My next book will be “Practical Math for Programmers”

tl;dr: I’m writing a new book, sign up for the announcements mailing list.

I’ve written exactly zero new technical blog posts this year because I’ve been spending all my writing efforts on my next book, Practical Math for Programmers (PMFP, subtitle: A Tour of Mathematics in Production Software).

I’ve written a little bit about it in my newsletter, Halfspace. There I rant, critique, brainstorm, and wax poetic about math and software. This blog will remain dedicated to technical details. If you like my casual writing—the complementary half of this space, hence the name—go sign up for that.

The concept for Practical Math grew from an ongoing frustration with academic mathematicians and computer scientists overselling the work in their research papers as “widely applicable.” I ranted about this at length in the newsletter, but it has bothered me and wasted my time throughout my education and career.

So this book will assemble a large list of applications of interesting math used in bona-fide production systems. I hope the title makes this goal abundantly obvious. It will serve the following purposes, beyond satiating my spite.

  • It will be a gateway drug to mathematics for people who are not interested in math (yet).
  • It will inspire the reader to draw analogies to problems they face in their own work.
  • It will be a clear and unambiguous answer to the question “when will we ever use this?”, hopefully satisfying skeptical students.
  • It will be much easier than pimbook to sample and read casually. It could be a bathroom reader for the stalls of software firms.

To achieve these, the book will follow three principles. It won’t be super heavy on mathematical details. No proofs, nice diagrams, and limited notation. It will also contain working demonstrations of each production application. Python 3, with a GitHub repo, full test suite, and code documentation to back it. And finally, each application section (I’m tentatively calling them “Tips”) will be only 3-5 pages long. I plan to write 60-75 Tips. Add essays, and this is a 200-300 page book. I’ve currently written drafts of 16 with two more in progress.

Though there will be technical details, the Tips will read be more like a survey than a textbook. And to make it more appealing and interesting, I’ll focus on the story behind the problem that the math is solving, the evidence that it is used in production, and the references you can read if you want details. In particular, I will interview the practitioners who actually use or manage the software built around the math, or cover the history from primary sources when I can. To achieve working demonstrations for 60+ applications (~3 years of blogging for me), I will limit myself to demonstrations that can fit in under 1 page of Python code, I will simplify the scope of the demonstrations (within reason), and port/adapt reference implementations in open source software when they are available.

And as I would hope, readers who are enticed by mathematics from reading Practical Math might be convinced to journey through A Programmer’s Introduction to Mathematics.

As for the content, the Tips are still largely unstructured, though themes are showing up around the following topics that will probably evolve into Chapter divisions.

  • Measuring things. Either because the concept is fuzzy or approximations are required. Problems around attribution, problems in counting, detection, and budgeting.
  • Dealing with uncertainty. Either uncertainty in measurements or defining useful probabilistic models for decision making.
  • Safety. In terms of data corruption, privacy, redundancy, etc.
  • Efficiency. Techniques for improving efficiency of things in place (without system redesigns). Or situations in which an weird bottleneck shows up that math helps smooth over.

And then there are topics that don’t fit anywhere, like Perlin noise generation or market design techniques. Maybe I’ll call that chapter Potpourri. If you have suggestions for applications you think would fit well in the book, please reach out! Especially if you have experience using the ideas in prod.

In between chapters I plan to write essays, such as one on the ethics of applying math to automated computer systems, and one reflecting on the sorts of problems that engineers face that could be avoided with just a modest amount of math, one on the value of analogy in engineering and math, and an updated and more professional version of “What’s in Production.”

I hope to have a first draft done this year (though I’m not on track), and publish some time in 2023.

The Gadget Decomposition in FHE

Lately I’ve been studying Fully Homomorphic Encryption, which is the miraculous ability to perform arbitrary computations on encrypted data without learning any information about the underlying message. It’s the most comprehensive private computing solution that can exist (and it does exist!).

The first FHE scheme by Craig Gentry was based on ideal lattices and was considered very complex (I never took the time to learn how it worked). Some later schemes (GSW = Gentry-Sahai-Waters) are based on matrix multiplication, and are conceptually much simpler. Even more recent FHE schemes build on GSW or use it as a core subroutine.

All of these schemes inject random noise into the ciphertext, and each homomorphic operation increases noise. Once the noise gets too big, you can no longer decrypt the message, and so every now and then you must apply a process called “bootstrapping” that reduces noise. It also tends to be the performance bottleneck of any FHE scheme, and this bottleneck is why FHE is not considered practical yet.

To help reduce noise growth, many FHE schemes like GSW use a technical construction dubbed the gadget decomposition. Despite the terribly vague name, it’s a crucial limitation on noise growth. When it shows up in a paper, it’s usually remarked as “well known in the literature,” and the details you’d need to implement it are omitted. It’s one of those topics.

So I’ll provide some details. The code from this post is on GitHub.

Binary digit decomposition

To create an FHE scheme, you need to apply two homomorphic operations to ciphertexts: addition and multiplication. Most FHE schemes admit one of the two operations trivially. If the ciphertexts are numbers as in RSA, you multiply them as numbers and that multiplies the underlying messages, but addition is not known to be possible. If ciphertexts are vectors as in the “Learning With Errors” scheme (LWE)—the basis of many FHE schemes—you add them as vectors and that adds the underlying messages. (Here the “Error” in LWE is synonymous with “random noise”, I will use the term “noise”) In LWE and most FHE schemes, a ciphertext hides the underlying message by adding random noise, and addition of two ciphertexts adds the corresponding noise. After too many unmitigated additions, the noise will grow so large it obstructs the message. So you stop computing, or you apply a bootstrapping operation to reduce the noise.

Most FHE schemes also allow you to multiply a ciphertext by an unencrypted constant $ A$, but then the noise scales by a factor of $ A$, which is undesirable if $ A$ is large. So you either need to limit the coefficients of your linear combinations by some upper bound, or use a version of the gadget decomposition.

The simplest version of the gadget decomposition works like this. Instead of encrypting a message $ m \in \mathbb{Z}$, you would encrypt $ m, 2m, 4m, …, 2^{k-1} m$ for some choice of $ k$, and then to multiply $ A < 2^k$ you write the binary digits of $ A = \sum_{i=0}^{k-1} a_i 2^i$ and you compute $ \sum_{i=0}^{k-1} a_i \textup{Enc}(2^i m)$. If the noise in each encryption is $ E$, and summing ciphertexts sums noise, then this trick reduces the noise growth from $ O(AE)$ to $ O(kE) = O(\log(A)E)$, at the cost of tracking $ k$ ciphertexts. (Calling the noise $ E$ is a bit of an abuse—in reality the error is sampled from a random distribution—but hopefully you see my point).

Some folks call the mapping $ \textup{PowersOf2}(m) = m \cdot (2^0, 2^1, 2^2, \dots, 2^{k-1})$, and for the sake of this article let’s call the operation of writing a number $ A$ in terms of its binary digits $ \textup{Bin}(A) = (a_0, \dots, a_{k-1})$ (note, the first digit is the least-significant bit, i.e., it’s a little-endian representation). Then PowersOf2 and Bin expand an integer product into a dot product, while shifting powers of 2 from one side to the other.

$ \displaystyle A \cdot m = \langle \textup{Bin}(A), \textup{PowersOf2}(m) \rangle$

This inspired the following “proof by meme” that I can’t resist including.

Working out an example, if the message is $ m=7$ and $ A = 100, k=7$, then $ \textup{PowersOf2}(7) = (7, 14, 28, 56, 112, 224, 448, 896)$ and $ \textup{Bin}(A) = (0,0,1,0,0,1,1,0)$ (again, little-endian), and the dot product is

$ \displaystyle 28 \cdot 1 + 224 \cdot 1 + 448 \cdot 1 = 700 = 7 \cdot 2^2 + 7 \cdot 2^5 + 7 \cdot 2^6$

A generalized gadget construction

One can generalize the binary digit decomposition to different bases, or to vectors of messages instead of a single message, or to include a subset of the digits for varying approximations. I’ve been puzzling over an FHE scheme that does all three. In my search for clarity I came across a nice paper of Genise, Micciancio, and Polyakov called “Building an Efficient Lattice Gadget Toolkit: Subgaussian Sampling and More“, in which they state a nice general definition.

Definition: For any finite additive group $ A$, an $ A$-gadget of size $ w$ and quality $ \beta$ is a vector $ \mathbf{g} \in A^w$ such that any group element $ u \in A$ can be written as an integer combination $ u = \sum_{i=1}^w g_i x_i$ where $ \mathbf{x} = (x_1, \dots , x_w)$ has norm at most $ \beta$.

The main groups considered in my case are $ A = (\mathbb{Z}/q\mathbb{Z})^n$, where $ q$ is usually $ 2^{32}$ or $ 2^{64}$, i.e., unsigned int sizes on computers for which we get free modulus operations. In this case, a $ (\mathbb{Z}/q\mathbb{Z})^n$-gadget is a matrix $ G \in (\mathbb{Z}/q\mathbb{Z})^{n \times w}$, and the representation $ x \in \mathbb{Z}^w$ of $ u \in (\mathbb{Z}/q\mathbb{Z})^n$ satisfies $ Gx = u$.

Here $ n$ and $ q$ are fixed, and $ w, \beta$ are traded off to make the chosen gadget scheme more efficient (smaller $ w$) or better at reducing noise (smaller $ \beta$). An example of how this could work is shown in the next section by generalizing the binary digit decomposition to an arbitrary base $ B$. This allows you to use fewer digits to represent the number $ A$, but each digit may be as large as $ B$ and so the quality is $ \beta = O(B\sqrt{w})$.

One commonly-used construction is to convert an $ A$-gadget to an $ A^n$-gadget using the Kronecker product. Let $ g \in A^w$ be an $ A$-gadget of quality $ \beta$. Then the following matrix is an $ A^n$-gadget of size $ nw$ and quality $ \sqrt{n} \beta$:

$ \displaystyle G = I_n \otimes \mathbf{g}^\top = \begin{pmatrix} g_1 & \dots & g_w & & & & & & & \\ & & & g_1 & \dots & g_w & & & & \\ & & & & & & \ddots & & & \\ & & & & & & & g_1 & \dots & g_w \end{pmatrix}$

Blank spaces represent zeros, for clarity.

An example with $ A = (\mathbb{Z}/16\mathbb{Z})$. The $ A$-gadget is $ \mathbf{g} = (1,2,4,8)$. This has size $ 4 = \log(q)$ and quality $ \beta = 2 = \sqrt{1+1+1+1}$. Then for an $ A^3$-gadget, we construct

Now given a vector $ (15, 4, 7) \in \mathbb{A}^3$ we write it as follows, where each little-endian representation is concatenated into a single vector.

$ \displaystyle \mathbf{x} = \begin{pmatrix} 1\\1\\1\\1\\0\\0\\1\\0\\1\\1\\1\\0 \end{pmatrix}$

And finally,

To use the definition more rigorously, if we had to write the matrix above as a gadget “vector”, it would be in column order from left to right, $ \mathbf{g} = ((1,0,0), (2,0,0), \dots, (0,0,8)) \in A^{wn}$. Since the vector $ \mathbf{x}$ can be at worst all 1’s, its norm is at most $ \sqrt{12} = \sqrt{nw} = \sqrt{n} \beta = 2 \sqrt{3}$, as claimed above.

A signed representation in base B

As we’ve seen, the gadget decomposition trades reducing noise for a larger ciphertext size. With integers modulo $ q = 2^{32}$, this can be fine-tuned a bit more by using a larger base. Instead of PowersOf2 we could define PowersOfB, where $ B = 2^b$, such that $ B$ divides $ 2^{32}$. For example, with $ b = 8, B = 256$, we would only need to track 4 ciphertexts. And the gadget decomposition of the number we’re multiplying by would be the little-endian digits of its base-$ B$ representation. The cost here is that the maximum entry of the decomposed representation is 255.

We can fine tune this a little bit more by using a signed base-$ B$ representation. To my knowledge this is not the same thing as what computer programmers normally refer to as a signed integer, nor does it have anything to do with the two’s complement representation of negative numbers. Rather, instead of the normal base-$ B$ digits $ n_i \in \{ 0, 1, \dots, B-1 \}$ for a number $ N = \sum_{i=0}^k n_i B^i$, the signed representation chooses $ n_i \in \{ -B/2, -B/2 + 1, \dots, -1, 0, 1, \dots, B/2 – 1 \}$.

Computing the digits is slightly more involved, and it works by shifting large coefficients by $ -B/2$, and “absorbing” the impact of that shift into the next more significant digit. E.g., if $ B = 256$ and $ N = 2^{11} – 1$ (all 1s up to the 10th digit), then the unsigned little-endian base-$ B$ representation of $ N$ is $ (255, 7) = 255 + 7 \cdot 256$. The corresponding signed base-$ B$ representation subtracts $ B$ from the first digit, and adds 1 to the second digit, resulting in $ (-1, 8) = -1 + 8 \cdot 256$. This works in general because of the following “add zero” identity, where $ p$ and $ q$ are two successive unsigned digits in the unsigned base-$ B$ representation of a number.

$ \displaystyle \begin{aligned} pB^{k-1} + qB^k &= pB^{k-1} – B^k + qB^k + B^k \\ &= (p-B)B^{k-1} + (q+1)B^k \end{aligned}$

Then if $ q+1 \geq B/2$, you’d repeat and carry the 1 to the next higher coefficient.

The result of all this is that the maximum absolute value of a coefficient of the signed representation is halved from the unsigned representation, which reduces the noise growth at the cost of a slightly more complex representation (from an implementation standpoint). Another side effect is that the largest representable number is less than $ 2^{32}-1$. If you try to apply this algorithm to such a large number, the largest digit would need to be shifted, but there is no successor to carry to. Rather, if there are $ k$ digits in the unsigned base-$ B$ representation, the maximum number representable in the signed version has all digits set to $ B/2 – 1$. In our example with $ B=256$ and 32 bits, the largest digit is 127. The formula for the max representable integer is $ \sum_{i=0}^{k-1} (B/2 – 1) B^i = (B/2 – 1)\frac{B^k – 1}{B-1}$.

max_digit = base // 2 - 1
max_representable = (max_digit 
    * (base ** (num_bits // base_log) - 1) // (base - 1)

A simple python implementation computes the signed representation, with code copied below, in which $ B=2^b$ is the base, and $ b = \log_2(B)$ is base_log.

def signed_decomposition(
  x: int, base_log: int, total_num_bits=32) -> List[int]:
    result = []
    base = 1 << base_log
    digit_mask = (1 << base_log) - 1
    base_over_2_threshold = 1 << (base_log - 1)
    carry = 0

    for i in range(total_num_bits // base_log):
        unsigned_digit = (x >> (i * base_log)) & digit_mask
        if carry:
            unsigned_digit += carry
            carry = 0

        signed_digit = unsigned_digit
        if signed_digit >= base_over_2_threshold:
            signed_digit -= base
            carry = 1

    return result

In a future article I’d like to demonstrate the gadget decomposition in action in a practical setting called key switching, which allows one to convert an LWE ciphertext encrypted with key $ s_1$ into an LWE ciphertext encrypted with a different key $ s_2$. This operation increases noise, and so the gadget decomposition is used to reduce noise growth. Key switching is used in FHE because some operations (like bootstrapping) have the side effect of switching the encryption key.

Until then!

Group Actions and Hashing Unordered Multisets

I learned of a neat result due to Kevin Ventullo that uses group actions to study the structure of hash functions for unordered sets and multisets. This piqued my interest because a while back a colleague asked me if I could think of any applications of “pure” group theory to practical computer programming that were not cryptographic in nature. He meant, not including rings, fields, or vector spaces whose definitions happen to be groups when you forget the extra structure. I couldn’t think of any at the time, and years later Kevin has provided me with a tidy example. I took the liberty of rephrasing his argument to be a bit simpler (I hope), but I encourage you to read Kevin’s post to see the reasoning in its original form.

Hashes are useful in programming, and occasionally one wants to hash an unordered set or multiset in such a way that the hash does not depend on the order the elements were added to the set. Since collection types are usually generic, one often has to craft a hash function for a set<T> or multiset<T> that relies on a hash function hash(t) of an unknown type T. To make things more efficient, rather than requiring one to iterate over the entire set each time a hash is computed, one might seek out a hash function that can be incrementally updated as new elements are added, and provably does not depend on the order of insertion.

For example, having a starting hash of zero, and adding hashes of elements as they are added (modulo $ 2^{64}$) has this incremental order-ignorant property, because addition is commutative and sums can be grouped. XOR-ing the bits of the hashes is similar. However, both of these strategies have downsides.

For example, if you adopt the XOR strategy for a multiset hash, then any element that has an even quantity in the multiset will be the same as if it were not in the set at all (or if it were in the set with some other even quantity). This is because x XOR x == 0. On the other hand, if you use the addition approach, if an element hashes to zero, its inclusion in any set has no effect on the hash. In Java the integer hash is the identity, so zero would be undetectable as a member of a multiset of ints in any quantity. Less drastically, a multiset with all even counts of elements will always hash to a multiple of 2, and this makes it easier to find hash collisions.

A natural question arises: given the constraint that the hash function is accumulative and commutative, can we avoid such degenerate situations? In principle the answer should obviously be no, just by counting. I.e., the set of all unordered sets of 64-bit integers has size $ 2^{2^{64}}$, while the set of 64-bit hashes has size merely $ 2^{64}$. You will have many many hash collisions, and would need a much longer hash to avoid them in principle.

More than just “no,” we can characterize the structure of such hash functions. They impose an abelian group structure on the set of hashes. And due to the classification theorem of finite abelian groups, up to isomorphism (and for 64-bit hashes) that structure consists of addition on blocks of bits with various power-of-2 moduli, and the blocks are XOR’ed together at the end.

To give more detail, we need to review some group theory, write down the formal properties of an accumulative hash function, and prove the structure theorem.

Group actions, a reminder

This post will assume basic familiarity with group theory as covered previously on this blog. Basically, this introductory post defining groups and actions, and this followup post describing the classification theorem for commutative (abelian) groups. But I will quickly review group actions since they take center stage today.

A group $ G$ defines some abstract notion of symmetries in a way that’s invertible. But a group is really meaningless by itself. They’re only useful when they “act” upon a set. For example, a group of symmetries of the square acts upon the square to actually rotate its points. When you have multiple group structures to consider, it makes sense to more formally separate the group structure from the set.

So a group action is formally a triple of a group $ G$, a set $ X$, and a homomorphism $ f:G \to S_X$, where $ S_X$ is the permutation group (or symmetry group) of $ X$, i.e., the set of all bijections $ X \to X$. The permutation group of a set encompasses every possible group that can act on $ X$. In other words, every group is a subgroup of a permutation group. In our case, $ G$ and $ f$ define a subgroup of symmetries on $ X$ via the image of $ f$. If $ f$ is not injective, some of the structure in $ G$ is lost. The homomorphism determines which parts of $ G$ are kept and how they show up in the codomain. The first isomorphism theorem says how: $ G / \textup{ker} f \cong \textup{im} f$.

This relates to our hashing topic because an accumulative hash—and a nicely behaved hash, as we’ll make formal shortly—creates a group action on the set of possible hash values. The image of that group action is the “group structure” imposed by the hash function, and the accumulation function defines the group operation in that setting.

Multisets as a group, and nice hash functions

An appropriate generalization of multisets whose elements come from a base set $ X$ forms a group. This generalization views a multiset as a “counting function” $ T: X \to \mathbb{Z}$. The empty set is the function that assigns zero to each element. A positive value of $ k$ implies the entry shows up in the multiset $ k$ times. And a negative value is like membership “debt,” which is how we represent inverses, or equivalently set difference operations. The inverse of a multiset $ T$, denoted $ -T$, is the multiset with all counts negated elementwise. Since integer-valued functions generally form a group under point-wise addition, these multisets also do. Call this group $ \textup{MSet}(X)$. We will freely use the suggestive notation $ T \cup \{ x \} $ to denote the addition of $ T$ and the function that is 1 on $ x$ and 0 elsewhere. Similarly for $ T – \{ x \}$.

$ \textup{MSet}(X)$ is isomorphic to the free abelian group on $ X$ (because an instance of a multiset only has finitely many members). Now we can define a hash function with three pieces:

  • An arbitrary base hash function $ \textup{hash}: X \to \mathbb{Z} / 2^n \mathbb{Z}$.
  • An arbitrary hash accumulator $ \textup{h}: \mathbb{Z} / 2^n \mathbb{Z} \times \mathbb{Z} / 2^n \mathbb{Z} \to \mathbb{Z} / 2^n \mathbb{Z}$
  • A seed, i.e., a choice for the hash of the empty multiset $ s \in \mathbb{Z} / 2^n \mathbb{Z}$

With these three data we want to define a multiset hash function $ h^*: \textup{MSet}(X) \to \mathbb{Z} / 2^n \mathbb{Z}$ recursively as

  • $ h^*(\left \{ \right \}) = s$
  • $ h^*(T \cup \left \{ x \right \}) = h(h^*(T), \textup{hash}(x))$
  • $ h^*(T – \left \{ x \right \}) = \dots$

In order for the second bullet to lead to a well-defined hash, we need the property that the accumulation order of individual elements does not matter. Call a hash accumulator commutative if, for all $ a, b, c \in \mathbb{Z} / 2^n \mathbb{Z}$,

$ \displaystyle h(h(a,b), c) = h(h(a,c), b)$

This extends naturally to being able to reorder any sequence of hashes being accumulated.

The third is a bit more complicated. We need to be able to use the accumulator to “de-accumulate” the hash of an element $ x$, even when the set that gave rise to that hash didn’t have $ x$ in it to start.

Call a hash accumulator invertible if for a fixed hash $ z = \textup{hash}(x)$, the map $ a \mapsto h(a, z)$ is a bijection. That is, accumulating the hash $ z$ to two sets with different hashes under $ h^*$ will not cause a hash collision. This defines the existence of an inverse map (even if it’s not easily computable). This allows us to finish the third bullet point.

  • Fix $ z = \textup{hash}(x)$ and let $ g$ be the inverse of the map $ a \mapsto h(a, z)$. Then $ h^*(T – \left \{ x \right \}) = g(h^*(T))$

Though we don’t have a specific definition for the inverse above, we don’t need it because we’re just aiming to characterize the structure of this soon-to-emerge group action. Though, in all likelihood, if you implement a hash for a multiset, it should support incremental hash updates when removing elements, and that formula would apply here.

This gives us the well-definition of $ h^*$. Commutativity allows us to define $ h^*(T \cup S)$ by decomposing $ S$ arbitrarily into its constituent elements (with multiplicity), and applying $ h^*(T \cup \{ x \})$ or $ h^*(T – \{ x \})$ in any order.

A group action emerges

Now we have a well-defined hash function on a free abelian group.

$ \displaystyle h^*: \textup{MSet}(X) \to \mathbb{Z} / 2^n \mathbb{Z}$

However, $ h^*$ is not a homomorphism. There’s no reason hash accumulation needs to mesh well with addition of hashes. Instead, the family of operations “combine a hash with the hash of some fixed set” defines a group action on the set of hashes. Let’s suppose for simplicity that $ h^*$ is surjective, i.e., that every hash value can be achieved as the hash of some set. Kevin’s post gives the more nuanced case when this fails, and in that case you work within $ S_{\textup{im}(h^*)}$ instead of all of $ S_{\mathbb{Z} / 2^n \mathbb{Z}}$.

The group action is defined formally as a homomorphism

$ \displaystyle \varphi : \textup{MSet}(X) \to S_{\mathbb{Z} / 2^n \mathbb{Z}}$

where $ \varphi(T)$ is the permutation $ a \mapsto h(a, h^*(T))$. Equivalently, we start from $ a$, pick some set $ S$ with $ h^*(S) = a$, and output $ h^*(T \cup S)$.

The map $ \varphi$ is a homomorphism. The composition of two accumulations is order-independent because $ h$ is commutative. This is how we view $ h$ as “the binary operation” in $ \textup{im} \varphi$, because combining two permutations $ a \mapsto h(a, h^*(T))$ and $ a \mapsto h(a, h^*(S))$ is the permutation $ a \mapsto h(a, h^*(S \cup T))$.

And now we can apply the first isomorphism theorem, that

$ \displaystyle \textup{MSet}(X) / \textup{ker} \varphi \cong \textup{im} \varphi \subset S_{\mathbb{Z} / 2^n \mathbb{Z}}$

This is significant because any quotient of an abelian group is abelian, and this quotient is finite because $ S_{\mathbb{Z} / 2^n \mathbb{Z}}$ is finite. This means that the group $ \textup{im} \varphi$ is isomorphic to

$ \displaystyle \textup{im} \varphi \cong \bigoplus_{i=1}^k \mathbb{Z}/2^{n_i} \mathbb{Z}$

where $ n = \sum_i n_i$, and where the operation in each component is the usual addition modulo $ n_i$. The $ i$-th summand corresponds to a block of $ n_i$ bits of the hash, and within that block the operation is addition modulo $ 2^{n_i}$. Here the “block” structure is where XOR comes in. Each block can be viewed as a bitmask with zeros outside the block, and two members are XOR’ed together, which allows the operations to apply to each block independently.

For example, the group might be $ \mathbb{Z} / 2^{4} \mathbb{Z} \times \mathbb{Z} / 2^{26} \mathbb{Z}\times \mathbb{Z} / 2^{2} \mathbb{Z}$ for a 32-bit hash. The first block corresponds to 32-bit unsigned integers whose top 4 bits may be set but all other bits are zero. Addition is done within those four bits modulo 16, leaving the other bits unchanged. Likewise, the second component has the top four bits zero and the bottom two bits zero, but the remaining 26 bits are summed mod $ 2^{24}$. XOR combines the bits from different blocks.

In one extreme case, you only have one block, and your group is just $ \mathbb{Z} / 2^n \mathbb{Z}$ and the usual addition combines hashes. In the other extreme, each bit is its own block, your group is $ (\mathbb{Z} / 2 \mathbb{Z})^n$, the operation is a bitwise XOR.

Note, if instead of $ 2^n$ we used a hash of some other length $ m$, then in the direct sum decomposition above, $ m$ would be the product of the sizes of the components. The choice $ m = 2^n$ maximizes the number of different structures you can have.

Implications for hash function designers

Here’s the takeaway.

First, if you’re trying to design a hash function that avoids the degeneracies mentioned at the beginning of this article, then it will have to break one of the properties listed. This could happen, say, by maintaining additional state.

Second, if you’re resigned to use a commutative, invertible, accumulative hash, then you might as well make this forced structure explicit, and just pick the block structure you want to use in advance. Since no clever bit shifting will allow you to outrun this theorem, you might as well make it simple.

Until next time!

Carnival of Mathematics #197

Welcome to the 197th Carnival of Mathematics!

197 is an unseemly number, as you can tell by the Wikipedia page which currently says that it has “indiscriminate, excessive, or irrelevant examples.” How deviant. It’s also a Repfigit, which means if you start a fibonacci-type sequence with the digits 1, 9, 7, and then continue with $ a_n = a_{i-3} + a_{i-2} + a_{i-1}$, then 197 shows up in the sequence. Indeed: 1, 9, 7, 17, 33, 57, 107, 197, …

Untangling the unknot

Kennan Crane et al showcased a new paper that can untangle tangled curves quickly, and can do things like generate Hilbert-type space-filling curves on surfaces. It’s a long thread with tons of links to videos and reading materials, covering energy functions, functional analysis, Sobolev methods, and a custom inner product.

Folding equilateral triangles without measuring

Dave Richeson shows off a neat technique for folding equilateral triangles using just paper and no measurements. Replies in the thread show the geometric series that converges to the right 60 degree angle.

Shots fired at UMAP and t-SNE

Lior Pachter et al. study what sorts of structure are preserved by dimensionality reduction techniques like UMAP (which I have also used in a previous article) by comparing it against a genomics dataset with understood structure. They make some big claims about how UMAP and t-SNE destroy important structure, and they show how to contrive the dimensionality reduction plot to look like an elephant even when there’s no elephantine structure in the data.

I’m not expert, but perhaps one best case scenario for UMAP enthusiasts would be that their analysis only applies when you go from very high dimensions down to 2 just so you can plot a picture. But if you stop at, say, $ \sqrt{n}$ dimensions, you might still preserve a lot of the meaningful structure. Either way, they make a convincing pitch for Johnson-Lindenstrauss’s random linear reductions, which I’ve also covered here. Their paper is on biorXiv.

Studying the Sieve

Ben Peters Jones took up Grant Sanderson’s math video challenge and released a series of videos studying the Sieve of Eratosthenes.

Additional submissions

Be sure to submit fun math you find in September to the next carvinal host!