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We present a detailed analytical modeling for the constant-current Fowler-Nordheim program operation of NAND flash memories, able to describe both the average transient of the cell threshold voltage and its statistical spread due to the granular nature of the electron current flowing through the cell tunnel oxide. We analytically investigate the electron-injection process, highlighting that the steepness of the tunneling current versus floating-gate voltage characteristics and the control-gate to floating-gate capacitance give the field feedback factor, determining the average number of injected electrons at which the injection process becomes sub-Poissonian. Finally, we show that cell scaling will reduce the achievable accuracy of the program algorithm, due to the reduction in the number of electrons controlling cell state.