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securityinthepresenceofkeyleakageunderastrongerversionoftheLPNassumptionWestressthatkeyleakagesecurityisincomparabletothenotionsstudiedhereFastpseudorandomobjectsPseudorandomgeneratorBasedontheh. linckejonaslundbergwelflowevxuse ABSTRACT This paper shows that existing software metric tools inter pret and implement the de64257nitions of objectoriented soft ware metrics di64256erently This delivers tooldependent met rics results and has even im INRIA / ENS, Paris. Ideal Lattices . and Ring-LWE. Ideal lattices. Cyclic . Lattices. A set L in . Z. n. is a . cyclic lattice . if:. 1.) For all . v,w. in L, . v+w. is also in L. 2.) For all v in L, -v is also in L. China Summer School on Lattices and Cryptography, June 2014. Starting Point: Linear Equations. Easy to solve a linear system of equations. Given . A. , . b. , find . s. S. olved using Gaussian elimination, Cramer rule, etc.. IN. LEFT WING EXTREMISM AFFECTED SURGUJA DISTRICT OF CHHATTISGARH. . Presented by:. . . Sundarraj. P. Superintendent of Police. . Surguja. District, Chhattisgarh. Vadim . Lyubashevsky. Cryptography. Allows for secure communication in the presence of malicious parties. 2. Cryptography. Allows for secure communication in the presence of malicious parties. 3. Cryptography. Benjamin Fuller. , . Xianrui. . Meng. , and Leonid Reyzin. December 2, 2013. Key Derivation from Noisy Sources. Physically . Unclonable. Functions (PUFs). Biometric Data. Goal of this talk: provide meaningful security for more sources. Huijia. Lin (USB), . Rafael Pass . (Cornell). Karn. . Seth . (Cornell -> Google). Sid . Telang. (Cornell -> Google). IO. Plethora of Applications. For example: SW14, BCP14, BZ14, GGHR14, BGL. Sanjeev. . Arora. , . Rong. . Ge. Princeton University. Learning Parities with Noise. Secret u = (1,0,1,1,1). u ∙ (0,1,0,1,1) = 0. u ∙ (1,1,1,0,1) = 1. u ∙ (0,1,1,1,0) = . 1. Learning Parities with Noise. Ideal Lattices . and Ring-LWE. Ideal lattices. Cyclic . Lattices. A set L in . Z. n. is a . cyclic lattice . if:. 1.) For all . v,w. in L, . v w. is also in L. 2.) For all v in L, -v is also in L. Maps from Lattices. Craig Gentry (IBM. ). Sergey . Gorbunov. (. MIT). Shai Halevi (IBM). https. ://eprint.iacr.org/2014/645. July 10, 2015. The Mathematics of Modern Cryptography. Multilinear Maps (MMAPs). S. ui. t. e for . A. lgebraic . L. attice. s. ). CCA KEM: . Kyber. Digital Signature: . Dilithium. www.pq-crystals.org. Roberto . Avanzi. – ARM. Joppe. . Bos. – NXP . Leo . Ducas. – CWI. Eike . China Summer School on Lattices and Cryptography, June 2014. Starting Point: Linear Equations. Easy to solve a linear system of equations. Given . A. , . b. , find . s. S. olved using Gaussian elimination, Cramer rule, etc.. SecurityCorrectnessPerformanceBibliographyParameterSelectioninRing-LWE-basedFullyHomomorphicEncryptionRachelPlayerInformationSecurityGroupRoyalHollowayUniversityofLondonbasedonjointworkswithMartinRAlb . Lyubashevsky. INRIA / ENS, Paris. Outline. LLL sketch. Application to Subset Sum. Application to SIS. Application to LWE. Lattice Reduction in Practice. Small Integer Solution. Problem (SIS). Learning With Errors.