A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap

August 2017 Notices of the AMs 709 In August 2015 the National Security Agency published a webpage announcing preliminary plans for transitioning to quantum-resistant algorithms (www.iad.gov /iad/programs/iad-initiatives/cnsa-suite.cfm). In December 2016 the National Institute of Standards and Technology (NIST) announced a call for proposals for quantum-resistant algorithms with a deadline of 30 November 2017 (www.nist.gov/pqcrypto). The effort to develop quantum-resistant technologies, and in particular post-quantum cryptosystems, is becoming a central research area in information security. Current research in post-quantum cryptography is based on state-of-thea r t c o m p u t a t i o n a l techniques such as algorithms in algebraic geometry, coding theory, and lattice theory. The mathematics utilized in PQC is diverse and sophisticated, including representation theory, harmonic analysis, mathematical physics, algebraic number theory, lattice theory, and algebraic geometry. Even the Riemann hypothesis is often used to deal with critical problems in complexity Over the past three decades, the family of public-key cryptosystems, a fundamental breakthrough in modern cryptography in the late 1970s, has become an increasingly integral part of our communication networks. The Internet, as well as other communication systems, relies principally on the Diffie-Hellman key exchange, RSA encryption, and digital signatures using DSA, ECDSA, or related algorithms. The security of these cryptosystems depends on the difficulty of certain number-theoretic problems, such as integer factorization or the discrete log problem. In 1994 Peter Shor showed that quantum computers can solve each of these problems in polynomial time, thus rendering the security of all cryptosystems based on such assumptions impotent. A large international community has emerged to address this issue in the hope that our public-key infrastructure may remain intact by utilizing new quantum-resistant primitives. In the academic world, this new science bears the moniker Post-Quantum Cryptography (PQC). Post-Quantum Cryptography—A New Opportunity and Challenge for the Mathematics Community