On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a {\em complete} Edwards curve, and that an Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if and only if its group order is divisible by 8 if $q \equiv -1 \pmod{4}$, and 16 if $q \equiv 1 \pmod{4}$. Furthermore, we give formulae for the proportion of d \in \F_q \setminus \{0,1\} for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.Comment: 27 page

Jonathan Edwards\u27 Life: More Than a Sermon

Jonathan Edwards, born, (1703-1758), was a great man. He is often known only for a sermon, Sinners in the Hands of an Angry God. This is unfortunate because followers of Christ should know this man\u27s life. This paper focuses on Jonathan Edwards as a godly family man and on his missiological work. An emphasis is not carefully analyzed by many. The research for this essay originated from the author\u27s desire to know more about Mr. Edwards. The texts studied are The works of Jonathan Edwards, along with many scholarly books and essays. The main modern books used are from Perry Miller and Elizabeth Dodds. All in all, the following research adds clarity and context to Edwards\u27 legacy and to its enduring value to Christians

Strobel\u27s The ecumenical Edwards: Jonathan Edwards and the theologians (Book Review)

Strobel, K. C. (Ed.). (2015). The ecumenical Edwards: Jonathan Edwards and the theologians. New York: Routledge. 270 pp. $109.95. ISBN 978131703456

The Beautiful Mystery: Examining Jonathan Edwards’ View of Marriage

In contemporary evangelical circles, Jonathan Edwards has gained wide popularity for his theological writings and vital role in the First Great Awakening. However, despite these often romanticized views, Edwards nonetheless stood in the midst of an eighteenth century society that began to develop new norms for sexual practice and new legal guidelines to support them. In order to combat what he saw to be a decaying moral culture, Edwards took a strong stance on marital issues, often to the displeasure of his congregation. What lay behind these convictions was a deep theological understanding of the sanctity of marriage. These views, although not new to the history of Christian thought, were uniquely reinvigorated by Edwards to a Calvinist generation that had recently abandoned them. It is both Edwards’ theology of marriage and reinforcement of its practice that not only make him a unique preacher for his time, but also a worthy study for Evangelicals today in the midst of modern marital controversies

Faster computation of the Tate pairing

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in the addition and doubling. Computing the coefficients of the functions and the sum or double of the points is faster than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also speed up pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.Comment: 15 pages, 2 figures. Final version accepted for publication in Journal of Number Theor

Edwards on the Will: A Century of American Theological Debate

Jonathan Edwards towered over his contemporaries--a man over six feet tall and a figure of theological stature--but the reasons for his power have been a matter of dispute. Edwards on the Will offers a persuasive explanation. In 1753, after seven years of personal trials, which included dismissal from his Northampton church, Edwards submitted a treatise, Freedom of the Will, to Boston publishers. Its impact on Puritan society was profound. He had refused to be trapped either by a new Arminian scheme that seemed to make God impotent or by a Hobbesian natural determinism that made morality an illusion. He both reasserted the primacy of God\u27s will and sought to reconcile freedom with necessity. In the process he shifted the focus from the community of duty to the freedom of the individual. Edwards died of smallpox in 1758 soon after becoming president of Princeton; as one obituary said, he was a most rational . . . and exemplary Christian. Thereafter, for a century or more, all discussion of free will and on the church as an enclave of the pure in an impure society had to begin with Edwards. His disciples, the New Divinity men--principally Samuel Hopkins of Great Barrington and Joseph Bellamy of Bethlehem, Connecticut--set out to defend his thought. Ezra Stiles, president of Yale, tried to keep his influence off the Yale Corporation, but Edwards\u27s ideas spread beyond New Haven and sparked the religious revivals of the next decades. In the end, old Calvinism returned to Yale in the form of Nathaniel William Taylor, the Boston Unitarians captured Harvard, and Edwards\u27s troublesome ghost was laid to rest. The debate on human freedom versus necessity continued, but theologians no longer controlled it. In Edwards on the Will, Guelzo presents with clarity and force the story of these fascinating maneuverings for the soul of New England and of the emerging nation. [From the publisher]https://cupola.gettysburg.edu/books/1074/thumbnail.jp

Edwards curves and CM curves

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the links with Kubert parameterizations of X0(2) and X0(4). We classify CM curves that admit an Edwards or Montgomery form over a finite field, and justify the use of isogenous curves when needed