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The work and findings in this paper are based on a short-term exploratory summer program for at-risk students conducted at California State University, Bakersfield, in the Summer of 2015. The summer program consists of a one-week exploratory program in one of three STEM fieldsChemistry, Engineering, and Mathematics. In this paper, the details of each of these programs are presented and the findings based on attitudinal surveys are discussed. Students indicate an increased interest in STEM research and an increased awareness of the skills and knowledge needed to succeed in a STEM field as a result of participating in this summer program.

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Abstract. In this work we re-examine two common modulus attacks on RSA. First, we show that Guo's continued fraction attack works much better in practice than previously expected. Given three instances of RSA with a common modulus N and private exponents each smaller than N 0.33 the attack can factor the modulus about 93% of the time in practice. The success rate of the attack can be increased up to almost 100% by including a relatively small exhaustive search. Next, we consider Howgrave-Graham and Seifert's lattice-based attack and show that a second necessary condition for the attack exists that limits the bounds (beyond the original bounds) once n ≥ 7 instances of RSA are used. In particular, by construction, the attack can only succeed when the private exponents are each smaller than N 0.5− , given sufficiently many instances, instead of the original bound of N 1− . In addition, we also consider the effectiveness of the attacks when mounted against multi-prime RSA and Tagaki's variant of RSA. For multi-prime RSA, we show three (or more) instances with a common modulus and private exponents smaller than N 1/3− is unsafe. For Takagi's variant, we show that three or more instances with a common modulus N = p r q is unsafe when all the private exponents are smaller than N 2/(3(r+1))− . The results, for both variants, is obtained using Guo's method and are successful almost always with the inclusion of a small exhaustive search. When only two instances are available, Howgrave-Graham and Seifert's attack can be mounted on multi-prime RSA when the private exponents are smaller than N (3+r)/7r− when there are r primes in the modulus.