Board on MathematicalView webinar videos Sciences and learn more & about Analytics BMSA at www.nas.edu/MathFrontiers MATHEMATICAL FRONTIERS 2019 Monthly Webinar Series, 2-3pm ET February 12: Machine Learning September 10: Logic and Foundations* for Materials Science* October 8: Mathematics of Quantum March 12: Mathematics of Privacy* Physics* April 9: Mathematics of Gravitational November 12: Quantum Encryption Waves* December 10: Machine Learning for Text May 14: Algebraic Geometry* June 11: Mathematics of Transportation* Made possible by support for BMSA from the July 9: Cryptography & Cybersecurity* National Science Foundation Division of Mathematical Sciences August 13: Machine Learning in and the Department of Energy Medicine* Advanced Scientific Computing Research * Webinar posted View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 2 MATHEMATICAL FRONTIERS Quantum Encryption Delaram Kahrobaei, Dustin Moody, Mark Green, University of York NIST UCLA (moderator) View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 3 MATHEMATICAL FRONTIERS Quantum Encryption Chair of Cyber Security University of York (UK) Adjunct Professor of Computer Science NYU: New York University Doctoral Faculty of Computer Science GC CUNY: The City University of New York Post-Quantum Prof. Delaram Kahrobaei Cryptography University of York ViewView we bwebinarinar vid videoseos a nandd l elearnarn mmoreore aboutabout BMSA BMSA at a twww.nas.edu www.nas.edu/MathFrontiers/MathFrontiers 4 4 Qu a n t um Com put at i o n Qu a n t um Com put er s are N o t a repl acem en t o f t he c l a ssi cal co m put er s I t wo r k s f a st er f o r so m e speci al c l a sses o f pr obl em s. View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 5 qu bi t A q u b i t i s t h e q u a n t u m a n al o gu e o f a c l a ssi cal bi t, a n d ca n be t ho u g ht o f a s v e c t o r s. T h e ba si s s t a t e s (ba si s vect o r s) ar e 1 0 | 0⟩ = a n d | 1⟩ = [ 0] [ 1] View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 6 Com bi n ed cu bi t s T hese ca n be co m bi n ed . I f we have t wo qu bi t s, t he st at es ar e 1 0 0 1 | 00⟩ = | 01⟩ = 0 0 0 0 0 0 0 0 | 10⟩ = | 11⟩ = 1 0 0 1 View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 7 Su per po si t i o n A pu r e qubi t st at e i s a l i n ear su per po si t i o n o f t hese ba si s st a t e s, w i t h p r o b a b i l i t y am pl i t u d es α, β ∈ ℂ | ψ⟩ = α| 0⟩ + β| 1⟩ View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 8 Ho w m a n y qu bi t s d o we have? A pa i r o f qu bi t s t hat ca n exi st a s ei t her 1s o r 0 s ca n em bo d y 4 po ssi bl e st at es. 3 qu bi t s ca n em bo d y 8 . View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 9 NSA NI ST Ann o u n cem en t f o r Sear c h o f Po st-Quan tum Pr i m i t i ves View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 10 T he qu a n t um - saf e pr i m i t i ves u n d er co n si d er at i o n by NI ST Mu l t i var i at e pr i m i t i ves (Syst em o f m u l t i var i at e po l yn o m i al equ at i o n s) Co d e- ba sed pr i m i t i ves (Deco d i n g pr o bl em i n a lin ear co d e) Lat t i ce- ba sed pr i m i t i ves (Sho r t o r c l o se vect o r pro bl em i n a l at t i ce) Ha sh- ba sed pr i m i t i ves (F i n d i n g co l l i si o n s o r pr ei m ag es i n cr ypt o g r a phi c ha sh f u n ct i o n s) I so g en y- ba sed k ey pr i m i t i ves (f i n d i n g a n u n k n o wn i so g en y bet ween a pa i r o f su per si n gu l ar el l i pt i c c u r ves) G r o u p- ba sed pr i m i t i ves (al g o r i t hm i c g r o u p theo r et i c pr o bl em s) View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 11 Cyber Secu r i t y: Cr ypt o g r a phy Key Exc ha n g es (RSA, DH) Sym m etri c Key En cr ypt ion Di g i t al Si g n at u r es Mu l t i l i n ear Ma ps Ful l y Ho momor phi c En cr ypt i o n s Aut hen t i cat i o n Se cr et Shar ing View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 12 Gro u p T heo r y i n Cr ypt o g r a phy Po l ycyc l i c G r o u ps: Ei ck , Ka hr obaei G r a ph (RAAG ) G r o u ps: F l o r es, Ka hr obaei, Ko ber da Hyper bo l i c G r o u ps: Chat t erji , Ka hr obaei Ar i t hm et i c G r o u ps: Ka hr obaei, Mal l a hi - Karai Ni l pot en t G r o u ps: Ka hr obaei, To r t o ra, To t a Eng el Gro u ps: Ka hr obaei, No ce Free n i l pot en t p- g r o u ps: Ka hr obaei, Shpi l rain Sem i g r o u p o f Mat r i ces o ver G r o u p- Ri n g s: Ka hr obaei, Ko u ppar is, Shpi l rain Sm al l Can cel l at i o n G r o u ps: Ha beeb, Ka hr obaei, Shpi l rain Free Met a bel i a n : G r o u ps: Shpi l rain, Z apat a , , Ka hr obaei, Ha beeb View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 13 Gro u p T heo r y i n Cr ypt o g r a phy Li n ear : G r o u ps: Ba um sl ag, F i n e, Xu Br a i d G r o u ps: Anshel - An shel - G o l dfel d , Ko - Lee Grigo r chuk G r o u ps: Pet r i d es Aut o m at a G r o u ps: G r igo r c huk , G r igo r iev G r o u ps o f Mat r i ces: Grigo r iev, Po n am eran co T ho m pso n : G r o u ps: Shpi l rain, Ushak o v View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 14 Al g o r i t hm i c G r o u p Theo r et i c Pr o bl em s u sed i n Cr ypt o g r a phy Di scr et e Lo g ar i t hm Pr obl em Su bg r o u p I so m o r phi sm Pr obl em G r o u p Ho m o m o r phi sm Pr obl em Po wer Con j u g ac y Sear ch Pr obl em G eo d esi c Len g t h Pr obl em Di st o r t ed Subg r o u p Con j u g ac y Sear ch Pr obl em Wo r d Dec i si o n Pr o bl em En d o m o r phi sm Sear c h Pr obl em Deco m po si t i o n Sear c h Pr obl em Subg r o u p mem ber shi p Pr obl em View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 15 Appr o aches t o Po st- qu a n t um G r o u p- Ba sed Cr ypt o g r a phy NP- Com pl et e a n d Un so l va bl e G r o u p- T heo r et i c Pr o bl em s An al ysi n g equ i val en ce t o Hi d d en Subg r o u p Pr obl em An al ysi n g G r o ver 's Sear ch Pr o bl em View webinar videos and learn more about BMSA at www.nas.edu/MathFrontiers 16 Hi d d en Su bg r o u p Pr obl em (HSP) Given a f ini t el y g en erat ed g r o u p G a n d a n ef f i cien t l y co m put a bl e f u nct i o n f f r o m G t o so m e f i n i t e set S su ch t hat f i s co n st a n t a n d d i st i n ct o n l ef t- co set s o f a su bg r o u p H o f f i n i t e i n d ex, f i n d a g en er at i n g set f o r H.