The Quadratic Sieve Lab Jeremy Johnson This worksheet illustrates some of the topics discussed in chapter 8 of the text; namely Dixon's algorithm and the quadratic sieve. The worksheet also shows some of the relevent Maple functions (legendre symbol for detecting quadratic residues, solving equations mod LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJhRidGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGRQ==performing Gaussian elimination mod 2).
<Text-field style="Heading 1" layout="Heading 1">Kraitchik's Improvement to Fermat's Algorithm</Text-field> Assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5is composite. If 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then 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and if some of the factors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5divide 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and some divide 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 be a non-trivial factor of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 restart; n := 13*17; IiRAIw== for k from ceil(sqrt(n)) to n-1 do r := k^2 mod n; if sqrt(r) = ceil(sqrt(r)) then print(k,r,igcd(k-sqrt(r),n)); end if; end do: NiUiIzoiIiUiIzg= NiUiI0ciJEAiIiM8 NiUiI0kiIzsiIzg= NiUiI1YiIyIpIiM8 NiUiI1giI08iIzg= NiUiI2UiI1wiIzw= NiUiI2ciI2siIzg= NiUiI3QiI0QiIzw= NiUiI3YiJCsiIiM4 NiUiIykpIiIqIiM8 NiUiIyEqIiRXIiIjOA== NiUiJC4iIiIiIiM8 NiUiJDAiIiQnPiIjOA== NiUiJDsiIiQnPiIjPA== NiUiJD0iIiIiIiM4 NiUiJEoiIiRXIiIjPA== NiUiJEwiIiIqIiM4 NiUiJFkiIiQrIiIjPA== NiUiJFsiIiNEIiM4 NiUiJGgiIiNrIiM8 NiUiJGoiIiNcIiM4 NiUiJHciIiNPIiM8 NiUiJHkiIiMiKSIjOA== NiUiJCI+IiM7IiM8 NiUiJCQ+IiRAIiIjOA== NiUiJDEjIiIlIiM8 NiUiJDIjIiQnPiIiIg== NiUiJDMjIiRwIiIjOA== NiUiJDQjIiRXIiIiIg== NiUiJDUjIiRAIiIiIg== NiUiJDYjIiQrIiIiIg== NiUiJDcjIiMiKSIiIg== NiUiJDgjIiNrIiIi NiUiJDkjIiNcIiIi NiUiJDojIiNPIiIi NiUiJDsjIiNEIiIi NiUiJDwjIiM7IiIi NiUiJD0jIiIqIiIi NiUiJD4jIiIlIiIi NiUiJD8jIiIiRiQ=
<Text-field style="Heading 1" layout="Heading 1">Dixon's Algorithm</Text-field> Kraitchik's idea provides a mechanism to find factors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5provided we can find two squares whose difference is a multiple of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlNGT0Y5Dixon's algorithm provides a systematic way to search for such squares. The idea is to find a sequence of integers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGT0YrRjw=such that 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 easy to factor. Note that 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 if 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is a square we have found the desired squares. Assume that 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then 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is a square if and only if all of the exponents 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are even. This is the case when all of the exponents are equal to zero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkbW9kRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIzIuRidGMkY1RjVGMg==It is unlikely that we will find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5such that this is the case; however, given the factorizations for a sequence of integers 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it is possible to use Gaussian elimination to construct such an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlNGT0Y5The idea is to look for a linear combination of the exponent vectors 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 sume to the zero vector 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when the components are taken mod 2. If such a linear combination is found then 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 it does not matter what values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5are used in this search, we only use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEic0YnRi9GMi1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORldGOQ==that are easy to factor (if when factoring a number it takes a long time, stop and try another number). Dixon's idea is to look for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEY5that can be factored using trial division with small primes (e.g. those less than 10,000). with(LinearAlgebra); 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 EasyFactor := proc(n,t) local v, np, e, i; v := Vector[row](t); np := n; for i from 1 to t do e := 0; while irem(np,ithprime(i)) = 0 do e := e + 1; np := np/ithprime(i); end do; v[i] := e; end do; if np = 1 then return v else return infinity; end if; end; Zio2JEkibkc2IkkidEdGJTYmSSJ2R0YlSSNucEdGJUkiZUdGJUkiaUdGJUYlRiVDJj5GKC0mSSdWZWN0b3JHRiU2I0kkcm93R0YlNiNGJj5GKUYkPyhGKyIiIkY2RiZJJXRydWVHJSpwcm90ZWN0ZWRHQyU+RioiIiE/KEYlRjZGNkYlLy1JJWlyZW1HRjg2JEYpLUkpaXRocHJpbWVHRiU2I0YrRjtDJD5GKiwmRipGNkY2RjY+RikqJkYpRjZGQSEiIj4mRihGQ0YqQCUvRilGNk9GKE9JKWluZmluaXR5R0Y4RiVGJUYl EasyFactor(13*2*3*5*3*5*17*11*11,5); SSlpbmZpbml0eUclKnByb3RlY3RlZEc= v := EasyFactor(2^3*3^2*5*11,5); 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 2^3*3^2*5*11; IiVnUg== p := Vector[row]([seq(ithprime(i),i=1..5)]); 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 NumberExpVector := proc(v,p) local i; mul(p[i]^v[i],i=1..Dimension(v)); end; Zio2JEkidkc2IkkicEdGJTYjSSJpR0YlRiVGJS1JJG11bEclKnByb3RlY3RlZEc2JCkmRiZGJyZGJEYnL0YoOyIiIi1fSS5MaW5lYXJBbGdlYnJhRzYkRitJKF9zeXNsaWJHRiVJKkRpbWVuc2lvbkdGJTYjRiRGJUYlRiU= NumberExpVector(v,p); IiVnUg== with(LinearAlgebra:-Modular); 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 n := ithprime(100)*ithprime(200); IidWO20= r := ceil(sqrt(n)); k := r; count := 0; L := []; K := []; IiQ5KQ== IiQ5KQ== IiIh NyI= NyI= while count < 10 do v := EasyFactor(k^2 mod n,10); if v <> infinity then count := count + 1; L := [convert(v,list),op(L)]; K := [k,op(K)]; print(v); end if; k := k + 1; end do: LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqU206XSI= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqayYpW10i LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqJXljLDo= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqITNULDo= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqSy85XSI= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqR2o4XSI= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqPydHLTo= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqM1Y8XSI= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqK15gXSI= LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqI1JXLDo= L; K; Nyw3LCIiJiIiIiIiI0YlRiUiIiFGJ0YnRidGJzcsIiIkRiVGJUYnRidGJ0YnRidGJUYlNywiIiVGJUYnRiVGJUYnRiVGJ0YnRic3LCIiKEYlRidGJUYnRidGJUYnRidGJzcsIiInRidGK0YnRidGJ0YnRidGJ0YnNyxGJEYnRidGJ0YlRidGJ0YnRidGJzcsRidGJkYlRidGJ0YlRidGJ0YlRiU3LEYlRiVGKUYnRidGJUYnRidGJUYnNyxGJ0YlRidGJ0YnRitGJ0YnRidGJzcsRiVGJkYmRidGJ0YnRidGJ0YnRiU= NywiJXQ5IiVQOSIlSjkiJUQ5IiVCOSIlNDkiJTQ4IiVXNyIlKD0iIiVjNg== A := Matrix([seq(L[i],i=1..10)]); 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 A2 := Mod(2,A,integer[]); 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 N := LinearAlgebra:-Modular[Basis](2,A2,column,false,row); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2Ki1JKG1hY3Rpb25HRiQ2JS1GUDYoLUknbXRhYmxlR0YkNjUtSSRtdHJHRiQ2Ly1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiMEYnRjkvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmJvLyUrZ3JvdXBhbGlnbkdGYm8vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGaW9GaW5GaW5GaW4tRmpuNigtRl1vNiRRIjFGJ0Y5RmBvRmNvRmVvRmdvRmpvRmluRmluRmluRmluRmluRmBvRmNvRmVvLyUmYWxpZ25HUSVheGlzRicvRmFvUSliYXNlbGluZUYnL0Zkb1EnY2VudGVyRicvRmZvUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGMS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRl5xLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmlxLyUmZnJhbWVHRmlxLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGPS8lLWVxdWFsY29sdW1uc0dGPS8lLWRpc3BsYXlzdHlsZUdGPS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZmcUY5L0krbXNlbWFudGljc0dGJFEqUm93VmVjdG9yRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGXHMvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnLyUpcnRhYmxlaWRHUSoxNTA2MDE3NjhGJy1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUZVNiUtRlA2KC1GWjY1LUZnbjYvRmluRmluRlxwRlxwRmluRlxwRmluRmluRmluRmluRmBvRmNvRmVvRmFwRmRwRmZwRmhwRmpwRlxxRl9xRmFxRmRxRmdxRmpxRlxyRl5yRmFyRmNyRmVyRmdyRmpyRjlGXHNGX3NGYnNGXHNGZXMvRmlzUSoxNDk5NTU3MTJGJ0ZbdC1GVTYlLUZQNigtRlo2NS1GZ242L0ZpbkZpbkZpbkZpbkZpbkZpbkZccEZccEZccEZccEZgb0Zjb0Zlb0ZhcEZkcEZmcEZocEZqcEZccUZfcUZhcUZkcUZncUZqcUZcckZeckZhckZjckZlckZnckZqckY5RlxzRl9zRmJzRlxzRmVzL0Zpc1EqMTQ5OTU1NzUyRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRj1GOUY5Rl9zRmJzRmd1Rmp1Rjk= v := A[7,1..10]+A[8,1..10]+A[9,1..10]+A[10,1..10]; 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 v/2; LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqN2YmKlwi p := Vector[row]([seq(ithprime(i),i=1..10)]); 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 r := NumberExpVector(v/2,p); IitdS1ciKikq ifactor(r); Ki4tSSFHNiI2IyIiIyIiIiktRiQ2IyIiJEYsRigpLUYkNiMiIiZGLEYoKS1GJDYjIiM4RixGKC1GJDYjIiNCRigtRiQ2IyIjSEYo k := K[7]*K[8]*K[9]*K[10]; Ii43aFJSV0Ij n; IidWO20= igcd(k-r,n); IiRUJg== ifactor(n); KiYtSSFHNiI2IyIkVCYiIiItRiQ2IyIlQjdGKA==
<Text-field style="Heading 1" layout="Heading 1">Factor Base</Text-field> with(numtheory); 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 n := 4999486012441; Ii5UQywnWyoqXA== count := 0; p := 2; FB := []; SolFB := []; IiIh IiIj NyI= NyI= while count < 30 do if legendre(n,p) = 1 then FB := [op(FB),p]; SolFB := [op(SolFB),Roots(x^2-n) mod p]; count := count + 1; print(p,Roots(x^2-n) mod p); end if; p := nextprime(p); end do: FB; SolFB; N0AiIiQiIiYiIigiIzwiIz4iI0oiI1YiI1oiI2YiI2giI24iJDIiIiRqIiIkIj0iJCQ+IiQoPiIkSCMiJFQjIiRqIyIkciMiJHgjIiQ2JCIkSiQiJFwkIiRmJCIkbiQiJCpRIiQoUiIkLCUiJDQl 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
<Text-field style="Heading 1" layout="Heading 1">Sieving</Text-field> EasyFactor := proc(n,t) local v, np, e, i; v := Vector[row](t); np := n; for i from 1 to t do e := 0; while irem(np,ithprime(i)) = 0 do e := e + 1; np := np/ithprime(i); end do; v[i] := e; end do; if np = 1 then return v else return infinity; end if; end; Zio2JEkibkc2IkkidEdGJTYmSSJ2R0YlSSNucEdGJUkiZUdGJUkiaUdGJUYlRiVDJj5GKC0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kkcm93R0YlNiNGJj5GKUYkPyhGKyIiIkY5RiZJJXRydWVHRjJDJT5GKiIiIT8oRiVGOUY5RiUvLUklaXJlbUdGMjYkRiktSSlpdGhwcmltZUdGMTYjRitGPUMkPkYqLCZGKkY5RjlGOT5GKSomRilGOUZDISIiPiZGKEZFRipAJS9GKUY5T0YoT0kpaW5maW5pdHlHRjJGJUYlRiU= n := 4999486012441; Ii5UQywnWyoqXA== ifactor(n); KiYtSSFHNiI2IyInaCoqKioiIiItRiQ2IyIoIm8qKlxGKA== seq(ithprime(i),i=1..80); NlxwIiIjIiIkIiImIiIoIiM2IiM4IiM8IiM+IiNCIiNIIiNKIiNQIiNUIiNWIiNaIiNgIiNmIiNoIiNuIiNyIiN0IiN6IiMkKSIjKikiIygqIiQsIiIkLiIiJDIiIiQ0IiIkOCIiJEYiIiRKIiIkUCIiJFIiIiRcIiIkXiIiJGQiIiRqIiIkbiIiJHQiIiR6IiIkIj0iJCI+IiQkPiIkKD4iJCo+IiQ2IyIkQiMiJEYjIiRIIyIkTCMiJFIjIiRUIyIkXiMiJGQjIiRqIyIkcCMiJHIjIiR4IyIkIkciJCRHIiQkSCIkMiQiJDYkIiQ4JCIkPCQiJEokIiRQJCIkWiQiJFwkIiRgJCIkZiQiJG4kIiR0JCIkeiQiJCRRIiQqUSIkKFIiJCwlIiQ0JQ== SQRT := floor(sqrt(n)); count := 0; M := 5000; IihgZkIj IiIh IiUrXQ== for i from 1 to 2*M do r := (SQRT-M+i); v := EasyFactor(abs(r^2-n),80); if v <> infinity then count := count + 1; print(i,ifactor(r^2-n)); end if; end do: NiQiJFYjLCQqLiktSSFHNiI2IyIiJiIiIyIiIi1GKDYjIiIoRi0tRig2IyIjPEYtLUYoNiMiJDIiRi0tRig2IyIkVCNGLS1GKDYjIiR4I0YtISIi NiQiJCVbLCQqMCktSSFHNiI2IyIiIyIiJSIiIiktRig2IyIiJEYxRi0tRig2IyIiKEYtLUYoNiMiIz5GLS1GKDYjIiNKRi0tRig2IyIjWkYtLUYoNiMiJFQjRi0hIiI= NiQiJFwnLCQqLi1JIUc2IjYjIiIkIiIiKS1GJzYjIiIoIiIjRistRic2IyIjSkYrLUYnNiMiI2ZGKy1GJzYjIiQoPkYrLUYnNiMiJG4kRishIiI= 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NiQiJWooKiowKS1JIUc2IjYjIiIkIiIjIiIiLUYnNiMiIiZGLCktRic2IyIiKEYrRiwtRic2IyIjPEYsLUYnNiMiI0pGLC1GJzYjIiNmRiwtRic2IyIkNiRGLA== NiQiJTMqKiowKS1JIUc2IjYjIiIjIiIlIiIiKS1GJzYjIiIkRipGLC1GJzYjIiImRiwtRic2IyIjPkYsLUYnNiMiI0pGLC1GJzYjIiQoPkYsLUYnNiMiJGojRiw= count; IiN0 P := Vector(2*M): Dimension(P); IiYrKyI= for i from 1 to Dimension(P) do r := SQRT - M + i; P[i] := evalf(log(abs(r^2-n))); end do: nops(FB); IiNJ evalf(log(SQRT + M)); JCIreTZDaTkhIik= SQRT - M + 1; IihhNEIj n mod 8; IiIi s := 2; IiIj while s <= Dimension(P) do P[s] := P[s] - evalf(log(8)); s := s + 2; end do: for i from 2 to nops(FB) do p := FB[i]; R := Roots(x^2-n) mod p; t := R[1][1]; s := t - (SQRT) + M mod p; if s = 0 then s := p; end if; print(t,s); while s <= Dimension(P) do P[s] := P[s] - evalf(log(p)); s := s + p; end do; t := R[2][1]; s := t - (SQRT) + M mod p; if s = 0 then s := p; end if; print(t,s); while s <= Dimension(P) do P[s] := P[s] - evalf(log(p)); s := s + p; end do; end do; IiIm NyQ3JCIiIkYkNyQiIiVGJA== IiIi IiIk NiQiIiIiIiQ= IiIl IiIi NiQiIiUiIiI= IiIo NyQ3JCIiIyIiIjckIiImRiU= IiIj IiIm NiQiIiMiIiY= IiIm IiIi NiQiIiYiIiI= IiM8 NyQ3JCIiJCIiIjckIiM5RiU= IiIk IiM2 NiQiIiQiIzY= IiM5 IiIm NiQiIzkiIiY= IiM+ NyQ3JCIiIkYkNyQiIz1GJA== IiIi IiIq NiQiIiIiIio= IiM9 IiIo NiQiIz0iIig= IiNK NyQ3JCIiJiIiIjckIiNFRiU= IiIm IiNI NiQiIiYiI0g= IiNF IiM+ NiQiI0UiIz4= IiNW NyQ3JCIjPiIiIjckIiNDRiU= IiM+ IiNO NiQiIz4iI04= IiND IiNT NiQiI0MiI1M= IiNa NyQ3JCIjPSIiIjckIiNIRiU= IiM9 IiM5 NiQiIz0iIzk= IiNI IiNE NiQiI0giI0Q= IiNm NyQ3JCIjOSIiIjckIiNYRiU= IiM5 IiNH NiQiIzkiI0c= IiNY IiIh NiQiI1giI2Y= IiNo NyQ3JCIjOyIiIjckIiNYRiU= IiM7 IiM7 NiQiIztGIw== IiNY IiNY NiQiI1hGIw== IiNu NyQ3JCIiJyIiIjckIiNoRiU= IiIn IiM+ NiQiIiciIz4= IiNo IiIo NiQiI2giIig= IiQyIg== NyQ3JCIjSyIiIjckIiN2RiU= IiNL IiNI NiQiI0siI0g= IiN2 IiNz NiQiI3YiI3M= IiRqIg== NyQ3JCIjUSIiIjckIiREIkYl IiNR IiNt NiQiI1EiI20= IiREIg== IiRgIg== NiQiJEQiIiRgIg== IiQiPQ== NyQ3JCIkaSIiIiI3JCIjPkYl IiRpIg== IiNN NiQiJGkiIiNN IiM+ IiNz NiQiIz4iI3M= IiQkPg== 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<Text-field style="Heading 1" layout="Heading 1">Question 1</Text-field> Use Dixon's algorithm to factor n := 4999486012441; Ii5UQywnWyoqXA== ifactor(n); KiYtSSFHNiI2IyInaCoqKioiIiItRiQ2IyIoIm8qKlxGKA==
<Text-field style="Heading 1" layout="Heading 1">Question 2</Text-field> Find a factor base of size 100 to use in applying the Quadratic Sieve to the integer 35419 05253 35205 94597 94529. n := 3541905253352059459794529; IjpIWHpmJWY/TmBfIT5hJA==
<Text-field style="Heading 1" layout="Heading 1">Question 3</Text-field> For each prime p in the factor base found in Exercise 8.10, solve the quadratic congruence x^2 = 35419 05253 35205 94597 94529 (mod p) n := 3541905253352059459794529; IjpIWHpmJWY/TmBfIT5hJA==
<Text-field style="Heading 1" layout="Heading 1">Extra credit</Text-field> Factor 3541905253352059459794529 using the quadratic sieve [book suggests multiple polynomial quadratic sieve]. Dixon's algorithm will be very slow [ you need the quadratic sieve to find the easily factorizable numbers faster].
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