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Index

>.(n), 32 J-l(n), 32 1/J(n), 30 1r(x), 109 u(n), 27 T(n), 27 ((s), 2 b-sequence, 121 kth (higher) power non-residue, 78 kth (higher) power residue, 78 kth Power Residuosity Problem

(kPRP), 181 kth Root Problem (kRTP), 181 kth power non-residue, 57 kth power residue, 57

additive group, 81 additive identity, 84 additive inverse, 84 algebraic computation Jaw, 94 algebraic equation, 22 algebraic integer, 163 algebraic number, 162 arithmetic function, 25 associativity, 80 asymmetric key cryptosystem, 196 authentication, 194

base-2 pseudoprimality test, 119

Carmichael number, 119 Carmichael's >.-function, 32, 50 Carmichael's theorem, 50 CFRAC factoring algorithm, 157 CFRAC method, 155 Chinese Remainder Theorem (CRT),

53 Chinese test, 120 closure, 80 common multiple, 9 commutative group, 81

commutative ring, 83 commutativity, 81 complete system of residues, 39 completely multiplicative function, 25 complex zeros, 112 composite number, 4 congruence classes, 37 congruent, 36 conic, 87 consecutive pairs of quadratic residues,

58 consecutive triples of quadratic

residues, 59 Continued FRACtion (CFRAC)

method, 140 convergent, 16 convergents, 23 Converse of Fermat's little theorem, 49 Converse of Wilson's theorem, 51 cryptanalysis, 193 cryptography, 193 cryptology, 193 cubic Diophantine equation, 88 cubic integer, 163 cyclic group, 81

deterministic encryption, 205 Diffie-Hellman-Merkle key-exchange,

211 Diophantine geometry, 87 discrete logarithm, 77 discrete logarithm problem, 170 dividend, 4 division algorithm, 4 division ring, 83 divisor, 3 domain, 25

ECPP (Elliptic Curve Primality Proving), 127

ECPP Algorithm, 128

234

ElGamal cryptosystem, 213 elliptic curve, 88, 214 elliptic curve analogue of Diffie­

Hellman, 216 elliptic curve analogue of ElGamal, 217 elliptic curve analogue of Massey-

Omura, 216 elliptic curve analogue of RSA, 217 elliptic curve cryptography (ECC), 214 elliptic function, 90 elliptic integral, 90 embedding messages on elliptic curves,

215 ENIGMA code, 194 equivalence classes, 37 equivalence relation, 37 Euclid, 5 Euclid's algorithm, 12 Euclid's Elements, 13 Euler probable prime, 124 Euler pseudoprime, 124 Euler's (totient) ¢-function, 30 Euler's criterion, 60 Euler's theorem, 50 even number, 4 extended Euclid's algorithm, 46

factor, 3 factoring by trial divisions, 141 fast group operations, 107 fast modular exponentiations, 104 fast point additions, 107 Fermat Last Theorem, 1 Fermat probable prime, 118 Fermat pseudoprime, 118 Fermat's factoring algorithm, 142 Fermat's little theorem, 49 field, 83 finite fields, 85 finite group, 81 finite order of a point on an elliptic

curve, 92 finite simple continued fraction, 17 fixed-point attack, 205 Fundamental Theorem of Arithmetic, 6

Galois field, 85 Gauss'sJemma, 63 geometric composition law, 91 Goldbach Conjecture, 1 greatest common divisor (gcd), 7 group, 80 group laws on elliptic curves, 92

height, 96 high-order congruence, 55 hybrid cryptosystem, 197

identity, 81 incongruent, 36 index calculus algorithm, 178 index calculus method, 178 index of a to the base g, 76 index of an integer modulo n, 76 infinite fields, 85 infinite group, 81

Index

infinite order of a point on an elliptic curve, 92

infinite simple continued fraction, 18 integer factorization problem, 139 integral domain, 83 inverse, 81 irrational numbers, 18

Jacobi symbol, 69

least (non-negative) residue of x modulo n, 38

least common multiple (lcm), 9 least non-negative residue, 36 least residue, 63 Legendre symbol, 61 Legendre's congruence, 153 Legendre, A. M., 60 Lehman's method, 139 Lenstra's Elliptic Curve Method

(ECM), 140, 149 linear congruence, 46 linear Diophantine equation, 22

Mobius 11-function, 32 Mobius inversion formula, 34 Massey-Omura cryptosystem, 213 Mersenne prime, 3 Miller-Rabin test, 122 modular arithmetic in ZjnZ, 41 modular inverse, 44 modulus, 36 multiple, 3 Multiple Polynomial Quadratic Sieve

(MPQS), 140 multiple polynomial quadratic sieve

(MPQS), 159 multiplicative function, 25 multiplicative group, 81 multiplicative identity, 84 multiplicative·inverse, 44, 84

Index

non-secret encryption, 195 non-singular elliptic curve, 89 non-witness, 124 non-zero field element, 84 nontrivial divisor, 4 nontrivial square root of 1, 121 nontrivial zeros, 112 Number Field Sieve (NFS), 140, 162,

179

odd number, 4 order of a modulo n, 72 order of a field, 85 order of a point on an elliptic curve, 92

partial quotients, 15 perfect number, 3 period, 20 periodic simple continued fraction, 20 Pocklington's theorem, 126, 135 point at infinity, 89 Pollard's p factoring algorithm, 147 Pollard's p-method, 140, 143 Pollard's "p- 1" factoring algorithm,

148 Pollard's "p- 1" method, 148 polynomial congruence, 55 polynomial congruential equation, 55 polynomial security, 206 primality testing problem, 99, 115 prime counting function, 109 prime factor, 6 prime field, 85 prime number, 4 Prime Number Theorem, 110 prime power, 85 primitive root of n, 73 privacy, 194 private key, 196 probabilistic encryption, 206, 207 probable prime, 118 proper divisor, 3 pseudoprime, 118 public-key, 196 public-key cryptography, 195 public-key cryptosystem, 196 purely periodic simple continued

fraction, 20

quadratic congruence, 56 quadratic integer, 163 quadratic irrational, 20 quadratic non-residue, 57 Quadratic reciprocity law, 65

235

quadratic residue, 57 Quadratic Residuosity Problem (QRP),

181, 206 Quadratic Sieve (QS), 158 quantum algorithm for discrete

logarithms, 179 quantum algorithm for integer

factorization, 168 quantum register, 169 quotient, 4

randomized encryption, 206 rank of an elliptic curve, 94 rational integers, 164 rational line, 87 rational number, 87 rational numbers, 17 rational point, 87 real base logarithm, 76 real number, 20 real zeros, 112 real-valued function, 25 reduced system of residues modulo n,

41 reflexive, 37 relatively prime, 8 remainder, 4 repeated doubling and addition, 107 repeated doubling method, 215 repeated squaring and multiplication,

103 residue, 36 residue class, 37 residue classes, 37 residue of x modulo n, 37 Riemann (-function, 2, 111 Riemann Hypothesis, 2 Riemann Hypothesis (RH), 112 ring, 82 ring with identity, 83 root finding problem, 204 RSA Assumption, 198 RSA cryptosystem, 197

secret-key, 196 secret-key cryptosystem, 196 semantic security, 206 Shanks' baby-step giant-step method

for discrete logarithms, 172 Shanks' class group method, 140 Shanks' SQUFOF method, 139 Sieve of Eratosthenes, 5 Silver-Pohlig-Hellman algorithm, 174

236

simple continued fraction, 15 Solovay-Strassen test, 124 Sophie Germain prime, 133 square root method, 173 SQuare RooT Problem (SQRTP), 181 strong probable prime, 122 strong pseudoprimality test, 120, 122 strong pseudoprime, 122 subgroup, 81 symmetric, 37

torsion subgroup, 94 transitive, 37 trial division, 140 trivial divisor, 4 trivial zeros, 112 Twin Prime Conjecture, 2

Wiener's attack, 205 Wilson's theorem, 51 witness, 124

Index

About the Author

SONG Y Y AN majored in both Computer Science and Math­ematics, and obtained a Doctorate in Mathematics (Number Theory) from the Department of Mathematics at the U ni­versity of York, England. His current research interests are in number theory, theoretical computer science, public-key cryp­tography and information/network security. His other publi­cations include Perfect, Amicable and Sociable Numbers: A

Computational Approach, World Scientific, 1996, and Number Theory for Computing, Springer-Verlag, 2nd Edition, 2002. Song Yan is currently with the School of Mathematical and Information Sciences at Coventry University, England, and can be contacted by s.yan@coventry.ac.uk.