4. Congruences, clocks, and calendars

‘Congruences, clocks, and calendars’ demonstrates how we might apply the idea of congruence, first introduced by Gauss in 1801, to problems such as testing which Mersenne numbers are primes and finding the day of the week on which a given date falls. Ancient Chinese puzzles depended on the solving of simultaneous linear congruences, inspiring mathematicians and giving rise to the Chinese Remainder Theorem. Exploring quadratic congruences leads towards the law of quadratic reciprocity, noted by Euler and Legendre and proved by Gauss. The problem, ‘Is 1066 a square or a non-square?’ can be solved by applying this law several times to reduce the numbers involved.

On an Almost-Universal Hash Function Family with Applications to Authentication and Secrecy Codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH[Formula: see text], which was shown to be [Formula: see text]-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH[Formula: see text], that we call GRDH, where we use an arbitrary integer [Formula: see text] instead of prime [Formula: see text] and let the keys [Formula: see text] satisfy the conditions [Formula: see text] ([Formula: see text]), where [Formula: see text] are given positive divisors of [Formula: see text]. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an [Formula: see text]-almost-[Formula: see text]-universal family of hash functions for some [Formula: see text] if and only if [Formula: see text] is odd and [Formula: see text] [Formula: see text]. Furthermore, if these conditions are satisfied then GRDH is [Formula: see text]-almost-[Formula: see text]-universal, where [Formula: see text] is the smallest prime divisor of [Formula: see text]. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [J. Math. Cryptol. 4 (2010) 121–148], and [J.UCS 15 (2009) 2937–2956].