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Elliptic Curve Systems
 IEEE P1363, Part 4: Elliptic Curve Systems
, 1995
"... . This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal publickey cryptosystem. Elliptic curve systems are publickey (asymmetric) cryptographic algorithms, typically used in conjunction with a hash algorithm to create digit ..."
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. This standard describes a method for data encryption and for digital signatures using the elliptic curve analogue of the ElGamal publickey cryptosystem. Elliptic curve systems are publickey (asymmetric) cryptographic algorithms, typically used in conjunction with a hash algorithm to create digital signatures, and for the secure distribution of secret keys for use in symmetrickey cryptosystems. Elliptic curve systems may also be used to transmit confidential information. Introduction The algebraic system defined on the points of an elliptic curve provides an alternate means to implement the ElGamal and ElGamallike public key encryption and signature protocols. These protocols are typically described in the literature in the algebraic system Z p , the integers modulo p, where p is a prime. For example, the NIST Digitial Signature Algorithm (DSA) is an ElGamallike signature scheme defined over Z p . Precisely the same protocol for signing could be defined over the points on an ell...
On Boolean Functions with Generalized Cryptographic Properties
 Properties, Indocrypt 2004, LNCS 3348
, 2004
"... By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation characteristics. ..."
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By considering a new metric, we generalize cryptographic properties of Boolean functions such as resiliency and propagation characteristics.
Applications of Exponential Sums in Communications Theory
, 1999
"... We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers. ..."
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We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers.
The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace
, 1999
"... The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . We derive an explicit formula for the number of irreducible degree n polynomials over GF(2) that have a given trace and subtrace. The trace and subtrace of an e ..."
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The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . We derive an explicit formula for the number of irreducible degree n polynomials over GF(2) that have a given trace and subtrace. The trace and subtrace of an element fi 2 GF(2 n ) are defined to be the coefficients of x n\Gamma1 and x n\Gamma2 , respectively, in the polynomial q(x) = n\Gamma1 Y i=0 (x + fi 2 i ). We also derive an explicit formula for the number of elements of GF(2 n ) of given trace and subtrace. Moreover, a new two equation Mobiustype inversion formula is proved. Keywords: Irreducible polynomial, minimal polynomial, trace, subtrace, Mobius inversion. 1 Introduction The trace of a degree n polynomial p(x) over GF(2) is the coefficient of x n\Gamma1 and the subtrace is the coefficient of x n\Gamma2 . It is well known that the formula L(n) = 1 n X djn (d)2 n=d (1) HewlettPackard Labs, Santa Rosa ema...
The Number of Irreducible Polynomials and Lyndon Words with Given Trace
 SIAM J. Discrete Mathematics
"... The trace of a degree n polynomial f(x) over GF (q) is the coefficient of x . Carlitz [Proc. AMS, 3 (1952) 693700] obtained an expression Iq (n; t), for the number of monic irreducible polynomials over GF (q) of degree n and trace t. Using a different approach, we derive a simple explicit expressio ..."
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The trace of a degree n polynomial f(x) over GF (q) is the coefficient of x . Carlitz [Proc. AMS, 3 (1952) 693700] obtained an expression Iq (n; t), for the number of monic irreducible polynomials over GF (q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for Iq (n; t). If t > 0, Iq (n; t) = ( )=(qn), where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count Lq (n; t), the number of qary Lyndon words whose characters sum to t mod q. This number is given by Lq (n; t) = ( )=(qn), where the sum is over all divisors d of n for which gcd(d; q)jt. Both results rely on a new form of Möbius inversion.
Quantum binary field inversion: improved circuit depth via choice of basis representation
 Quantum Information & Computation
, 2013
"... Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal ba ..."
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Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of ‘ghostbit basis’ representations can be used to implement inverters with a quantum circuit of depth O(m log(m)). To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the ItohTsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields F2m based on an elementary gate set that is useful for faulttolerant implementation. 1
Computing Composed Products of Polynomials
 IN FINITE FIELDS: THEORY, APPLICATIONS, AND ALGORITHMS
, 1999
"... If f(x) and g(x) are polynomials in Fq [x] of degrees m and n respectively, then the composed sum of f and g, denoted f g, is the degree mn polynomial whose roots are all sums of roots of f with roots of g. Likewise, the composed multiplication of f and g, denoted f ffi g, is the degree mn poly ..."
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If f(x) and g(x) are polynomials in Fq [x] of degrees m and n respectively, then the composed sum of f and g, denoted f g, is the degree mn polynomial whose roots are all sums of roots of f with roots of g. Likewise, the composed multiplication of f and g, denoted f ffi g, is the degree mn polynomial whose roots are all products of roots of f with roots of g. In 1987, Brawley and Carlitz defined a more general notion of polynomial composition, denoted by f \Pi g, for which f g and f ffi g are special cases. They prove that when f and g are irreducible with degrees m and n coprime, then f \Pi g is irreducible of degree mn. This gives us a way to obtain irreducibles of relatively large degree using irreducibles of smaller degrees. In this paper, we describe several methods of computing polynomial compositions of the above form and compare their time complexities.
HAMMING DISTANCE FROM IRREDUCIBLE POLYNOMIALS OVER F2
"... Abstract. We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we p ..."
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Abstract. We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree n polynomials of Hamming distance one from a randomly chosen set of ⌊2 n /n ⌋ odd density polynomials is asymptotically (1 − e −4)2 n−1, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant c such that every polynomial has Hamming distance at most c from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over F2 for degrees 1 ≤ n ≤ 32, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this “empirical ” study. 1.