“Practical Math” Preview: Collect Sensitive Survey Responses Privately

This is a draft of a chapter from my in-progress book, Practical Math for Programmers: A Tour of Mathematics in Production Software.

Tip: Determine an aggregate statistic about a sensitive question, when survey respondents do not trust that their responses will be kept secret.


import random

def respond_privately(true_answer: bool) -> bool:
    '''Respond to a survey with plausible deniability about your answer.'''
    be_honest = random.random() < 0.5
    random_answer = random.random() < 0.5
    return true_answer if be_honest else random_answer

def aggregate_responses(responses: List[bool]) -> Tuple[float, float]:
    '''Return the estimated fraction of survey respondents that have a truthful
    Yes answer to the survey question.
    yes_response_count = sum(responses)
    n = len(responses)
    mean = 2 * yes_response_count / n - 0.5
    # Use n-1 when estimating variance, as per Bessel's correction.
    variance = 3 / (4 * (n - 1))
    return (mean, variance)

In the late 1960’s, most abortions were illegal in the United States. Daniel G. Horvitz, a statistician at The Research Triangle Institute in North Carolina and a leader in survey design for social sciences, was tasked with estimating how many women in North Carolina were receiving illegal abortions. The goal was to inform state and federal policymakers about the statistics around abortions, many of which were unreported, even when done legally.

The obstacles were obvious. As Horvitz put it, “a prudent woman would not divulge to a stranger the fact that she was party to a crime for which she could be prosecuted.” [Abernathy70] This resulted in a strong bias in survey responses. Similar issues had plagued surveys of illegal activity of all kinds, including drug abuse and violent crime. Lack of awareness into basic statistics about illegal behavior led to a variety of misconceptions, such as that abortions were not frequently sought out.

Horvitz worked with biostatisticians James Abernathy and Bernard Greenberg to test out a new method to overcome this obstacle, without violating the respondent’s privacy or ability to plausibly deny illegal behavior. The method, called randomized response, was invented by Stanley Warner in 1965, just a few years earlier. [Warner65] Warner’s method was a bit different from what we present in this Tip, but both Warner’s method and the code sample above use the same strategy of adding randomization to the survey.

The mechanism, as presented in the code above, requires respondents to start by flipping a coin. If heads, they answer the sensitive question truthfully. If tails, they flip a second coin to determine how to answer the question—heads resulting in a “yes” answer, tails in a “no” answer. Naturally, the coin flips are private and controlled by the respondent. And so if a respondent answers “Yes” to the question, they may plausibly claim the “Yes” was determined by the coin, preserving their privacy. The figure below describes this process as a diagram.

A branching diagram showing the process a survey respondent takes to record their response.

Another way to describe the outcome is to say that each respondent’s answer is a single bit of information that is flipped with probability 1/4. This is half way between two extremes on the privacy/accuracy tradeoff curve. The first extreme is a “perfectly honest” response, where the bit is never flipped and all information is preserved. The second extreme has the bit flipped with probability 1/2, which is equivalent to ignoring the question and choosing your answer completely at random, losing all information in the aggregate responses. In this perspective, the aggregate survey responses can be thought of as a digital signal, and the privacy mechanism adds noise to that signal.

It remains to determine how to recover the aggregate signal from these noisy responses. In other words, the surveyor cannot know any individual’s true answer, but they can, with some extra work, estimate statistics about the underlying population by correcting for the statistical bias. This is possible because the randomization is well understood. The expected fraction of “Yes” answers can be written as a function of the true fraction of “Yes” answers, and hence the true fraction can be solved for. In this case, where the random coin is fair, that formula is as follows (where \mathbf{P} stands for “the probability of”).

\displaystyle \mathbf{P}(\textup{Yes answer}) = \frac{1}{2} \mathbf{P}(\textup{Truthful yes answer}) + \frac{1}{4}

And so we solve for \mathbf{P}(\textup{Truthful yes answer})

\displaystyle \mathbf{P}(\textup{Truthful yes answer}) = 2 \mathbf{P}(\textup{Yes answer}) - \frac{1}{2}

We can replace the true probability \mathbf{P}(\textup{Yes answer}) above with our fraction of “Yes” responses from the survey, and the result is an estimate \hat{p} of \mathbf{P}(\textup{Truthful yes answer}). This estimate is unbiased, but has additional variance—beyond the usual variance caused by picking a finite random sample from the population of interest—introduced by the randomization mechanism.

With a bit of effort, one can calculate that the variance of the estimate is

\displaystyle \textup{Var}(\hat{p}) = \frac{3}{4n}

And via Chebyshev’s inequality, which bounds the likelihood that an estimator is far away from its expectation, we can craft a confidence interval and determine the needed sample sizes. Specifically, the estimate \hat{p} has additive error at most q with probability at most \textup{Var}(\hat{p}) / q^2. This implies that for a confidence of 1-c, one requires at least n \geq 3 / (4 c q^2) samples. For example, to achieve error 0.01 with 90 percent confidence (c=0.1), one requires 7,500 responses.

Horvitz’s randomization mechanism didn’t use coin flips. Instead they used an opaque box with red or blue colored balls which the respondent, who was in the same room as the surveyor, would shake and privately reveal a random color through a small window facing away from the surveyor. The statistical principle is the same. Horvitz and his associates surveyed the women about their opinions of the privacy protections of this mechanism. When asked whether their friends would answer a direct question about abortion honestly, over 80% either believed their friends would lie, or were unsure. [footnote: A common trick in survey methodology when asking someone if they would be dishonest is to instead ask if their friends would be dishonest. This tends to elicit more honesty, because people are less likely to uphold a false perception of the moral integrity of others, and people also don’t realize that their opinion of their friends correlates with their own personal behavior and attitudes. In other words, liars don’t admit to lying, but they think lying is much more common than it really is.] But 60% were convinced there was no trick involved in the randomization, while 20% were unsure and 20% thought there was a trick. This suggests many people were convinced that Horvitz’s randomization mechanism provided the needed safety guarantees to answer honestly.

Horvitz’s survey was a resounding success, both for randomized response as a method and for measuring abortion prevalence. [Abernathy70] They estimated the abortion rate at about 22 per 100 conceptions, with a distinct racial bias—minorities were twice as likely as whites to receive an abortion. Comparing their findings to a prior nationwide study from 1955—the so-called Arden House estimate—which gave a range of between 200,000 and 1.2 million abortions per year, Horvitz’s team estimated more precisely that there were 699,000 abortions in 1955 in the United States, with a reported standard deviation of about 6,000, less than one percent. For 1967, the year of their study, they estimated 829,000.

Their estimate was referenced widely in the flurry of abortion law and court cases that followed due to a surging public interest in the topic. For example, it is cited in the 1970 California Supreme Court opinion for the case Ballard v. Anderson, which concerned whether a minor needs parental consent to receive an otherwise legal abortion. [Ballard71, Roemer71] It was also cited in amici curiae briefs submitted to the United States Supreme Court in 1971 for Roe v. Wade, the famous case that invalidated most U.S. laws making abortion illegal. One such brief was filed jointly by the country’s leading women’s rights organizations like the National Organization for Women. Citing Horvitz for this paragraph, it wrote, [Womens71]

While the realities of law enforcement, social and public health problems posed by abortion laws have been openly discussed […] only within a period of not more than the last ten years, one fact appears undeniable, although unverifiable statistically. There are at least one million illegal abortions in the United States each year. Indeed, studies indicate that, if the local law still has qualifying requirements, the relaxation in the law has not diminished to any substantial extent the numbers in which women procure illegal abortions.

It’s unclear how the authors got this one million number (Horvitz’s estimate was 20% less for 1967), nor what they meant by “unverifiable statistically.” It may have been a misinterpretation of the randomized response technique. In any event, randomized response played a crucial role in providing a foundation for political debate.

Despite Horvitz’s success, and decades of additional research on crime, drug use, and other sensitive topics, randomized response mechanisms have been applied poorly. In some cases, the desired randomization is inextricably complex, such as when requiring a continuous random number. In these cases, a manual randomization mechanism is too complex for a respondent to use accurately. Trying to use software-assisted devices can help, but can also produce mistrust in the interviewee. See [Rueda16] for additional discussion of these pitfalls and what software packages exist for assisting in using randomized response. See [Fox16] for an analysis of the statistical differences between the variety of methods used between 1970 and 2010.

In other contexts, analogues to randomized response may not elicit the intended effect. In the 1950’s, Utah used death by firing squad as capital punishment. To avoid a guilty conscience of the shooters, one of five marksmen was randomly given a blank, providing him some plausible deniability that he knew he had delivered the killing shot. However, this approach failed on two counts. First, once a shot was fired the marksman could tell whether the bullet was real based on the recoil. Second, a 20% chance of a blank was not enough to dissuade a guilty marksman from purposely missing. In the 1951 execution of Elisio Mares, all four real bullets missed the condemned man’s heart, hitting his chest, stomach, and hip. He died, but it was neither painless nor instant.

Of many lessons one might draw from the botched execution, one is that randomization mechanisms must take into account both the psychology of the participants as well as the severity of a failed outcome.


  title = {{Randomized Response and Related Methods: Surveying Sensitive Data}},
  author = {James Alan Fox},
  edition = {2nd},
  year = {2016},
  doi = {10.4135/9781506300122},

  author = {Abernathy, James R. and Greenberg, Bernard G. and Horvitz, Daniel G.
  title = {{Estimates of induced abortion in urban North Carolina}},
  journal = {Demography},
  volume = {7},
  number = {1},
  pages = {19-29},
  year = {1970},
  month = {02},
  issn = {0070-3370},
  doi = {10.2307/2060019},
  url = {https://doi.org/10.2307/2060019},

  author = {Stanley L. Warner},
  journal = {Journal of the American Statistical Association},
  number = {309},
  pages = {63--69},
  publisher = {{American Statistical Association, Taylor \& Francis, Ltd.}},
  title = {Randomized Response: A Survey Technique for Eliminating Evasive
           Answer Bias},
  volume = {60},
  year = {1965},

  title = {{Ballard v. Anderson}},
  journal = {California Supreme Court L.A. 29834},
  year = {1971},
  url = {https://caselaw.findlaw.com/ca-supreme-court/1826726.html},

  title = {{Motion for Leave to File Brief Amici Curiae on Behalf of Women’s
           Organizations and Named Women in Support of Appellants in Each Case,
           and Brief Amici Curiae.}},
  booktitle = {{Appellate Briefs for the case of Roe v. Wade}},
  number = {WL 128048},
  year = {1971},
  publisher = {Supreme Court of the United States},

  author = {R. Roemer},
  journal = {Am J Public Health},
  pages = {500--509},
  title = {Abortion law reform and repeal: legislative and judicial developments
  volume = {61},
  number = {3},
  year = {1971},

  title = {Chapter 10 - Software for Randomized Response Techniques},
  editor = {Arijit Chaudhuri and Tasos C. Christofides and C.R. Rao},
  series = {Handbook of Statistics},
  publisher = {Elsevier},
  volume = {34},
  pages = {155-167},
  year = {2016},
  booktitle = {Data Gathering, Analysis and Protection of Privacy Through
               Randomized Response Techniques: Qualitative and Quantitative Human
  doi = {https://doi.org/10.1016/bs.host.2016.01.009},
  author = {M. Rueda and B. Cobo and A. Arcos and R. Arnab},

Why Theoretical Computer Scientists Aren’t Worried About Privacy

There has been a lot of news recently on government surveillance of its citizens. The biggest two that have pervaded my news feeds are the protests in Turkey, which in particular have resulted in particular oppression of social media users, and the recent light on the US National Security Agency’s widespread “backdoor” in industry databases at Google, Verizon, Facebook, and others. It appears that the facts are in flux, as some companies have denied their involvement in this program, but regardless of the truth the eye of the public has landed firmly on questions of privacy.

Barack Obama weighed in on the controversy as well, being quoted as saying,

You can’t have 100% security and 100% privacy, and also zero inconvenience.

I don’t know what balance the US government hopes to strike, but what I do know is that privacy and convenience are technologically possible, and we need not relinquish security to attain it.

Before I elaborate, let me get my personal beliefs out of the way. I consider the threat of terrorism low compared to the hundreds of other ways I can die. I should know, as I personally have been within an \varepsilon fraction of my life for all \varepsilon > 0 (when I was seven I was hit by a bus, proclaimed dead, and revived). So I take traffic security much more seriously than terrorism, and the usual statistics will back me up in claiming one would be irrational to do otherwise. On the other hand, I also believe that I only need so much privacy. So I don’t mind making much of my personal information public, and I opt in to every one of Google’s tracking services in the hopes that my user experience can be improved. Indeed it has, as services like Google Now will, e.g., track my favorite bands for me based on my Google Play listening and purchasing habits, and alert me when there are concerts in my area. If only it could go one step further and alert me of trending topics in theoretical computer science! I have much more utility for timely knowledge of these sorts of things than I do for the privacy of my Facebook posts. Of course, ideologically I’m against violating privacy as a matter of policy, but this is a different matter. One can personally loathe a specific genre of music and still recognize its value and one’s right to enjoy it.

But putting my personal beliefs aside, I want to make it clear that there is no technological barrier to maintaining privacy and utility. This may sound shocking, but it rings true to the theoretical computer scientist. Researchers in cryptography have experienced this feeling many times, that their wildest cryptographic dreams are not only possible but feasible! Public-key encryption and digital signatures, secret sharing on a public channel, zero-knowledge verification, and many other protocols have been realized quite soon after being imagined. There are still some engineering barriers to implementing these technologies efficiently in large-scale systems, but with demand and a few years of focused work there is nothing stopping them from being used by the public. I want to use this short post to describe two of the more recent ideas that have pervaded the crypto community and provide references for further reading.

Differential Privacy and Fully Homomorphic Encryption

There are two facts which are well known in theoretical computer science that the general public is not aware of. The first is about the privacy of databases:

There is a way to mine information from a database without the ability to inspect individual entries in the database.

This is known as differential privacy. The second is no less magical:

There are secure encryption schemes which allow one to run programs on encrypted data and produce encrypted results, without the ability to decrypt the data.

This is known as fully homomorphic encryption. 

The implications of these two facts should be obvious: search engines need not know our queries but can still fetch us search results and mine our information to serve ads, Facebook need not have access to our personal data but may still accurately predict new friends, grocery stores can even know what products to place side by side without knowing what any individual customer has purchased. Banks could process our transactions without knowing the amounts involved, or even the parties involved. Perhaps most importantly, governments can have access to databases (in the form of differentially private queries) and mine for the existence of threats without violating any individual user’s privacy. If they get an indication of a terrorist threat, then they can use the usual channels (court orders) to get access to specific individual data.

It’s easy to argue that these techniques will never become mainstream enough for individuals to benefit from it. Indeed, we’ve had cryptography for many years but few average users actively encrypt their communication for a lack of convenience. And then there are questions of policy: why would any company relinquish the ability to work directly with user data? And the cost of rearchitecturing existing services to utilize these technologies would be enough to dissuade most business leaders.

But the point of all this is that these are problems of policy that could in principle be solved without waiting for governments and corporations to get their act together. With enough demand for such services and with enough technologically-minded entrepreneurs (I’m looking at you, Silicon Valley), it would be just a matter of time before the world was differentially private. Mathematics cannot be revoked or legislated away.

Fully Homomorphic Encryption

A fully homomorphic encryption scheme is a normal encryption scheme (two functions “enc” and “dec” to encrypt and decrypt) with one additional function, which we’ll call “eval.” Very roughly, eval accepts as input the text of a program and a ciphertext, and produces as output a ciphertext such that the following diagram commutes:


That is, m is our message, and \textup{eval} runs f on the encrypted version of our message. In practice this happens by lifting two operations, multiplication and addition, from plaintexts (which are usually number-representations of letters) to ciphertexts (again usually numbers). Once this is done one can simulate the functionality of an arbitrary circuit on the encrypted data without decrypting it. Those readers who have been following our category theory series will recognize these sorts of diagrams as being functorial. [Actually, at the time of this writing we have yet to look at functors, but we will soon!] So perhaps a better term would be “functorial encryption.”

I should emphasize: a truly homomorphic encryption scheme has the ability to run any computable function on the encrypted data. There is no loss of functionality in preserving the privacy from the program runner. The main use of this is to maintain privacy while deferring large computations to the cloud. We do this all the time, e.g. a search query, but it also applies to big websites like Reddit, which operate entirely on Amazon Web Services.

Fully homomorphic encryption was first envisaged by Rivest, Adleman (two of the inventors of RSA), and Dertouzos in the late seventies, mainly because the RSA encryption scheme is close to being homomorphic (one can multiply ciphertexts, but not add them). In 2009, Craig Gentry released the first working fully-homomorphic scheme based on the mathematical theory of ideal lattices, and later that year he (with a group of other researchers) came up with a second system that is arguably as simple as RSA; it operates on integers with modular arithmetic.

Gentry has produced a lot of research since then in homomorphic encryption, but the interested reader should probably start with his tutorial paper describing his arithmetic-based system. From there, there are existing implementations in Python (using Sage) and C++, both of which are freely available on github.

Differential Privacy

The main idea of differential privacy is that one can add noise to statistical data to protect the identities of individual records. Slightly more rigorously, a randomized algorithm f is said to be \varepsilon-differentially private if for all possible datasets (inputs) D_1, D_2 which differ on a single record, and all possible collections of outputs y of f, the probability of correctly guessing D_1 from y is not significantly different from that of D_2. In particular, their quotient is at most e^{\varepsilon} (this choice of using e is arbitrary, but makes the analysis nicer).

The motivation for differential privacy came from two notable events in which companies released “anonymized” data which was partially de-anonymized because it was too specific. The first was the million-dollar Netflix Prize contest to develop a better recommendation algorithm, and the second was the release of the Massachusetts Group Insurance Commission medical database. As such, many companies are very strict with how they handle their user data, and information sharing the medical community is practically nonexistent.

There are many known differentially private algorithms, and they’re much stronger than one would imagine at first. One can run random forests of decision trees, network trace analysis, query-click analysis, certain forms of clustering, and a whole host of combinatorial optimization problems. For a gentle introduction to differential privacy, see Christine Task’s lecture video, a Practical Beginner’s Guide to Differential Privacy. There is also an influential survey from Microsoft Research of Dwork. These go into much more detail about the abilities and inabilities of differential privacy than I could do here.

If there’s one thing to take away from this discussion, it’s that efficient protocols for ensuring privacy are out there waiting to be implemented in software. So while we complain and listen to others complain about governments violating our liberties (and indeed, this discussion is extremely important to have), let’s do a little mathematics, do a little computer science, and figure out how to make privacy the standard of measure in software.

Until next time!