It is shown how to distribute a secret to n persons such that each person can verify that he has received correct information about the secret without talking with other persons. Any k of these persons can later find the secret (1 ? k ? n), whereas fewer than k persons get no (Shannon) information about the secret. The information rate of the scheme is 1/2 and the distribution as well as the verification requires approximately 2k modular multiplications pr. bit of the secret. It is also shown how a number of persons can choose a secret "in the well" and distribute it veritably among themselves.

https://link.springer.com/content/pdf/10.1007%2F3-540-46766-1_9.pdf

Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing

An Introduction to Probability Theory and its Applications, Volume I

INTRODUCTION THE BASIC THEORY Weierstrass Equations The Group Law Projective Space and the Point at Infinity Proof of Associativity Other Equations for Elliptic Curves Other Coordinate Systems The j-Invariant Elliptic Curves in Characteristic 2 Endomorphisms Singular Curves Elliptic Curves mod n TORSION POINTS Torsion Points Division Polynomials The Weil Pairing The Tate-Lichtenbaum Pairing Elliptic Curves over Finite Fields Examples The Frobenius Endomorphism Determining the Group Order A Family of Curves Schoof's Algorithm Supersingular Curves The Discrete Logarithm Problem The Index Calculus General Attacks on Discrete Logs Attacks with Pairings Anomalous Curves Other Attacks Elliptic Curve Cryptography The Basic Setup Diffie-Hellman Key Exchange Massey-Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures The Digital Signature Algorithm ECIES A Public Key Scheme Based on Factoring A Cryptosystem Based on the Weil Pairing Other Applications Factoring Using Elliptic Curves Primality Testing Elliptic Curves over Q The Torsion Subgroup: The Lutz-Nagell Theorem Descent and the Weak Mordell-Weil Theorem Heights and the Mordell-Weil Theorem Examples The Height Pairing Fermat's Infinite Descent 2-Selmer Groups Shafarevich-Tate Groups A Nontrivial Shafarevich-Tate Group Galois Cohomology Elliptic Curves over C Doubly Periodic Functions Tori Are Elliptic Curves Elliptic Curves over C Computing Periods Division Polynomials The Torsion Subgroup: Doud's Method Complex Multiplication Elliptic Curves over C Elliptic Curves over Finite Fields Integrality of j-Invariants Numerical Examples Kronecker's Jugendtraum DIVISORS Definitions and Examples The Weil Pairing The Tate-Lichtenbaum Pairing Computation of the Pairings Genus One Curves and Elliptic Curves Equivalence of the Definitions of the Pairings Nondegeneracy of the Tate-Lichtenbaum Pairing ISOGENIES The Complex Theory The Algebraic Theory Velu's Formulas Point Counting Complements Hyperelliptic Curves Basic Definitions Divisors Cantor's Algorithm The Discrete Logarithm Problem Zeta Functions Elliptic Curves over Finite Fields Elliptic Curves over Q Fermat's Last Theorem Overview Galois Representations Sketch of Ribet's Proof Sketch of Wiles's Proof APPENDIX A: NUMBER THEORY APPENDIX B: GROUPS APPENDIX C: FIELDS APPENDIX D: COMPUTER packages REFERENCES INDEX Exercises appear at the end of each chapter.

Elliptic Curves: Number Theory and Cryptography

A q-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted.We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2t − 1, we design three query LDCs of length N = exp(O(n1/t)), for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp(O(n10 − 7)) compared to exp(O(n1/2)) in the previous constructions. It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp(nO(1/log log n)) for infinitely many n.We also obtain analogous improvements for Private Information Retrieval (PIR) schemes. We give 3-server PIR schemes with communication complexity of O(n10 − 7) to access an n-bit database, compared to the previous best scheme with complexity O(n1/5.25). Assuming again that there are infinitely many Mersenne primes, we get 3-server PIR schemes of communication complexity nO(1/log logn)) for infinitely many n.Previous families of LDCs and PIR schemes were based on the properties of low-degree multivariate polynomials over finite fields. Our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set.

/pdf/towards-3-query-locally-decodable-codes-of-subexponential-3kj282s4d2.pdf

Towards 3-query locally decodable codes of subexponential length

The established methodologies for studying computational complexity can be applied to the new problems posed by very large-scale integrated (VLSI) circuits. This thesis develops a "VLSI model of computation" and derives upper and lower bounds on the silicon area and time required to solve the problems of sorting and discrete Fourier transformation. In particular, the area A and time T taken by any VLSI chip using any algorithm to perform an N-point Fourier transform must satisfy AT('2) (GREATERTHEQ)
cN('2)log('2)N, for some fixed c > 0. A more general result for both sorting and Fourier transformation is that AT('2x) = (OMEGA)(N('1+x)log('2x)N), for any x in the
range 0 (LESSTHEQ) x (LESSTHEQ) 1. Also, the energy dissipated by a VLSI chip during the solution of either of these problems is at least (OMEGA)(N('3/2)log N). The tightness of these bounds is demonstrated by the existence of nearly optimal circuits for both sorting and Fourier transformation. The circuits based on the shuffle-exchange interconnection pattern are fast but large: T = O(log('2)N) for Fourier transformation, T = O(log('3)N) for sorting; both have area A of
at most O(N('2)/log('1/2)N). The circuits based on the mesh interconnection pattern are slow but small: T = O(N('1/2)loglog N), A = O(N log('2)N).

A complexity theory for VLSI

/pdf/the-pseudoprimes-to-25-10-x1kyj3enz7.pdf

The pseudoprimes to 25⋅10⁹

We have determined the irregular primes below 125000 and tabulated their distribution Two primes of index five of irregularity were found, namely 78233 and 94693 Fermat's Last Theorem has been verified for all exponents up to 125000 We computed the cyclotomic invariants,Mp, Ap , and found that -O 0 for all p < 125000 The complete factorizations of the numerators of the Bernoulli numbers B2k for 2k < 60 and of the Euler numbers E2k for 2k < 42 are given

/pdf/the-irregular-primes-to-125000-4t0oux2kje.pdf

The irregular primes to 125000

In electronic subscription and pay TV systems, data can be organized and encrypted using symmetric key algorithms according to predefined time periods and user privileges and then broadcast to users. This requires an efficient way of managing the encryption keys. In this scenario, time-bound key management schemes for a hierarchy were proposed by Tzeng and Chien in 2002 and 2005, respectively. Both schemes are insecure against collusion attacks. In this paper, we propose a new key assignment scheme for access control, which is both efficient and secure. Elliptic-curve cryptography is deployed in this scheme. We also provide the analysis of the scheme with respect to security and efficiency issues.

/pdf/an-efficient-time-bound-hierarchical-key-management-scheme-3ovnkecads.pdf

An Efficient Time-Bound Hierarchical Key Management Scheme for Secure Broadcasting

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers We list some large prime divisors of Mersenne numbers Mp in the range 17000 <p < 105

/pdf/divisors-of-mersenne-numbers-46sdm6m1d1.pdf

Divisors of Mersenne numbers

We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about 0(k) log k log 0(k), where 0 is Euler's phi-function.

/pdf/greatest-of-the-least-primes-in-arithmetic-progressions-q5kyt95c9b.pdf

Samuel S. Wagstaff

Papers

The pseudoprimes to 25⋅10⁹

The irregular primes to 125000

An Efficient Time-Bound Hierarchical Key Management Scheme for Secure Broadcasting

Divisors of Mersenne numbers

Greatest of the least primes in arithmetic progressions having a given modulus