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Lecture Notes on Cryptography
, 2001
"... This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MI ..."
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This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5, and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E. Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the notion of provable security and its usage for the design of secure protocols. Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science. Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography and network security course at UCSD.
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
Some results on pseudosquares
 Math. Comp
, 1996
"... Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe ..."
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Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp √ e p/2, an inequality that must hold under the extended Riemann Hypothesis. 1.
Identifying the Matrix Ring: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS
, 2010
"... We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We d ..."
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We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.
The second largest prime divisor of an odd perfect number exceeds ten thousand
 Math. Comp
, 1999
"... Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) =2n. It is well known that a number is even and perfect if and only if it has the form 2 p−1 (2 p − 1) where 2 p − 1isprime. No odd perfect numbers are known, nor has any proof ..."
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Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) =2n. It is well known that a number is even and perfect if and only if it has the form 2 p−1 (2 p − 1) where 2 p − 1isprime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed 10 6, and Hagis showed that the second largest must exceed 10 3. In this paper, we improve the latter bound. In particular, we prove the statement in the title of this paper. 1.
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by s ..."
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Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but  Supported in part by NSF grant CCR9734918. y Supported in part by NSF grant CCR9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...
Improving Privacy in Cryptographic Elections
, 1986
"... This report describes two simple extensions to the paper A Robust and Verifiable Cryptographically Secure Election Scheme presented in the 1985 Symposium on the Foundations of Computer Science. The first extension allows the "government" to be divided into an arbitrary number of " ..."
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This report describes two simple extensions to the paper A Robust and Verifiable Cryptographically Secure Election Scheme presented in the 1985 Symposium on the Foundations of Computer Science. The first extension allows the "government" to be divided into an arbitrary number of "tellers". With this extension, trust in any one teller is sufficient to assure privacy, even if the remaining tellers conspire in an attempt to breach privacy. The second extension allows a government to reveal (and convince voters of) the winner in an election without releasing the actual tally. Combining these two extensions in a uniform manner remains an open problem. 1 Introduction and Background In [CoFi85], a protocol was presented which gives a method of holding a mutually verifiable secretballot election. The participants are the voters, a government, a trusted "beacon" which generates publically readable random bits, and a trusted global clock. The protocol has four basic phases. In phas...
Sharpening PRIMES is in P for a large family of numbers
 Math. Comp
, 2005
"... We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algo ..."
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We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.
The role of smooth numbers in number theoretic algorithms
 In International Congress of Mathematicians
, 1994
"... A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin pr ..."
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A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin properties of smooth numbers
Automorphisms of Finite Rings and Applications to Complexity of Problems
 STACS’05, Springer LNCS 3404
, 2005
"... In mathematics, automorphisms of algebraic structures play an important role. ..."
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In mathematics, automorphisms of algebraic structures play an important role.