Simple unsolved math problem, 3

Posted on 2016/10/22 by wdjoyner

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, 1 + 2 + 3 = 6 implies 6 is a perfect number.

Unsolved Problem: Are there any odd perfect numbers?

The belief, by some, that there are none goes back over 500 years (wikipedia).

If you want to check out some recent research into this problem, see oddperfect.org.

b5527a1273f40d19e7fa821caa0208b9

(Another unsolved problem: Are there an infinite number of even perfect numbers?)

Simple unsolved math problem, 2

Posted on 2016/10/18 by wdjoyner

In 1911, Otto Toeplitz asked the following question.

Inscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square?

This question, also known as the square peg problem or the Toeplitz’ conjecture, is still unsolved in general. (It is known in lots of special cases.)

Inscribed square, by Claudio Rocchini

Thanks to Mark Meyerson (“Equilateral triangles and continuous curves”,Fundamenta Mathematicae, 1980) and others, the analog for triangles is true. For any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C. (In particular, the triangle can be equilateral.) The survey page by Mark J. Nielsen has more information on this problem.

Added 2016-11-23: See also this recent post by T. Tao.

Added 2020-07-01: This has apparently been solved by Joshua Greene and Andrew Lobb! See their ArXiV paper (https://arxiv.org/abs/2005.09193).

Simple unsolved math problem, 1

Posted on 2016/10/16 by wdjoyner

In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.

First, we define the function $f$ on the set of positive integers:

If the number $n$ is even, divide it by two: $f(n)=n/2$ .
If the number $n$ is odd, triple it and add one: $f(n)=3n+1$ .

In modular arithmetic notation, define the function $f$ as follows:
$f(n)= {n/2},\ if \ n\equiv 0 \pmod 2$ , and $f(n)= {3n+1},\ if \ n\equiv 1 \pmod 2$ . Believe it or not, this is the restriction to the positive integers of the complex-valued map $(2+7z-(2+5z)\cos(\pi z))/4$ .

The 3n+1 conjecture is: The sequence
$n,\ f(n),\ f^2(n)=f(f(n)),\ f^3(n)=f(f^2(n)),\ \dots$
will eventually reach the number 1, regardless of which positive integer $n$ is chosen initially.

This is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.

From xkdc by Randall Munroe – https://xkcd.com/710/

Hill verses Hamming

Posted on 2016/08/05 by wdjoyner

It’s easy to imagine the 19th century Philadelphia wool dealer Frank J. Primrose as a happy man. I envision him shearing sheep during the day, while in the evening he brings his wife flowers and plays games with his little children until bedtime. However, in 1887 Frank J. Primrose was not a happy man. This is because in June of that year, he had telegraphed his agent in Kansas instructions to buy a certain amount of wool. However, the telegraph operator made a single mistake in transmitting his message and Primrose unintentionally bought far more wool than he could possibly sell. Ordinarily, such a small error has little consequence, because errors can often be detected from the context of the message. However, this was an unusual case and the mistake cost him about a half-million dollars in today’s money. He promptly sued and his case eventually made its way to the Supreme Court. The famous 1894 United States Supreme Court case Primrose v. Western Union Telegraph Company decided that the telegraph company was not liable for the error in transmission of a message.

Thus was born the need for error-correcting codes.

Introduction

Lester Hill is most famously known for the Hill cipher, frequently taught in linear algebra courses today. We describe this cryptosystem in more detail in one of the sections below, but here is the rough idea. In this system, developed and published in the 1920’s, we take a $k\times k$ matrix K, composed of integers between 0 and 25, and encipher plaintext p by $p\longmapsto c=Kp$ , where the arithmetical operations are performed mod 26. Here K is the key, which should be known only to the sender and the intended receiver, and c is the ciphertext transmitted to the receiver.

On the other hand, Richard Hamming is known for the Hamming codes, also frequently taught in a linear algebra course. This will be describes in more detail in one of the sections below, be here is the basic idea. In this scheme, developed in the 1940’s, we take a $k\times k$ matrix G over a finite field F, constructed in a very particular way, and encode a message m by $m\longmapsto c=mG$ , where the arithmetical operations are performed in F. The matrix G is called the generator matrix and c is the codeword transmitted to the receiver.

Here, in a nutshell, is the mystery at the heart of this post.

These schemes of Hill and Hamming, while algebraically very similar, have quite different aims. One is intended for secure communication, the other for reliable communication. However, in an unpublished paper [H5], Hill developed a hybrid encryption/error-detection scheme, what we shall call “Hill codes” (described in more detail below).

Why wasn’t Hill’s result published and therefore Hill, more than Hamming, known as a pioneer of error-correcting codes?

Perhaps Hill himself hinted at the answer. In an overly optimistic statement, Hill wrote (italics mine):

Further problems connected with checking operations in finite fields will be treated in another paper. Machines may be devised to render almost quite automatic the evaluation of checking elements $c_1,\dots,c_q$ according to any proposed reference matrix of the general type described in Section 7, whatever the finite field in which the operations are effected. Such machines would enable us to dispense entirely with tables of any sort, and checks could be determined with great speed. But before checking machines could be seriously planned, the following problem — which is one, incidentally, of considerable interest from the standpoint of pure number theory — would require solution.

– Lester Hill, [H5]

By my interpretation, this suggests Hill wanted to answer the question below before moving on. As simple looking as it is, this problem is still, as far as I know, unsolved at the time of this writing.

Question 1 (Hill’s Problem):
Given k and q, find the largest r such that there exists a $k\times r$ van der Monde matrix with the property that every square submatrix is non-singular.

Indeed, this is closely related to the following related question from MacWilliams-Sloane [MS77], also still unsolved at this time. (Since Cauchy matrices do give a large family of matrices with the desired property, I’m guessing Hill was not aware of them.)

Question 2: Research Problem (11.1d)
Given k and q, find the largest r such that there exists a $k\times r$ matrix having entries in GF(q) with the property that every square submatrix is non-singular.

In this post, after brief biographies, an even more brief description of the Hill cipher and Hamming codes is given, with examples. Finally, we reference previous blog posts where the above-mentioned unpublished paper, in which Hill discovered error-correcting codes, is discussed in more detail.

Short biographies

Who is Hill? Recent short biographies have been published by C. Christensen and his co-authors. Modified slightly from [C14] and [CJT12] is the following information.

Lester Sanders Hill was born on January 19, 1890 in New York. He graduated from Columbia University in 1911 with a B. A. in Mathematics and earned his Master’s Degree in 1913. He taught mathematics for a few years at Montana University, then at Princeton University. He served in the United States Navy Reserves during World War I. After the WWI, he taught at the University of Maine and then at Yale, from which he earned his Ph.D. in mathematics in 1926. His Ph.D. advisor is not definitely known at this writing but I think a reasonable guess is Wallace Alvin Wilson.

In 1927, he accepted a position with the faculty of Hunter College in New York City, and he remained there, with one exception, until his resignation in 1960 due to illness. The one exception was for teaching at the G.I. University in Biarritz in 1946, during which time he may have been reactivated as a Naval Reserves officer. Hill died January 9, 1961.

Thanks to an interview that David Kahn had with Hill’s widow reported in [C14], we know that Hill loved to read detective stories, to tell jokes and, while not shy, enjoyed small gatherings as opposed to large parties.

Who is Hamming? His life is much better known and details can be readily found in several sources.

Richard Wesley Hamming was born on February 11, 1915, in Chicago. Hamming earned a B.S. in mathematics from the University of Chicago in 1937, a masters from the University of Nebraska in 1939, and a PhD in mathematics (with a thesis on differential equations)
from the University of Illinois at Urbana-Champaign in 1942. In April 1945 he joined the Manhattan Project at the Los Alamos Laboratory, then left to join the Bell Telephone Laboratories in 1946. In 1976, he retired from Bell Labs and moved to the Naval Postgraduate School in Monterey, California, where he worked as an Adjunct Professor
and senior lecturer in computer science until his death on January 7, 1998.

Hill’s cipher

The Hill cipher is a polygraphic cipher invented by Lester S. Hill in 1920’s. Hill and his colleague Wisner from Hunter College filed a patent for a telegraphic device encryption and error-detection device which was roughly based on ideas arising from the Hill cipher. It appears nothing concrete became of their efforts to market the device to the military, banks or the telegraph company (see Christensen, Joyner and Torres [CJT12] for more details). Incidently, Standage’s excellent book [St98] tells the amusing story of the telegraph company’s failed attempt to add a relatively simplistic error-detection to telegraph codes during that time period.

Some books state that the Hill cipher never saw any practical use in the real world. However, research by historians F. L. Bauer and David Kahn uncovered the fact that the Hill cipher saw some use during World War II encrypting three-letter groups of radio call signs [C14]. Perhaps insignificant, at least compared to the practical value of Hamming codes, none-the-less, it was a real-world use.

The following discussion assumes an elementary knowledge of matrices. First, each letter is first encoded as a number, namely

$A \leftrightarrow 0, B \leftrightarrow 1, \dots, Z \leftrightarrow 25$ . The subset of the integers $\{0, 1, \dots , 25\}$ will be denoted by Z/26Z. This is closed under addition and multiplication (mod 26), and sums and products (mod 26) satisfy the usual associative and distributive properties. For R = Z/26Z, let GL(k,R) denote the set of invertible matrix transformations $T:R^k\to R^k$ (that is, one-to-one and onto linear functions).

The construction

Suppose your message m consists of n capital letters, with no spaces. This may be regarded an n-tuple M with elements in R = Z/26Z. Identify the message M as a sequence of column vectors ${\bf p}\in R^k$ . A key in the Hill cipher is a $k\times k$ matrix K, all of whose entries are in R, such that the matrix K is invertible. It is important to keep K and k secret.

The encryption is performed by computing ${\bf c} = K{\bf p},$ and rewriting the resulting vector as a string over the same alphabet. Decryption is performed similarly by computing ${\bf p} = K^{-1} {\bf c}.$ .

Example 1: Suppose m is the message “BWGN”. Transcoding into numbers, the plaintext is rewritten $p_0=1, p_1=22, p_2=6, p_3=13$ . Suppose the key is
$K=\left(\begin{array}{rr} 1 & 3 \\ 5 & 12 \end{array}\right).$
Using Hill’s encryption above gives $c_0=7,c_1=3,c_2=24,c_3=3$ . (Verification is left to the reader as an exercise.)

Security concerns: For example, this cipher is linear and can be broken by a known plaintext attack.

Hamming codes

Richard Hamming is a pioneer of coding theory, introducing the binary
Hamming codes in the late 1940’s. In the days when an computer error could crash the computer and force the programmer to retype his punch cards, Hamming, out of frustration, designed a system whereby the computer could automatically correct certain errors. The family of codes named after him can easily correct one error.

Hill’s unpublished paper

While he was a student at Yale, Hill published three papers in Telegraph and Telephone Age [H1], [H2], [H3]. In these papers Hill described a mathematical method for checking the accuracy of telegraph communications. There is some overlap with these papers and [H5], so it seems likely to me that Hill’s unpublished paper [H5] dates from this time (that is, during his later years at Yale or early years at Hunter).

In [H5], Hill describes a family of linear block codes over a finite field and an algorithm for error-detection (which can be easily extended to error-correction). In it, he states the construction of what I’ll call the “Hill codes,” (defined below), gives numerous computational examples, and concludes by recording Hill’s Problem (stated above as Question 1). It is quite possibly Hill’s best work.

Here is how Hill describes his set-up.

Our problem is to provide convenient and practical accuracy checks upon
a sequence of n elements $f_1, f_2, \dots, f_r$ in a finite algebraic
field F. We send, in place of the simple sequence $f_1, f_2, \dots, f_r$ , the amplified sequence $f_1, f_2, \dots, f_r, c_1, c_2, \dots, c_k$
consisting of the “operand” sequence and the “checking” sequence.

– Lester Hill, [H5]

Then Hill continues as follows. Let F=GF(p) denote the finite field having p elements, where $p>2$ is a prime number. The checking sequence contains k elements of F as follows:
$c_j = \sum_{i=1}^r a_{i}^jf_i,$
for $j = 1, 2, \dots, k$ . The checks are to be determined by means of a
fixed matrix
$A = \left( \begin{array}{cccc} a_{1} & a_{2} & \dots & a_{r} \\ a_{1}^2 & a_{2}^2 & \dots & a_{r}^2 \\ \vdots & & & \vdots \\ a_{1}^k & a_{2}^k & \dots & a_{r}^k \\ \end{array} \right)$
of elements of F, the matrix having been constructed according to the criteria in Hill’s Problem above. In other words, if the operand sequence (i.e., the message) is the vector ${\bf f} = (f_1, f_2, \dots, f_r)$ , then the amplified sequence (or codeword in the Hill code) to be transmitted is

${\bf c} = {\bf f}G,$
where $G = \left( I_r, A \right)$ and where $I_r$ denotes the
$r\times r$ identity matrix. The Hill code is the row space of G.

We conclude with one more open question.

Question 3:
What is the minimum distance of a Hill code?

The minimum distance of any Hamming code is 3.

Do all sufficiently long Hill codes have minimum distance greater than 3?

Summary

Most books today (for example, the excellent MAA publication written by Thompson [T83]) date the origins of the theory of error-correcting codes to the late 1940s, due to Richard Hamming. However, this paper argues that the actual birth is in the 1920s due to Lester Hill. Topics discussed include why Hill’s discoveries weren’t publicly known until relatively recently, what Hill actually did that trumps Hamming, and some open (mathematical) questions connected with Hill’s work.

For more details, see these previous blog posts.

Acknowledgements: Many thanks to Chris Christensen and Alexander Barg for
helpful and encouraging conversations. I’d like to explicitly credit Chris Christensen, as well as historian David Kahn, for the original discoveries of the source material.

Bibliography

[C14] C. Christensen, Lester Hill revisited, Cryptologia 38(2014)293-332.

[CJT12] ——, D. Joyner and J. Torres, Lester Hill’s error-detecting codes, Cryptologia 36(2012)88-103.

[H1] L. Hill, A novel checking method for telegraphic sequences, Telegraph and
Telephone Age (October 1, 1926), 456 – 460.

[H2] ——, The role of prime numbers in the checking of telegraphic communications, I, Telegraph and Telephone Age (April 1, 1927), 151 – 154.

[H3] ——, The role of prime numbers in the checking of telegraphic
communications, II, Telegraph and Telephone Age (July, 16, 1927), 323 – 324.

[H4] ——, Lester S. Hill to Lloyd B. Wilson, November 21, 1925. Letter.

[H5] ——, Checking the accuracy of transmittal of telegraphic communications by means of operations in finite algebraic fields, undated and unpublished notes, 40 pages.
(hill-error-checking-notes-unpublished)

[MS77] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.

[Sh] A. Shokrollahi, On cyclic MDS codes, in Coding Theory and Cryptography: From Enigma and Geheimschreiber to Quantum Theory, (ed. D. Joyner), Springer-Verlag, 2000.

[St98] T. Standage, The Victorian Internet, Walker & Company, 1998.

[T83] T. Thompson, From Error-Correcting Codes Through Sphere Packings to Simple Groups, Mathematical Association of America, 1983.

Splitting fields of representations of generalized symmetric groups, 6

Posted on 2016/01/06 by wdjoyner

This post shall list some properties of the Schur index $m_F(G)$ in the case where $G = S_n\ wr\ C_\ell$ is a generalized symmetric group and $F$ is either the reals or rationals.

Let $\eta_k(z)=z^k$ , for $z\in C_\ell$ , $1\leq k\leq \ell$ .

Theorem: Let $G = S_n\ wr\ C_\ell$ . Let $\mu=(\eta_{e_1},...,\eta_{e_n})\in (C_\ell^n)^*$ , for some $e_j\in \{0,...,\ell-1\}$ , and let $\rho\in (S_n)_\mu^*$ . Let
$\chi$ denotes the character of $\theta_{\mu,\rho}$ .

Suppose that one of the following conditions holds:
1. $4|\ell$ and $\overline{e_1+...+e_n}$ divides $\overline{\ell/4}$ in ${\mathbb{Z}}/\ell {\mathbb{Z}}$ , or
2. $(e_1+...+e_n,\ell)=1$ ,
Then $m_{\Bbb{Q}}(\chi)=1$ .
Suppose that one of the following conditions holds:
1. $(n,\ell)=1$ , $4|\ell$ , and $(e_1+...+e_n)x\equiv \ell /4\ ({\rm mod}\ \ell)$ is not solvable, or
2. $(n,\ell)=1$ and $(e_1+...+e_n,\ell)>1$ .
Then $m_{\mathbb{Q}}(\chi\eta_1)=1$ .

This theorem is proven in this paper. Benard has shown that $m_{\mathbb{Q}}(\chi)=1$ , for all $\chi$ as in the above theorem.

Since the Schur index over ${\mathbb{Q}}$ of any irreducible character $\chi$ of a generalized symmetric group $G$ is equal to $1$ , each such character is associated to a representation $\pi$ all of whose matrix coefficients belong to the splitting field ${\mathbb{Q}}(\chi)$ .

What is the splitting field ${\mathbb{Q}}(\chi)$ , for $\chi\in G^*$ ?

This will be addressed in the next post.

Splitting fields of representations of generalized symmetric groups, 5

Posted on 2016/01/03 by wdjoyner

It is a result of Benard (Schur indices and splitting fields of the unitary reflection groups, J. Algebra, 1976) that the Schur index over ${\mathbb{Q}}$ of any irreducible character of a generalized symmetric group is equal to $1$ . This post recalls, for the sake of comparison with the literature, other results known about the Schur index in this case.

Suppose that $G$ is a finite group and $\pi \in G^*$ is an irreducible representation of $G$ , $\pi :G\rightarrow Aut(V)$ , for some complex vector space $V$ . We say that $\pi$ may be realized over a subfield $F\subset {\mathbb{C}}$ if there is an $F$ -vector space $V_0$ and an action of $G$ on $V_0$ such that $V$ and ${\mathbb{C}}\otimes V_0$ are equivalent representations of $G$ , where $G$ acts on ${\mathbb{C}}\otimes V_0$ by “extending scalars” in $V_0$ from $F$ to ${\mathbb{C}}$ . Such a representation is called an $F$ -representation. In other words, $\pi$ is an $F$ -representation provided it is equivalent to a representation which can be written down explicitly using matrices with entries in $F$ .

Suppose that the character $\chi$ of $\pi$ has the property that

$\chi(g)\in F, \ \ \ \ \ \ \forall g\in G,$

for some subfield $F\subset {\mathbb{C}}$ independent of $g$ . It is unfortunately true that, in general, $\pi$ is not necessarily an $F$ -representation. However, what is remarkable is that, for some $m\geq 1$ , there are $m$ representations, $\pi_1,...,\pi_m$ , all equivalent to $\pi$ , such that $\pi_1\oplus ...\oplus \pi_m$ is an $F$ -representation. The precise theorem is the following remarkable fact.

Theorem: (Schur) Let $\chi$ be an irreducible character and let $F$ be any field containing the values of $\chi$ . There is an integer $m \geq 1$ such that $m\chi$ is the character of an $F$ -representation.

The smallest $m\geq 1$ in the above theorem is called the Schur index and denoted $m_F(\chi)$ .

Next, we introduce some notation:

let ${\mathbb{R}}(\pi) = {\mathbb{R}}(\chi)$ denote the extension field of ${\mathbb{R}}$ obtained by adjoining all the values of $\chi(g)$ \ ($g\in G$), where $\chi$ is the character of $\pi$ ,
let $\nu(\pi) = \nu(\chi)$ denote the Frobenius-Schur indicator of $\pi$ (so $\nu(\pi)= {1\over |G|}\sum_{g\in G} \chi(g^2)$ ),
let $m_{\mathbb{R}}(\pi) = m_{\mathbb{R}}(\chi)$ denote the Schur multiplier of $\pi$ (by definition, the smallest integer $m\geq 1$ such that $m\chi$ can be realized over ${\mathbb{R}}$ (this integer exists, by the above-mentioned theorem of Schur).

The following result shows how the Schur index behaves under induction (see Proposition 14.1.8 in G. Karpilovsky,
Group representations, vol. 3, 1994).

Proposition: Let $\chi$ be an irreducible character of $G$ and let $\psi$ denote an irreducible character of a subgroup $H$ of $G$ . If $= 1$ then $m_{\Bbb{R}}(\chi)$ divides $m_{\Bbb{R}}(\psi)$ .

A future post shall list some properties of the Schur index in the case where $G$ is a generalized symmetric group and $F$ is either the reals or rationals.

Remarks on mathematical research, according to Ira Glass

Posted on 2015/04/09 by wdjoyner

Ira Glass, of This American Life (http://www.thisamericanlife.org/), did an interview where he talked at length about writing news stories. They are here (links are to short youtube videos):

Ira Glass on Storytelling, part 1 of 4
Ira Glass on Storytelling, part 2 of 4
Ira Glass on Storytelling, part 3 of 4
Ira Glass on Storytelling, part 4 of 4

I thought a lot of what he said was based on general principles which applied to mathematical research as well. Here is perhaps what he would have said if he was talking about mathematics, sometimes with direct quotes from his interview:

There are two building blocks to a idea for a paper

The problem or question. This is sometimes an issue in the intersection of two fields or a question of why some object of interest behaves the way you think it does, based on an example you know.
The revelation. This might be a key example or technique that will hopefully reveal the answer to your question.

You can have a great question, but if they don’t turn out to have any useful techniques or examples to work with, your idea is uninteresting. Conversely you can have a significant revelation with a fantastically powerful method, but if the problem or examples themselves are uninteresting, again you’ve got a weak idea.

You have to set aside just as much time looking for good ideas as you do producing them. In other words, the work of thinking up a good idea to write about is as much work and time as writing it up.

Not enough gets said about the importance of abandoning crap.

Most of your research ideas are going to be crap. That’s okay because the only way you can surface great ideas is by going through a lot of crappy ones. The only reason you want to be doing this is to make something memorable and special.

“The thing I’d like to say to you with all my heart is that most everybody I know who does interesting creative work went through a phase of years where with their good taste, they could tell what they were doing wasn’t as good as they wanted it to be … it didn’t have that special thing they wanted it to have … Everybody goes through that phase … and the most important thing you can do is do a lot of work.”

Ira Glass.

Boolean functions from the graph-theoretic perspective

Posted on 2012/09/08 by wdjoyner

This is a very short introductory survey of graph-theoretic properties of Boolean functions.

I don’t know who first studied Boolean functions for their own sake. However, the study of Boolean functions from the graph-theoretic perspective originated in Anna Bernasconi‘s thesis. More detailed presentation of the material can be found in various places. For example, Bernasconi’s thesis (e.g., see [BC]), the nice paper by P. Stanica (e.g., see [S], or his book with T. Cusick), or even my paper with Celerier, Melles and Phillips (e.g., see [CJMP], from which much of this material is literally copied).

For a given positive integer $n$ , we may identify a Boolean function

$f:GF(2)^n\to GF(2),$
with its support

$\Omega_f = \{x\in GF(2)^n\ |\ f(x)=1\}.$

For each $S\subset GF(2)^n$ , let $\overline{S}$ denote the set of complements $\overline{x}=x+(1,\dots,1)\in GF(2)^n$ , for $x\in S$ , and let $\overline{f}=f+1$ denote the complementary Boolean function. Note that

$\Omega_f^c=\Omega_{\overline{f}},$

where $S^c$ denotes the complement of $S$ in $GF(2)^n$ . Let

$\omega=\omega_f=|\Omega_f|$

denote the cardinality of the support. We call a Boolean function even (resp., odd) if $\omega_f$ is even (resp., odd). We may identify a vector in $GF(2)^n$ with its support, or, if it is more convenient, with the corresponding integer in $\{0,1, \dots, 2^n-1\}.$ Let

$b:\{0,1, \dots, 2^n-1\} \to GF(2)^n$

be the binary representation ordered with least significant bit last (so that, for example, $b(1)=(0,\dots, 0, 1)\in GF(2)^n$ ).

Let $H_n$ denote the $2^n\times 2^n$ Hadamard matrix defined by $(H_n)_{i,j} = (-1)^{b(i)\cdot b(j)}$ , for each $i,j$ such that $0\leq i,j\leq n-1$ . Inductively, these can be defined by

$H_1 = \left( \begin{array}{cc} 1 & 1\\ 1 & -1 \\ \end{array} \right), \ \ \ \ \ \ H_n = \left( \begin{array}{cc} H_{n-1} & H_{n-1}\\ H_{n-1} & -H_{n-1} \\ \end{array} \right), \ \ \ \ \ n>1.$
The Walsh-Hadamard transform of $f$ is defined to be the vector in ${\mathbb{R}}^{2^n}$ whose $k$ th component is

$({\mathcal{H}} f)(k) = \sum_{i \in \{0,1,\ldots,2^n-1\}}(-1)^{b(i) \cdot b(k) + f(b(i))} = (H_n (-1)^f)_k,$

where we define $(-1)^f$ as the column vector where the $i$ th component is

$(-1)^f_i = (-1)^{f(b(i))},$

for $i = 0,\ldots,2^n-1$ .

Example
A Boolean function of three variables cannot be bent. Let $f$ be defined by:

$\begin{array}{c|cccccccc} x_2 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ x_1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ x_0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ \hline (-1)^f & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ {\mathcal{H}}f & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$
This is simply the function $f(x_0,x_1,x_2)=x_0$ . It is even because

$\Omega_f = \{ (0,0,1), (0,1,1), (1,0,1), (1,1,1) \},\ \mbox{ so } \ \omega = 4.$

Here is some Sage code verifying this:

sage: from sage.crypto.boolean_function import *
sage: f = BooleanFunction([0,1,0,1,0,1,0,1])
sage: f.algebraic_normal_form()
x0
sage: f.walsh_hadamard_transform()
(0, -8, 0, 0, 0, 0, 0, 0)

(The Sage method walsh_hadamard_transform is off by a sign from the definition we gave.) We will return to this example later.

Let $X=(V,E)$ be the Cayley graph of $f$ :

$V = GF(2)^n,\ \ \ \ E = \{(v,w)\in V\times V\ |\ f(v+w)=1\}.$
We shall assume throughout and without further mention that $f(0)\not=1,$ so $X$ has no loops. In this case, $X$ is an $\omega$ -regular graph having $r$ connected components, where

$r = |GF(2)^n/{\rm Span}(\Omega_f)|.$

For each vertex $v\in V$ , the set of neighbors $N(v)$ of $v$ is given by

$N(v)=v+\Omega_f,$

where $v$ is regarded as a vector and the addition is induced by the usual vector addition in $GF(2)^n$ . Let $A = (A_{ij})$ be the $2^n\times 2^n$ adjacency matrix of $X$ , so

$A_{ij} = f(b(i)+b(j)), \ \ \ \ \ 0\leq i,j\leq 2^n-1.$

Example:
Returning to the previous example, we construct its Cayley graph.

First, attach afsr.sage from [C] in your Sage session.

     sage: flist = [0,1,0,1,0,1,0,1]
     sage: V = GF(2)ˆ3
     sage: Vlist = V.list()
     sage: f = lambda x: GF(2)(flist[Vlist.index(x)])
     sage: X = boolean_cayley_graph(f, 3)
     sage: X.adjacency_matrix()
     [0 1 0 1 0 1 0 1]
     [1 0 1 0 1 0 1 0]
     [0 1 0 1 0 1 0 1]
     [1 0 1 0 1 0 1 0]
     [0 1 0 1 0 1 0 1]
     [1 0 1 0 1 0 1 0]
     [0 1 0 1 0 1 0 1]
     [1 0 1 0 1 0 1 0]
     sage: X.spectrum()
     [4, 0, 0, 0, 0, 0, 0, -4]
     sage: X.show(layout="circular")

In her thesis, Bernasconi found a relationship between the spectrum of the Cayley graph $X$ ,

${\rm Spectrum}(X) = \{\lambda_k\ |\ 0\leq k\leq 2^n-1\},$

(the eigenvalues $\lambda_k$ of the adjacency matrix $A$ ) to the Walsh-Hadamard transform $\mathcal H f = H_n (-1)^f$ . Note that $f$ and $(-1)^f$ are related by the equation $f=\frac 1 2 (e - (-1)^f),$ where $e=(1,1,...,1)$ . She discovered the relationship

$\lambda_k = \frac 1 2 (H_n e - \mathcal H f)_k$

between the spectrum of the Cayley graph $X$ of a Boolean function and the values of the Walsh-Hadamard transform of the function. Therefore, the spectrum of $X$ , is explicitly computable as an expression in terms of $f$ .

References:

[BC] A. Bernasconi and B. Codenotti, Spectral analysis of Boolean functions as a graph eigenvalue problem, IEEE Trans. Computers 48(1999)345-351.

[CJMP] Charles Celerier, David Joyner, Caroline Melles, David Phillips, On the Hadamard transform of monotone Boolean functions, Tbilisi Mathematical Journal, Volume 5, Issue 2 (2012), 19-35.

[S] P. Stanica, Graph eigenvalues and Walsh spectrum of Boolean functions, Integers 7(2007)\# A32, 12 pages.

Here’s an excellent video of Pante Stanica on interesting applications of Boolean functions to cryptography (30 minutes):

Some favorite quotes on math, science, learning

Posted on 2008/04/21 by wdjoyner

Here is a collection of some favorite quotes from scientists and writers. For more, see this post.

There are some things which cannot be learned quickly,
and time, which is all we have,
must be paid heavily for their acquiring.
They are the very simplest things,
and because it takes a man’s life to know them
the little new that each man gets from life
is very costly and the only heritage he has to leave.
Ernest Hemingway
(From A. E. Hotchner, Papa Hemingway, Random House, NY, 1966)

I believe that a scientist looking at nonscientific problems is just as dumb as the next guy.
Richard Feynman

The best thing for being sad is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honor trampled in the sewers of baser minds. There is only one thing for it then to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting.
T. H. White in The Once and Future King

Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold. Leo Tolstoy

Education is what survives when what has been learnt has been forgotten.
B. F. Skinner

The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.
Norbert Wiener, in Ex-Prodigy: My Childhood and Youth

Science is a differential equation. Religion is a boundary condition.
Alan Turing

Everything is vague to a degree you do not realize till you have tried to make it precise.
Bertrand Russell

For every complicated problem there is a solution that is simple, direct, understandable, and wrong.
H. L. Mencken

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
John Louis von Neumann

To be what we are, and to become what we are capable of becoming, is the only end in life.
Baruch Spinoza

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