Partisan Gerrymandering and the Efficiency

OPINION OPINION Post-Quantum A Formula Goes to Court: Cryptography—A New Partisan Gerrymandering Opportunity and Challenge forand the Efficiency Gap the Mathematics CommunityMira Bernstein and Moon Duchin Note: The opinions expressed here are not necessarily those of Notices. Jintai Ding and Daniel Smith-Tone Responses on the Notices webpage are invited. Note: The opinions expressed here are not necessarily those of Notices. Responses on the Notices webpage are invited. Over the past three decades, the family of public-key In August 2015 the National Security Agency published cryptosystems, a fundamental breakthrough in modern a webpage announcing preliminary plans for transi- cryptography in the late 1970s, has become an increas- tioning to quantum-resistant algorithms (www.iad.gov /iad/programs/iad-initiatives/cnsa-suite.cfm). ingly integral part of our communication networks. The Gerrymandering is drawing political boundary lines with votes by your opponents and thus reduce your opponents’ In December 2016 the National Institute of Standards Internet, as well as other communication systems, relies an ulterior motive. This idea has great currency right now, share of seats relative to their share of the votes cast. and Technology (NIST) announced a call for proposals principally on the Diffie-Hellman key exchange, RSA with particular attention paid to manipulating shapes In this note we will focus on partisan gerry- for quantum-resistant algorithms with a deadline of 30 encryption, and digital signatures using DSA, ECDSA, or of US congressional and legislative districts in order to mandering. The Supreme Court has heard cases on November 2017 (www.nist.gov/pqcrypto). The effort related algorithms. The security of these cryptosystems obtain a preferred outcome. Gerrymandering comes in partisan gerrymandering three times,1 and each time the to develop quantum-resistant technologies, and in partic- many flavors, including racial gerrymandering, where justices have balked. They disagreed among themselves depends on the difficulty of certain number-theoretic ular post-quantum cryptosystems, is becoming a central a minority group is subject to dilution of its voting about whether partisan gerrymandering was within the problems, such as integer factorization or the discrete research area in information security. strength; partisan gerrymandering, where one political Court’s purview at all and about what standards might log problem. In 1994 Peter Shor showed that quantum Current research in post-quantum cryptography is party controls the process and tries to exaggerate its own be used to detect it. In the most recent case, the swing computers can solve each of these problems in polynomial based on state-of-the- political dominance; and incumbent gerrymandering, vote belonged to Justice Anthony Kennedy, who wrote a time, thus rendering the security of all cryptosystems art computational where officials try to create safe seats for incumbents on lengthy opinion explaining why he was unsatisfied with based on such assumptions impotent. techniques such as al- both sides of the aisle. All kinds of gerrymandering use the standards proposed by the plaintiffs to demonstrate gorithms in algebraic ge- Even the A large international community has emerged to ad- some of the same techniques, especially packing, where that unconstitutional gerrymandering had occurred. ometry, coding theory, dress this issue in the hope that our public-key infrastruc- Riemannyou seek to stuff your opponents into districts with very Kennedy also gave some guidelines for what would be and lattice theory. The ture may remain intact by utilizing new quantum-resistant high percentages, and cracking, where you disperse your needed to satisfy him in the future. For the past ten mathematics utilized primitives. In the academic world, this new science bears hypothesisopponents is into several districts in numbers too small years, legal scholars and political scientists have pored in PQC is diverse and the moniker Post-Quantum Cryptography (PQC). to predominate. Both of these techniques generate wasted over Kennedy’s opinion, trying to design a standard that sophisticated, including often used to he might find convincing. Jintai Ding is professor of mathematical sciences at the University representation theory, In 2015, one such team, Nicholas Stephanopoulos and of Cincinnati. His e-mail address is [email protected]. harmonic analysis, math- deal with critical Mira Bernstein is research assistant professor in science, tech- Eric McGhee [3], advanced a new idea to quantify partisan Daniel Smith-Tone is professor of mathematics at the Univer- ematical physics, alge- nology, and society at Tufts University. Her e-mail address is gerrymandering, tailored to Kennedy’s guidelines. They sity of Louisville and a research mathematician at the National problems in braic number theory, lat- [email protected]. propose a simple numerical score called the efficiency Institute of Standards and Technology. His e-mail address is tice theory, and algebraic Moon Duchin is associate professor of mathematics at Tufts Uni- [email protected]. complexity gap (퐸퐺) to detect and reject unfair congressional and geometry. Even the Rie- versity. Her e-mail address is [email protected]. legislative maps that are rigged to keep one party on For permission to reprint this article, please contact: mann hypothesis is often analysis.See an article on “Gerrymandering, Sandwiches, and Topology” top in a way that is unresponsive to voter preferences. [email protected]. used to deal with critical in the Graduate Section (page 1010). 퐸퐺 can be computed based on voting data from a single DOI: http://dx.doi.org/10.1090/noti1546 problems in complexity For permission to reprint this article, please contact: [email protected]. 1Davis v. Bandemer (1986); Vieth v. Jubelirer (2004); LULAC v. AUGUST 2017 NOTICES OF THE AMS DOI: http://dx.doi.org/10.1090/noti1573 709 Perry (2006) 1020 Notices of the AMS Volume 64, Number 9 OPINION election: if the result exceeds a certain threshold, then the Wasted votes, in the 퐸퐺 formulation, are any votes cast districting plan has been found to have discriminatory for the losing side or votes cast for the winning side partisan effect. in excess of the 50 percent needed to win. That is, the Last November, for the first time in thirty years, number of votes wasted by 퐴-voters in district 푑푖 is a federal court invalidated a legislative map as an 퐴 푇 퐴 2 푇 − 푖 , 푑 ∈ 풟 , 푇 unconstitutional partisan gerrymander [4]. A center- 퐴 푖 2 푖 퐴 퐴 푖 푊푖 = { 퐴 퐵 = 푇푖 − 푆푖 ⋅ . piece in the district court’s ruling was the high efficiency 푇푖 , 푑푖 ∈ 풟 2 gap in Wiconsin’s 2012–2016 elections in favor of Re- Quick observation: the total number of wasted votes publicans. Now the case is under appeal to the Supreme 퐴 퐵 in a district, 푊푖 = 푊푖 + 푊푖 , is always half of the turnout Court, and many observers are hoping that 퐸퐺 will finally 푇푖. The question is how the wasted votes are distributed. provide a legally manageable standard for detecting par- If (nearly) all the wasted votes belong to the winning tisan gerrymanders in time for the 2020 census. If this side, it’s a packed district. If (nearly) all the wasted votes happens, it will signal a seismic shift in American politics. belong to the losing side, it’s a competitive district. And In this note, we engage in the following exercise: first, if there are several adjacent districts where most of the we give a self-contained analysis of 퐸퐺 for a mathematical wasted votes are on the losing side, then it may be a audience, describing what it measures and what it does cracked plan. not. Then, we examine how our findings relate to the We can now define the efficiency gap associated with original proposal, the press coverage, and the court districting plan 풟: decisions to date. We close by looking to the future career of this consequential formula. 푆 퐴 퐵 퐴 퐵 푊푖 − 푊푖 푊 − 푊 퐸퐺 = ∑ = . 푇 푇 What is the Efficiency Gap? 푖=1 The US electoral scene is dominated by the use of Thus 퐸퐺 is a signed measure of how much more vote geographically-defined districts in which one represen- share is wasted by supporters of party 퐴 than 퐵. If tative is chosen by a plurality vote. We will follow the 퐸퐺 is large and positive, the districting plan is deemed 퐸퐺 literature by simplifying to the case of only two unfair to 퐴. On the other hand, 퐸퐺 ≈ 0 indicates a fair political parties, 퐴 and 퐵. Let’s say that our state has 푆 plan, inasmuch as the two parties waste about an equal congressional or legislative seats, and denote the set of number of votes. Stephanopoulos–McGhee write that this districts by 풟 = {푑1, … , 푑푆}. In a particular election, write definition “captures, in a single tidy number, all of the 풟 = 풟퐴 ⊔ 풟퐵 where 풟푃 is the subset of districts won by packing and cracking decisions that go into a district 푃 plan.” party 푃 ∈ {퐴, 퐵}. In what follows, if a value 푋푖 is defined with respect to party 푃 and district 푑푖, we will write For a toy example, consider the figure below, which 푆 푆 shows three possible districting plans for the same 푃 푃 퐴 퐵 퐴 퐵 distribution of voters. Each box represents a voter; 퐴- 푋 = ∑ 푋푖 ; 푋푖 = 푋푖 +푋푖 ; and 푋 = 푋 +푋 = ∑ 푋푖. 푖=1 푖=1 voters are marked ⋆ and 퐵-voters are blank. Since 퐴-voters 푃 make up half of the population, intuitively we expect a Let 푆푖 be 1 if party 푃 won in district 푑푖 and 0 if not, 퐴 퐵 so that 푆푃 = |풟푃| is the number of seats won statewide “fair” plan to have 푆 = 푆 . Plans I and III both do this and 푃 it is easy to check that they have 퐸퐺 = 0.

Load more