as "one or more" forgeries gives the reader a very wrong idea.

So, okay, instead of looking at the cost of one forgery,
we should look at the cost of breaking one key, right?
No:
the cost of breaking one key
often wildly overestimates the per-key cost of breaking many keys.
I'll again take an ECC example:
people typically say that breaking 160-bit ECC costs 2^{80};
but breaking a batch of a million 160-bit ECC keys costs a total of just 2^{90},
i.e., 2^{70} per key.

Both of these mistakes—confusing the cost of one forgery with the cost per forgery, and confusing the cost of breaking one key with the cost per key—cause serious problems for the "attacker economist" approach.

The situation is completely different
if key sizes are chosen
so that forging one message under one key is *infeasible*.
Forging more messages is certainly no easier than forging one message,
and is therefore also infeasible;
the reduced cost per forgery doesn't matter.
Of course,
one still has to be careful to check whether attacking a batch
increases the success probability of feasible attacks,
exactly the AES-128 issue described at the start of this blog post.

**Can an attacker actually carry out 2 ^{80} or 2^{90} or 2^{100} operations?**
Here are some back-of-the-envelope numbers
that help put attack possibilities into perspective.
Mass-market GPUs use state-of-the-art chip technology,
are optimized for floating-point operations,
and perform about 2

- 2
^{26}watts (2^{84}mults/year): the power substation for one of NSA's computer centers. - 2
^{30}watts (2^{88}mults/year): the power available to a botnet that has broken into millions of computers around the Internet. - 2
^{44}watts (2^{102}mults/year): the power actually used by the human race at this instant. - 2
^{56}watts (2^{114}mults/year): the power that the Earth's surface receives from the Sun. - 2
^{57}watts (2^{115}mults/year): the power that the Earth's atmosphere receives from the Sun.

Of course, the operations in a typical cryptanalytic attack are more complicated than floating-point multiplications. On the other hand, serious attackers can build special-purpose chips streamlined for their attacks, and computer technology continues to improve. There's no substitute for detailed analyses of actual attack costs, such as papers at the various SHARCS workshops.

Anyway,
the overall scale of these numbers explains
why cryptographers distinguish between 2^{80} security and 2^{128} security.
2^{80} might sound like a big number,
but it's not that big compared to the 2^{55} nanoseconds in a year.
Serious attackers today have enough computer equipment
to be performing more than 2^{25} useful operations each nanosecond,
for a total of more than 2^{80} useful operations in a year.
On the other hand, this is still much, much, much smaller than 2^{128}.

**Batch factorization.**
Suppose that someone ignores my advice
and deploys a system providing "2^{80} security".
Probably the security claim is based on
an attack algorithm that was optimized for breaking a single key.
How much better can the attacker do by attacking a large batch of keys, say 2^{20} keys?

Earlier I explained how well batch algorithms do for guessing secret cipher keys and for breaking ECC. Here are the results for 80-bit cipher keys and 160-bit ECC keys:

Attack cost | Average number of keys found out of 2^{20} secret 80-bit cipher keys |
Average number of keys found out of 2^{20} secret 160-bit ECC keys |
---|---|---|

2^{50} | 1/2^{10} | 1/2^{60} |

2^{60} | 1 | 1/2^{40} |

2^{70} | 2^{10} | 1/2^{20} |

2^{80} | 2^{20} | 1 |

2^{90} | overkill | 2^{20} |

2^{100} | overkill | overkill |

But what about RSA?

RSA is vulnerable to a sophisticated attack strategy called **combining congruences**.
The basic idea of combining congruences isn't hard to explain,
but there's a massive literature introducing
an apparently neverending series of speedups to the basic idea
and combining these speedups in all sorts of tricky ways.
The state-of-the-art algorithms have so many adjustable knobs and hard-to-analyze traps
that there's no consensus on how they will perform
beyond the range of academic experiments.

If you want to know how big your RSA key needs to be for, say, 2^{128} security,
you might ask the summary site
http://www.keylength.com/en/compare/,
which collects recommendations from several sources.
These recommendations range from 2048 bits to 6974 bits.
What's the right answer?

I mentioned earlier that people typically report security around 2^{80} for RSA-1024.
What exactly does this 2^{80} mean?
Shamir and Tromer
wrote in 2003
that "it appears possible to break a 1024-bit RSA key in one year
using a device whose cost is about $10M (previous predictions were in the trillions of dollars)";
Franke, Kleinjung, Paar, Pelzl, Priplata, and Stahlke
wrote in 2005
that there were "concerns about the feasibility" of the Shamir–Tromer design
and presented a different design costing twenty times as much.
What's the right answer?

The only real consensus is that the number of "operations" in NFS
scales asymptotically as L^{1.901...+o(1)}, where

- L is exp((log N)
^{1/3}(log log N)^{2/3}); - N is the RSA modulus being factored; and
- o(1) is something that converges to 0 as N increases.

The difference between o(1) and 0 is important.
The L^{1.901...+o(1)} formula can't be safely used to predict cost for any particular size of N,
and can't be safely used to extrapolate from one size of N to another.
If someone finds a new algorithm replacing 1.901... with, say, 1.8,
then that algorithm might be worse for every size of N relevant to cryptography.
However,
there will be *some* size of N where the new algorithm is faster,
and maybe that size *is* small enough to be relevant to cryptography,
so obviously the performance of the new algorithm has to be explored in detail.

In a 1993 paper,
Coppersmith introduced a variant of NFS
that uses only L^{1.638...+o(1)} "operations" to factor N,
after a precomputation that uses L^{2.006...+o(1)} "operations".
The precomputation doesn't depend on the target N;
it depends only on roughly how big N is.
Coppersmith described the precomputation as the construction of a "factorization factory"
that can then be applied to any target.
If you have a batch of (say) L^{0.5+o(1)} targets N,
then the factory-construction cost is negligible,
and the total number of "operations" in Coppersmith's algorithm
is L^{1.638...+o(1)} per target.

I've been writing "operations" in quotes here
because this is counting random access `x[i]=j` as a single "operation",
no matter how large the array `x` is:
bigger than cache, bigger than DRAM, bigger than your 2TB disk, etc.
This metric does a poor job of predicting real-world algorithm performance.
Any attacker who can afford a gigantic storage array
will also invest in many parallel processing elements (such as GPUs),
and will find that parallel computations are much less expensive than random access to storage.

In a
2001 paper,
I reoptimized NFS "sieving" and "linear algebra" for a realistic circuit model of computation,
and concluded that the cost of single-target NFS actually scales as L^{1.976...+o(1)}.
In a
2014 paper,
Lange and I reoptimized Coppersmith's ideas for a realistic circuit model of computation,
and concluded that the cost per integer of factoring L^{0.5+o(1)} integers
actually scales as L^{1.704...+o(1)}.
It turns out that Coppersmith's data flow,
first precomputing a "factory" and then applying the "factory" to one target N at a time,
isn't realistic:
our new "batch NFS" algorithm scraps Coppersmith's "factory"
and handles a large batch of targets simultaneously.

Let me again emphasize that these exponents
don't say anything about any particular size of N.
What's clear is merely that batching reduces cost *for sufficiently large N*,
i.e., all N past some cutoff;
it isn't clear what the cutoff is,
and it isn't clear what the actual cost reduction is for any specific size.
Lange and I have started digging into
the more subtle question of what this means for
the huge pile of 1024-bit keys that attackers have been collecting.

**More examples of difficulties for the "attacker economist" philosophy.**
"It makes no sense for an adversary to spend (say)

Silverman estimated that spending $10 million
on 4300 "500-MIPS machines" with "4.4 Gbytes per machine"
would let the attacker factor a 768-bit key in "about 670 months",
i.e., 56 years,
i.e., 2^{71.7} instructions.
Silverman predicted
"a date of 2019 for breaking such a key by public effort."

There's one important point that Silverman got right: accessing large volumes of memory is expensive. But what does this mean for an algorithm that was optimized for the number of "operations", counting (expensive) memory access as one "operation" and also counting (cheap) arithmetic as one "operation"? Silverman emphasized the expense of the memory access, apparently not realizing the possibility of reoptimizing algorithms for actual cost. Silverman also got many other important details wrong.

The factorization of the RSA-768 challenge was in fact publicly announced in 2009,
ten years earlier than Silverman's "2019" prediction.
The paper
reported that the factorization had taken two years on "many hundreds of machines";
that it would have taken about 1500 core-years on 2.2GHz Opterons with 2GB of RAM,
i.e., 2^{66.5} cycles;
and that
"it is not unreasonable to expect that 1024-bit RSA moduli can be factored well within
the next decade by an academic effort such as ours".

Of course, any serious attacker can be presumed to have built efficient special-purpose hardware tuned for integer factorization, while an "academic effort" does not have this luxury. Why does Silverman define security by the costs of a "public effort", if this is overstating the costs incurred by the actual attackers?

Furthermore,
if 768 bits are beyond the cutoff for batch NFS,
then the attacker can factor many 768-bit keys more cost-effectively than factoring just one.
As I said before,
it isn't known yet what the cutoff actually is,
so one can't justifiably claim that there's a cost advantage,
but one also can't justifiably claim that there *isn't* a cost advantage.
As Lange and I wrote in our paper on batch NFS:

Users comparing the value of an RSA key to the cost of an attack machine need to know theper-keycost of batch NFS. This has not been seriously studied. What the literature has actually studied in detail is the cost of NFS attackingone key at a time; this is not the same question.

Why was Silverman comparing the costs and benefits of "breaking a key"
if the actual job facing the attacker is to break *many* keys?
There's an implicit assumption that each key has to be attacked separately;
batch algorithms challenge this assumption.

Now let's move up from 768-bit keys to 1024-bit keys. Silverman claimed that it would require "at least 27 years to reach a point where 1024-bit keys are vulnerable to a public effort" and specifically predicted a date of "around 2037" for factoring a 1024-bit key. Since the time from Silverman's paper to the public RSA-768 factorization was only half of what Silverman had predicted, and the people announcing that factorization had said that it was "not unreasonable to expect" a public RSA-1024 factorization before 2020, one would think that anyone quoting Silverman's 1024-bit predictions would have some serious explaining to do. But Silverman's "around 2037" was repeated—even bumped to "around 2040"—in a 2011 paper by Grigg and Gutmann:

[Silverman estimated in 2000 that] a single 1,024-bit key could be factored by around 2040. That’s a massive effort for one key, with every one of the millions of other 1,024-bit keys in use today still being safe until the same effort gets applied to them, as well.

Grigg and Gutmann don't mention Silverman's "public effort" hypothesis; don't discuss the advantages of special-purpose hardware; don't discuss the problems with Silverman's extrapolation methodology; don't mention Silverman's highly inaccurate 768-bit extrapolation; and don't mention that Silverman's 1024-bit extrapolation has been disputed. But what I find particularly interesting in the Grigg–Gutmann quote is the leap from one key to many keys, ignoring the possibility of more cost-effective batch attacks. Even if it's true that factoring a 1024-bit key is a "massive effort" that won't happen before 2040, there's no justification for the claim that factoring N keys will require N times the effort.

To summarize, here's what's actually known about the per-key cost of breaking RSA keys: "It's complicated."

**Batch non-elliptic discrete logs.**
What's even more complicated
is the cost of solving the classic discrete-logarithm problem.
I'm not talking about elliptic curves here:
I'm talking about discrete logs in the multiplicative group of a finite field.
I'll focus on the simple case of a prime field,
exactly the problem of finding a secret key given a public key in the original DH system.

In the literature on combining congruences,
there is a long history of speedup ideas being adapted from factorization algorithms
to "index calculus" discrete-log algorithms (and occasionally vice versa).
NFS is no exception.
A 1992 paper by Gordon explained how to use NFS to compute discrete logs mod a prime
in L^{2.080...+o(1)} "operations";
a 1993 paper by Schirokauer reduced 2.080... to 1.922...;
a 2003 paper by Matyukhin reduced 1.922... to 1.901...,
the same exponent as factorization.

Typical index-calculus algorithms have two stages.
The first stage takes the prime as input and computes discrete logs of many small numbers.
The second stage takes a target as an extra input and computes the discrete log of that target.
The usual picture before NFS was that
the second stage was much faster than the first stage.
For NFS this picture was restored in a 2003 paper by Joux and Lercier.
A 2006 paper by Commeine and Semaev improved and analyzed the Joux–Lercier idea,
obtaining exponent 1.901... for the prime and exponent just 1.442... for each target.
If the attacker is targeting a batch of L^{0.5+o(1)} discrete logs modulo the same prime
then the cost of each discrete log with the Commeine–Semaev algorithm
is just L^{1.442...+o(1)}, much faster than factorization.

Furthermore, the first stage handles the prime p very similarly to how NFS factors N, and it's easy to see that Coppersmith's precomputation ideas are also applicable here. Barbulescu's thesis in 2013 presented a discrete-log algorithm using the following number of "operations":

- L
^{2.006...+o(1)}for precomputation depending only on the rough size of p. - L
^{1.638...+o(1)}to compute discrete logs of many small numbers mod p. - L
^{1.231...+o(1)}, improving on the Commeine–Semaev algorithm, to compute the discrete log of a particular target.

The techniques in my 2014 "Batch NFS" paper with Lange, replacing "operations" with a realistic cost metric, should be combinable with the batch version of this algorithm. In other words, computing a batch of many discrete logs mod many primes should cost much less than handling each discrete log separately. Of course, as in the case of integer factorization, figuring out the costs for (say) 1024-bit primes is a harder problem than figuring out the asymptotic cost exponents.

**The Logjam attack.**
A paper this year
from Adrian, Bhargavan, Durumeric, Gaudry, Green, Halderman, Heninger, Springall,
Thomé, Valenta, VanderSloot, Wustrow, Zanella-Béguelin, and Zimmermann
reports an attack called "Logjam" allowing man-in-the-middle attackers
to "compromise connections to over 7% of Top Million HTTPS sites".
The attack combines several unnecessary features of HTTPS:

- Real-world clients often support non-elliptic "DHE" along with, or instead of, elliptic "ECDHE". It would be interesting to trace the reasons for this: for example, were people worried about ECC patents?
- DHE allows the server to choose primes as small as it wants: 1024 bits or even smaller. This is obviously a concession to efficiency.
- There's also a separate "DHE-EXPORT" option, limited to 512-bit primes and often supported by real-world servers. It's clear that this is attributable entirely to the U.S. government's efforts last century to impose weak cryptography upon the general public.
- Neither the client nor the server will notice if a man-in-the-middle attacker replaces the client's DHE request with DHE-EXPORT: the server and client will both send 512-bit ephemeral public keys. Gutmann has claimed that this is a downgrade attack against "broken implementations" and "bad programming", but it's actually a flaw in the TLS protocol specification.

The attacker now computes the server's 512-bit discrete log, and easily fools the client into thinking that the attacker is the server.

Obstacle: If computing a 512-bit discrete log takes too long, the client will give up on the connection, and the attack will fail. Grigg and Gutmann say that breaking a "dinky 512-bit key" will take "months" or "weeks"; maybe they were talking about RSA, but everyone says that 512-bit discrete-log keys are at least as hard to break as 512-bit RSA keys, right?

Solution: The Logjam paper reports 512-bit discrete-log computations "in about a minute", and it's clear that this can be improved with more hardware. "An adversary who performs a large precomputation for a prime p can then quickly calculate arbitrary discrete logs in that group, amortizing the cost over all targets that share this parameter," the paper says. "Although this fact is well known among mathematical cryptographers, it seems to have been lost among practitioners deploying cryptosystems."

There are clearly many different ways that the attack could have been prevented. I'll focus on three responses that involve changing the cryptographic primitives:

- In the server, generate (and regenerate as frequently as possible) site-specific DHE and DHE-EXPORT primes. In other words, blame the success of the attack on the fact that some widely distributed primes were reused for a huge number of connections.
- In the client, require a larger minimum size for DHE (and disable DHE-EXPORT). In other words, blame the success of the attack on the fact that the primes were too small.
- In the client and the server, disable DHE (and DHE-EXPORT) in favor of ECDHE. In other words, blame the success of the attack on the fact that people were using non-elliptic DH instead of elliptic DH.

From a security perspective, the first response misses the point. Yes, there's a batch attack against multiple discrete logs for the same prime, but there are also batch attacks against many other cryptographic systems. What we actually need to know is whether attacks are feasible; there's no substitute for quantitative analysis of attacks against particular systems. The superficial statement "primes are shared" is a poor predictor of "batch attacks exist", and "batch attacks exist" is a poor predictor of "attacks are feasible".

The second response sounds much more reasonable from a security perspective, but security ends up being compromised by performance problems. The Logjam paper found that 84% of servers in 2015 used a "1024-bit or smaller group", despite NIST's recommendation to move up to 2048-bit groups by 2010; if it weren't for performance problems then nobody would ever have been using 1024-bit groups in the first place.

The third response produces a reasonable level of security: ECDHE, as actually implemented, doesn't allow any small groups. Properly implementing ECDHE with NIST P-256 isn't easy, but all of the critical pitfalls here are eliminated by next-generation ECC. The performance of NIST P-256 is already acceptable for many sites, and I expect the performance of next-generation ECC to be acceptable for almost all sites.

One can mix and match responses: for example, use ECDHE with a server-specific elliptic curve. But it's better to use ECDHE with a larger shared curve. Compared to a server-specific curve, a larger shared curve saves bandwidth; in most situations ends up saving CPU time; and has larger benefits against all known attacks.

**Are we accurately simulating the attacker?**
Experts have publicly proposed and analyzed a huge number of ideas
for attacking the problem of computing an elliptic-curve discrete logarithm.
For most elliptic curves
these ideas have produced only marginal improvements:
the best attacks publicly known today against "conservative" elliptic curves
(NIST P-256, Curve25519, etc.)
aren't much better than the best attacks
that were publicly known at the dawn of ECC thirty years ago.
Is it safe to conclude that no much better algorithm exists,
and in particular that the attacker doesn't know a much better algorithm?

Maybe, maybe not. Maybe the public search for a better algorithm hasn't been sufficiently comprehensive, or it happened to wander off in the wrong directions. Maybe there's a much better algorithm lurking just beyond the public search, an algorithm that required too big a leap for anyone to find. Maybe the attacker has hired smarter people, or searched longer, or simply gotten lucky.

The same risk is even more obvious for some other famous problems. For example, the problem of factoring a given 1024-bit RSA modulus is an amazingly productive research topic with a long and continuing history of public improvements. Very few experts are willing to point to a factorization algorithm and say "I bet this factorization algorithm is within a factor 2 of the best possible".

Of course, we try hard to minimize the risk for cryptographic users. We focus efforts on a small number of cryptanalytic problems, and we advertise these problems to a huge community of algorithm designers. We aim for overkill in studying every conceivable solution to these problems. We track how thoroughly these problems have actually been studied, and we track the remaining prospects for improvement. We direct users towards the problems with the strongest track records.

But what happens if **the algorithmic problems facing the attackers
aren't actually the algorithmic problems we're studying**?
In particular, what happens if the attack problems
are *easier* than the problems we're studying?

Maybe we're starting from the wrong cost metric for algorithms. For example, maybe we're using the oversimplified "operations" cost metric that I mentioned before, counting random access to a large array as a single "operation" when in fact it's much more expensive than an arithmetic operation. (Sometimes people say that counting "operations" is an underestimate of actual cost and is therefore safe; but this is part of what went wrong with the Silverman predictions.) Maybe we're optimizing algorithms for mass-market easy-to-program Intel CPUs, while the attacker is optimizing for special-purpose chips with much more parallelism and with fancy networks on chip. Maybe the attacker has a quantum computer, making some operations feasible while we normally think of those operations as being insanely expensive.

Or maybe we're setting our cost limits too low. Maybe we're studying the problem of setting public factorization records using academic computer clusters, while the attacker is studying the problem of factoring 1024-bit keys as quickly as possible using billion-dollar clusters.

Or maybe,
in an effort to advertise attractively simple problems,
we've oversimplified the problem of attacking the actual cryptosystem.
Maybe we're studying the problem of recovering a signature key
while the attacker is studying the actual problem of forging signatures.
Maybe we're studying the problem of attacking *one* key
while the attacker is studying the actual problem of attacking a large *batch* of keys.
This oversimplification is surprisingly common,
and is exactly the reason for this blog post.