264
Index
N
Noise distribution, 26
Number-theoretic transform (NNT), 40, 42
INTT, 24
learning with errors, 25–26
P
Partial Gibbs derivative, 89
Permutation matrices
bent functions, 83–86
Gibbs dyadic derivative, 86
bent functions, 89–92
definition, 87
diagonal matrix, 87
discrete Walsh functions, 88
Gibbs coefficients, 87
Gibbs differentiator, 87
Gibbs matrix, 88
matrix notation, 87
partial Gibbs derivative, 89
non-zero values, 83
Walsh spectral domain, 83
Walsh transform matrix, 85–86
See also Gibbs permutation matrices
Positive polarity Reed–Muller (PPRM)
expression, 218
Post-quantum cryptography (PQC), 22
Primitive functions, 136, 144
construction of, 140
input variables, 141–143
Probability heuristics, 119–120
Program-level explanation, 10
Pseudo-Kronecker Expressions (PKRMs), 178,
188
Q
Quantum algorithms, 22
Quantum computation
logic synthesis, 185
optimization techniques, 185
quantum circuit, 186
reversible logic synthesis, 186–188
R
Rademacher-Walsh spectrum, 197–198
Reed–Muller spectra, 84, 89, 98
Residue number system (RNS)
advantage of, 237
backward conversion, 239
Boolean minimization, 251–252
Chinese reminder theorem, 238
computation, 238
computer arithmetic, 238
EDA tool, 239
features, 238
hardware computation, 237
characteristics, 245
modular product, 249–251
modulo function computation, 245–248
implementations, 237
MA, 237
Mentor Graphics, 252
mixed-radix approach, 239–240
modulus function calculation
automaton model, 240
bit-by-bit input reduction, 241
cryptography, 240
hardware realization, 240
limitation, 240
memory elements, 240
modular exponentiation, 243
periodic property, 242–243
powers of 2, 243–244
type 2 n ± k, 244–245
parallel data processing, 238
structure, 238, 239
Synopsys, 252–257
Xilinx, 252
Reversible Boolean functions
binary reversible functions, 222
CFs, 217, 220
elements, 230
linear variables, 227–228
NPNP-classes, 228
P-equivalence classes, 231–234
projection functions, 229–231
variable assignments, 229, 230, 232
coordinate functions, 217
EXOR-sum of products expression, 218
extrapolation, cycle structures
balanced functions, 224–225
NPNP classes, 223, 224
PPRMs expressions, 223, 224
RevFunCFLVs, 226
homogeneous property, 218
linear and nonlinear, 219
logic synthesis, 218
multiplexer-based field-programmable gate
arrays, 221
NPNP-equivalent, 220–221
P-equivalent functions, 219, 222
PPRM expression, 218
SC function, 219
Reversible logic synthesis, 186–188, 192
RNS, see Residue number system
Index
N
Noise distribution, 26
Number-theoretic transform (NNT), 40, 42
INTT, 24
learning with errors, 25–26
P
Partial Gibbs derivative, 89
Permutation matrices
bent functions, 83–86
Gibbs dyadic derivative, 86
bent functions, 89–92
definition, 87
diagonal matrix, 87
discrete Walsh functions, 88
Gibbs coefficients, 87
Gibbs differentiator, 87
Gibbs matrix, 88
matrix notation, 87
partial Gibbs derivative, 89
non-zero values, 83
Walsh spectral domain, 83
Walsh transform matrix, 85–86
See also Gibbs permutation matrices
Positive polarity Reed–Muller (PPRM)
expression, 218
Post-quantum cryptography (PQC), 22
Primitive functions, 136, 144
construction of, 140
input variables, 141–143
Probability heuristics, 119–120
Program-level explanation, 10
Pseudo-Kronecker Expressions (PKRMs), 178,
188
Q
Quantum algorithms, 22
Quantum computation
logic synthesis, 185
optimization techniques, 185
quantum circuit, 186
reversible logic synthesis, 186–188
R
Rademacher-Walsh spectrum, 197–198
Reed–Muller spectra, 84, 89, 98
Residue number system (RNS)
advantage of, 237
backward conversion, 239
Boolean minimization, 251–252
Chinese reminder theorem, 238
computation, 238
computer arithmetic, 238
EDA tool, 239
features, 238
hardware computation, 237
characteristics, 245
modular product, 249–251
modulo function computation, 245–248
implementations, 237
MA, 237
Mentor Graphics, 252
mixed-radix approach, 239–240
modulus function calculation
automaton model, 240
bit-by-bit input reduction, 241
cryptography, 240
hardware realization, 240
limitation, 240
memory elements, 240
modular exponentiation, 243
periodic property, 242–243
powers of 2, 243–244
type 2 n ± k, 244–245
parallel data processing, 238
structure, 238, 239
Synopsys, 252–257
Xilinx, 252
Reversible Boolean functions
binary reversible functions, 222
CFs, 217, 220
elements, 230
linear variables, 227–228
NPNP-classes, 228
P-equivalence classes, 231–234
projection functions, 229–231
variable assignments, 229, 230, 232
coordinate functions, 217
EXOR-sum of products expression, 218
extrapolation, cycle structures
balanced functions, 224–225
NPNP classes, 223, 224
PPRMs expressions, 223, 224
RevFunCFLVs, 226
homogeneous property, 218
linear and nonlinear, 219
logic synthesis, 218
multiplexer-based field-programmable gate
arrays, 221
NPNP-equivalent, 220–221
P-equivalent functions, 219, 222
PPRM expression, 218
SC function, 219
Reversible logic synthesis, 186–188, 192
RNS, see Residue number system