3 of 8 Link to original question; polish.

This answers the original question in the particular case of $p=2^{64}$. It quickly computes the modular inverse of any odd 64-bit integer, on most modern C compilers with 64-bit support. It is constant-time if basic operations are (which is often a worry for multiplication).

#include <stdint.h>                     // for uint64_t and uint32_t

// given odd a, compute x such that a*x = 1 over 64 bits.
uint64_t invmod(uint64_t a)
    uint32_t b = (uint32_t)a;           // low 32 bits of a
    uint32_t x = (((b+0x2)&0x4)<<1)+b;  // low  4 bits of inverse
    x =    (0x2-b*x)*x;                 // low  8 bits of inverse
    x =    (0x2-b*x)*x;                 // low 16 bits of inverse
    x =    (0x2-b*x)*x;                 // low 32 bits of inverse
    return (0x2-a*x)*x;                 //     64 bits of inverse

Justification: $a\cdot x\equiv1\pmod{2^k}\implies a\cdot(2-a\cdot x)\cdot x\equiv1\pmod{2^{2\cdot k}}$

Note: The code uses intermediary 32-bit variable b only in order to optimize speed on 32-bit CPUs; we could otherwise use a, or/and make x wider than 32-bit.