I use a combination of analytic and algebraic techniques to study L-series associated to number fields and automorphic forms on GL(n).

One of my early accomplishments was one of the first applications of a higher rank group (GL(3)), to break down a stone wall that had impeded the foundations of the spectral theory of ordinary automorphic forms.  This is the the paper Coefficients of Maass forms and the Siegel Zero, in the Annals of Mathematics, which has been cited 301 times, and its Appendix, which has been cited 71 times.

For the past 25 years or so, one of my main themes has been the development of the theory of multiple Dirichlet series as a technique to tie together and study families of L-series.   I have doen this in collaboration with Dorian Goldfeld, Daniel Bump, Sol Freedberg, Ben Brubaker, Gautem Chinta and a number of other people.   The analysis of these series led to a tie in with finite Dynkin diagrams, and ultimately an entirely unexpected connection with combinatorial representation theory, which was the subject of a one semester program at ICERM in the Spring of 2013, bringing together researchers in both fields.  

The ultimate goal of this program is the Lindelof Hypothesis, which is related to the question of finding upper bounds for automorphic L-series on the center line of the critical strip.    The barrier to this is the extension of the theory of automorphic forms over groups of finite rank groups, to Kac-Moody groups and algebras.    This is a subject presently under active study by many people.

I also study lattice based public key cryptography, in particular NTRU. In 1996, I, together with Jill Pipher and Joe Silverman, introduced a new form of cryptography that we called NTRU.  In 1996, while checking to see if the name NTRU had been used before, we found one entry under a google search: The Northern Transvaal Rugby Union.  A google search now, on 10/22/2017, turned up 236,000 results.   We introduced the notion of a lattice with a multiplicative structure into cryptography, and all lattice based cryptographic constructions, that are remorely practical, now are built on such "ring lattices".    Our original paper: NTRU: A Ring Based Public Key Crypto-System, has had over 1300 citations. 

All citations can be found here: https://scholar.google.com/citations?view_op=list_works&hl=en&user=9OrDVa4AAAAJ

I use a combination of analytic and algebraic techniques to study L-series associated to number fields and automorphic forms on GL(n). For the past 25 years or so, one of my main themes has been the development of the theory of multiple Dirichlet series as a technique to tie together and study families of L-series. I also study lattice based public key cryptography, in particular NTRU.

Current funding is:

NSF-SATC, TWC: Medium: Collaborative: Development and Evaluation of Next Generation Homomorphic Encryption Schemes,  under the direction of Jeffrey Hoffstein, Joseph H. Silverman

This award starts August,2016 and ends July 31, 2018.

Cubic twists of ${\rm GL}(2)$ automorphic $L$-functions B. Brubaker, S. Friedberg\ and\ J. Hoffstein, Invent. Math. {\bf 160} (2005), no.~1, 31--58; MR2129707 (2005m:11091)

Asymptotics for sums of twisted $L$-functions and applications G. Chinta, S. Friedberg\ and\ J. Hoffstein, in {\it Automorphic representations, $L$-functions and applications: progress and prospects}, 75--94, de Gruyter, Berlin, 2005; MR2192820

Sums of twisted ${\rm GL}(3)$ automorphic $L$-functions D. Bump, S. Friedberg\ and\ J. Hoffstein, in {\it Contributions to automorphic forms, geometry, and number theory}, 131--162, Johns Hopkins Univ. Press, Baltimore, MD, 2004; MR2058607 (2005f:11098)

Nonvanishing twists of GL(2) automorphic $L$-functions B. Brubaker\ et al., Int. Math. Res. Not. {\bf 2004}, no.~78, 4211--4239; MR2111362 (2005h:11099)

Multiple Dirichlet series and moments of zeta and $L$-functions A. Diaconu, D. Goldfeld\ and\ J. Hoffstein, Compositio Math. {\bf 139} (2003), no.~3, 297--360; MR2041614 (2005a:11124)

NTRUSign: digital signatures using the NTRU lattice J. Hoffstein\ et al., in {\it Topics in cryptology---CT-RSA 2003}, 122--140, Lecture Notes in Comput. Sci., 2612, Springer, Berlin, 2003; see MR0000000 (2005b:94045)

Random small Hamming weight products with applications to cryptography J. Hoffstein\ and\ J. H. Silverman, Discrete Appl. Math. {\bf 130} (2003), no.~1, 37--49; MR2008405 (2005a:94057)

NSS: an NTRU lattice-based signature scheme J. Hoffstein, J. Pipher\ and\ J. H. Silverman, in {\it Advances in cryptology---EUROCRYPT 2001 (Innsbruck)}, 211--228, Lecture Notes in Comput. Sci., 2045, Springer, Berlin, 2001; MR1895435 (2003e:94069)

Optimizations for NTRU J. Hoffstein\ and\ J. Silverman, in {\it Public-key cryptography and computational number theory (Warsaw, 2000)}, 77--88, de Gruyter, Berlin, 2001; MR1881629 (2003f:94060)

MiniPASS: Authentication and digital signatures in a constrained environment J. Hoffstein and J.H. Silverman, Workshop on Cryptographic Hardware and Embedded Systems (CHESS 2000), C.K. Koc and C. Paar

Automorphic forms and sums of squares over function fields J. Hoffstein, K. D. Merrill\ and\ L. H. Walling, J. Number Theory {\bf 79} (1999), no.~2, 301--329; MR1728153 (2001d:11054)

The distribution of the quadratic symbol in function fields and a faster mathematical stream cipher J. Hoffstein\ and\ D. Lieman, in {\it Cryptography and computational number theory (Singapore, 1999)}, 59--68, Birkh\"auser, Basel, 2001; MR1944719 (2003m:11193)

Polynomial rings and efficient public key authentication J. Hoffstein\ and\ J. H. Silverman, in {\it Cryptography and computational number theory (Singapore, 1999)}, 269--286, Birkh\"auser, Basel, 2001; MR1944732 (2003k:94028)

Polynomial Rings and Efficient Public Key Authentication II J. Hoffstein and J.H. Silverman, Proceedings of a Conference on Cryptography and Number Theory (CCNT '99) \ed I. Shparlinski

NTRU: a ring-based public key cryptosystem J. Hoffstein, J. Pipher\ and\ J. H. Silverman, in {\it Algorithmic number theory (Portland, OR, 1998)}, 267--288, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998; see MR0000000 (2000g:11002)

Nonvanishing of $L$-series and the combinatorial sieve J. Hoffstein\ and\ W. Luo, Math. Res. Lett. {\bf 4} (1997), no.~2-3, 435--444; MR1453073 (98d:11052)

On some applications of automorphic forms to number theory D. Bump, S. Friedberg\ and\ J. Hoffstein, Bull. Amer. Math. Soc. (N.S.) {\bf 33} (1996), no.~2, 157--175; MR1359575 (97a:11072)

The symmetric cube D. Bump, D. Ginzburg\ and\ J. Hoffstein, Invent. Math. {\bf 125} (1996), no.~3, 413--449; MR1400313 (97j:11023)

Omega results for automorphic $L$-functions J. Hoffstein\ and\ P. Lockhart, in {\it Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996)}, 239--250, Proc. Sympos. Pure Math., Part 2, Amer. Math. Soc., Providence, RI, 1999; MR1703761 (2000j:11071)

Average values of cubic $L$-series D. Farmer, J. Hoffstein\ and\ D. Lieman, in {\it Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996)}, 27--34, Proc. Sympos. Pure Math., Part 2, Amer. Math. Soc., Providence, RI, 1999; MR1703757 (2000e:11064)

Nonvanishing theorems for automorphic $L$-functions on ${\rm GL}(2)$ S. Friedberg\ and\ J. Hoffstein, Ann. of Math. (2) {\bf 142} (1995), no.~2, 385--423; MR1343325 (96e:11072)

Siegel zeros and cusp forms J. Hoffstein\ and\ D. Ramakrishnan, Internat. Math. Res. Notices {\bf 1995}, no.~6, 279--308; MR1344349 (96h:11040)

Coefficients of Maass forms and the Siegel zero J. Hoffstein\ and\ P. Lockhart, Ann. of Math. (2) {\bf 140} (1994), no.~1, 161--181; MR1289494 (95m:11048)

Eisenstein series and theta functions on the metaplectic group J. Hoffstein, in {\it Theta functions: from the classical to the modern}, 65--104, Amer. Math. Soc., Providence, RI, 1993; MR1224051 (94h:11050)

On the number of Fourier coefficients that determine a modular form D. Goldfeld\ and\ J. Hoffstein, in {\it A tribute to Emil Grosswald: number theory and related analysis}, 385--393, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993; MR1210527 (94b:11037)

Theta functions on the $n$-fold metaplectic cover of ${\rm SL}(2)$---the function field case J. Hoffstein, Invent. Math. {\bf 107} (1992), no.~1, 61--86; MR1135464 (92k:11049)

Average values of $L$-series in function fields J. Hoffstein\ and\ M. Rosen, J. Reine Angew. Math. {\bf 426} (1992), 117--150; MR1155750 (93c:11022)

An estimate for the Hecke eigenvalues of Maass forms D. Bump\ et al., Internat. Math. Res. Notices {\bf 1992}, no.~4, 75--81; MR1159448 (93d:11047)

$p$-adic Whittaker functions on the metaplectic group D. Bump, S. Friedberg\ and\ J. Hoffstein, Duke Math. J. {\bf 63} (1991), no.~2, 379--397; MR1115113 (92d:22024)

Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives D. Bump, S. Friedberg\ and\ J. Hoffstein, Ann. of Math. (2) {\bf 131} (1990), no.~1, 53--127; MR1038358 (92e:11053)

On explicit integral formulas for ${\rm GL}(n,R)$-Whittaker functions E. Stade, Duke Math. J. {\bf 60} (1990), no.~2, 313--362; MR1047756 (92a:11060)

Nonvanishing theorems for $L$-functions of modular forms and their derivatives D. Bump, S. Friedberg\ and\ J. Hoffstein, Invent. Math. {\bf 102} (1990), no.~3, 543--618; MR1074487 (92a:11058)

The Kubota symbol for ${\rm Sp}(4,Q(i))$ D. Bump, S. Friedberg\ and\ J. Hoffstein, Nagoya Math. J. {\bf 119} (1990), 173--188; MR1071906 (91m:11036)

On Artin's conjecture and the class number of certain CM fields I,II J. Hoffstein\ and\ N. Jochnowitz, Duke Math. J. {\bf 59} (1989), no.~2, 553--563, 565--584; MR1016903 (90h:11104)

On Waldspurger's theorem D. Bump, S. Friedberg\ and\ J. Hoffstein, in {\it Automorphic forms and analytic number theory (Montreal, PQ, 1989)}, 25--36, Univ. Montr\'eal, Montreal, QC, 1990; MR1111008 (92e:11037)

A nonvanishing theorem for derivatives of automorphic $L$-functions with applications to elliptic curves D. Bump, S. Friedberg\ and\ J. Hoffstein, Bull. Amer. Math. Soc. (N.S.) {\bf 21} (1989), no.~1, 89-93; MR0983456 (90b:11063)

On Shimura's correspondence D. Bump\ and\ J. Hoffstein, Duke Math. J. {\bf 55} (1987), no.~3, 661--691; MR0904946 (89c:11072)

$L$-series of automorphic forms on ${\rm GL}(3,R)$ J. Hoffstein\ and\ M. R. Murty, in {\it Th\'eorie des nombres (Quebec, PQ, 1987)}, 398--408, de Gruyter, Berlin, 1989; MR1024578 (90m:11075)

Some Euler products associated with cubic metaplectic forms on ${\rm GL}(3)$ D. Bump\ and\ J. Hoffstein, Duke Math. J. {\bf 53} (1986), no.~4, 1047--1072; MR0874680 (88d:11044)

Cubic metaplectic forms on ${\rm GL}(3)$ D. Bump\ and\ J. Hoffstein, Invent. Math. {\bf 84} (1986), no.~3, 481--505; MR0837524 (87i:11059)

Eisenstein series of ${1\over 2}$-integral weight and the mean value of real Dirichlet $L$-series D. Goldfeld\ and\ J. Hoffstein, Invent. Math. {\bf 80} (1985), no.~2, 185--208; MR0788407 (86m:11029)

Some conjectured relationships between theta functions and Eisenstein series on the metaplectic group D. Bump\ and\ J. Hoffstein, in {\it Number theory (New York, 1985/1988)}, 1--11, Lecture Notes in Math., 1383, Springer, Berlin, 1989; MR1023915 (90m:11076)

Real zeros of Eisenstein series J. Hoffstein, Math. Z. {\bf 181} (1982), no.~2, 179--190; MR0674270 (83m:10030)

On automorphic functions of half-integral weight with applications to elliptic curves D. Goldfeld, J. Hoffstein\ and\ S. J. Patterson, in {\it Number theory related to Fermat's last theorem (Cambridge, Mass., 1981)}, 153--193, Progr. Math., 26, Birkh\"auser, Boston, Mass., 1982; MR0685295 (84i:10031)

On the Siegel-Tatuzawa theorem J. Hoffstein, Acta Arith. {\bf 38} (1980/81), no.~2, 167--174; MR0604232 (82k:10050)

Some analytic bounds for zeta functions and class numbers J. Hoffstein, Invent. Math. {\bf 55} (1979), no.~1, 37--47; MR0553994 (80k:12019)

Some results related to minimal discriminants J. Hoffstein, in {\it Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979)}, 185--194, Lecture Notes in Math., 751, Springer, Berlin, 1979; MR0564928 (81d:12005)